QA IV I UC-NRLF $C IbS 755 IMPRIMITIVE SUBSTITUTION GkOITS OF DEGEEE SIXTEEN DISSERTATION PRESENTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE GRADUATE SCHOOL OF THE OHIO STATE UNIVERSITY. By Grace Marie Bareis THE OHIO STATE UNIVERSITY 1909 /?/ 1 ■ft ., r ^ TABLE OF CONTENTS. Introduction and notation 3 Lists of groups 1. Two systems of imprimitivity 5 2. Four systems of imprimitivity 38 3. Eight systems of imprinrdtivity 44 PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. IMPRIMITIVE SUBSTITUTION GROUPS OF DEGREE SIXTEEN. Sixteen symbols can be divided in three ways into sets containing an equal number of symbols, viz., into two 8'«, four 4*s, or eiglit 2's. Tn the following rist a group is c -^aed ande. the firr* systems of int to which its head belongs. "For example a gr'^""' havi four systems of imprimitv' It., is listed amor ** In previous lists letters of the alphabet have Uauuiiy . i . elcinei.LS. ^n this list letters Arith subscripts are used, the letters to distinguish between systems and the numbers as subscripts to dis- tinguish between elements within the system. The sixteen symbols used differ therefore with the number of systems. For two systems of imprimitivity they are fll 02 03 04 as a« 07 Og 61 62 63 &4 65 &6 bi bs for four systems they are oi 02 03 04 61 62 63 64 ci C2 C3 C4 di 3/2 dz di for eight systems they are oi 02 61 62 ci Ci d\ d^ e\ e^fxfi g\ g^ h h The transformations necessary for the final comparison of groups of different systems are very simple. The transitive constituents of the heads are distinguished by number- ing them as they are numbered by Professor G. A. Miller in his "Memoir on the Substitution Groups whose Degree does not Exceed Eight" [American Journal of Mathematics, vol. 21 (1899)]]. For example (0102030405060708)321, is the eleventh group of order 32 and degree 8 in Professor Miller's list. Each distinct head is written and is followed by substitutions which generate groups that multiplied into the given head produce distinct imprimitive groups. In this list the following notation is used: (1) Two systems, t = 0161. a2&2.0363. 0464.0565.0666 -0767.0868 3 492H1^^» 4 Grace Marie Bareis (2) Four systems, t = aibi . Ozbi . ttsbs . ttibi . Cidi . C2d^ • Csds . Cidi ti = ttibiCi. 026202. asbzCz.ttihiCi tz = ai6i . 0262 . tts&s • 04^4 Of these t ti generate a group simply isomorphic to the alternating group of degree four and t ti U generpte one simply isomorphic to the symmetric group of degree four. (3) Eitrht systems, tx = aibiCidieifigi.(hb2C2d^e2f2g2 h = aieijidibiCigih.(he2f2d^b2C2g2h tz = bidiCigieifi.b2diC2g2e2f2 U = hiai.bidi.cigi.eifi.h(h.b2d2.C2g2.e2f2 h = b1g1d1C1.e1f1.b2g2d2C2.e2f2 U = ai6i^i . 0262^2 <7 = 01^1.02^2 Of these h h generate a group simply isomorphic to the symmetric group of degree eight; ^1 U to the alternating group; h U tz^ U to the primitive group of order 1344; ^1 k tz to the 336; ^1 1^^ tz"" to I681; ^1 1^ tz"^ to I682; ti t^ to the 56 group. The generators of the tails involve only the t's, or they involve sub- stitutions that do not interchange the systems multiplied by fs. To illustrate [(aia2a3a4«5«6a7a8)644(&l&2&3&4&5&6&7fe8)64 J32 , 32 t, QiazChd^ ' bibzb2b4 . t are the two groups that have for their common head a 32, 32 isomorphism of the fourth group of order 64 in Professor Miller's list, and for tails this head multiplied by t and 010^0204. bibzb2bi.t respectively. This notation is still further abridged to L(^12345678)644(&12345678)644 J32 , 32 t, ai324 • Ol324 • t Similarly E(«1234)4(&1234)4(Cl234)4((^1234)4]] t ti tz, t ti ttn . 6l2 • C12 . C?12 • tz are the two groups having for their common head the direct product of the non-cyclic transitive groups of degree four and having for tails this head multiplied by each of the 23 substitutions different from identity of the groups generated by t ti tz and t ti a^ . 612 . cn . dn . tz respectively. Systems Order No. 3251404800 1 1625702400 1,2 812851200 1 3612672 1 2654208 1 1327104 1,2 3,4 5,6 663552 1,4 5 6 7 8 9 331776 1,2 3,4 5 6,9 10 294912 1 225792 1 165888 1,4 5 6,7 8 147456 1,2 3,4 OF Intransitivity of Heads 8, 8. [(ai2345678) all (6l2345678) all] t [(ai2346678) all (6i2345678) all] pos t, Un-t [(ai2345678) POS (6l2345678) pos] t L(<*12345678)l344(Ol2345678)l344j t L(ai2345678)ll62(6l2345678)ll52j t L(<*12345678)ll62(&12346678)ll62j dim t, ai6.26-37-48' ^ L(<*12346678)ll62(&12345678)ll62j676, 576 ^ Cll2-t L(<^12345678)ll52(Ol2345678)ll52j676. 576 ^ 012-^ L (^12345678) 1152 (Ol2345678) 1152 J288. 288 ^ ^12 • ^, fll5•26•37•48•^ ai626-37-48- ^ L(<*12346678)l]52(&12346678)ll52_|288, 288 t L(Q12345678)i152(012345678)i152J288. 288 t L(ai2345678)ll52(Ol2345678)ll52J!88, 288 t L(<^12345678)576j(&12345678)5762J t L(012345678)576,(012345678)576,J i L(rtl2346678)ll62(6l2345678)ll62jl44. 144 ^ fll2-56-^ L(^12345678)i162(&12345678)i152Ji44, 144 i, ai2-56-^ «C(^12346678)676j(6l2345678)5762J288. 288 fll6-26.37.48 • ^ |_(ai2345678)5762(&12346678)576,j288, 288 ^ 012-56 -^ fli2 • t'i2 • t) ctse • O12 . ^ L(Ol2345678)576,(&12345678)576,j288, 288 fll626'37.48 • ^ L(012345678)384 (^12345678)384 J t C('^)2345678)336(Ol2346678)336j t L(012345678)576,(6i2345678)5762Ji44, 144'. 012-56 •', Ol6-2«-87'48-^ ai2.^2.' L(012345678)576,(^2345678)576jJi44. 144 t L(Ol2345«78)676, (^12345678)576, Jl44, 144 01625-37-48 • '» QM-bn.t L(Ol2345678)288(0l2346678)288j ' L(ai2345678)384 (^2345678)384 Jl92. 192 t, dlS-U't L(Ol2345678)384(Ol2346678)384jl92, 192 ^ dn-i "*N Grace Marie Bareis >rder No. 5,6 112896 1,2 110592 1 82944 1,2 80640 1 73728 1 2,5 6 7 8 9 10 11 56448 1 2 55296 1,2 3,4 49152 1 40320 1,2 36864 1 2 3,4 5,6 7,8 9, 10 11, 12 32768 1 27648 1,2 24576 1,2 3,4 5,7 21504 L(^12345678)384(Ol2345678)384l92, 192 t, dn-t L(Cfl2345678)336(Ol2345678)336jl68, 168^, CH5642378-^ L(ctl2345678)5762(Ol2345678)o762j96, 96 t L (^12345678)288 (Ol2345678)288jl44, 144^, CH5.26.37-48 • ^ [(ai2345678) all (6l2345678) all]i, i t L(^12345678)l922(Ol2345678)l922J t L(^12345678)384 (0x2345678)384 J96, 96 ^, dvi-t, «13-24-^, ftl423 • t L(^12345678)384(0l2345678)384j96, 96 ^ L(^12345678)384(Ol2345678)384j96, 96^ L(^12345678)384(Ol2345678)384j96, 96^ L(^12345678)i923(Oi2345678)i923J t L(^12345678)l924(&12345678)l924J t L (^12345678) 1925(012345678) 1925 J" ^ L(^12345678)l68i(^12345678)l68iJ t L(^12345678)l682(Ol2345678)l682J t L(^12345678)5762(^12345678)5762J48, 48 ^, fll6-25-37-48- ^ L(CH2345678)5762(^12345678)5762J48, 48^, ai6-25-37-48- ^ L ('^12345678)384 (Ol2345678)384j64, 64 t [(ai2345678) POS (6l2345678) POS^l, 1 t, U^.b^.t L(«12345678)ll52(&.2345678)ll52dl6, 16 t L(fl^l2345678)l922(^12345678)l922j9e, 96 fll5-26-37.48- ^ L(^12345678)l922(Ol2345678)l922j96, 96^, ai2.56-^ L(*12345678)l922(^12345678)l922j96, 96^, ai5-26-37-48 • ^ L(^12345678)l923(Ol2345678)l923j96, 96^, ClVl'i L(ai2345678)l924(&12345678)l924j96, 96^, 013.24-^ L(C[l2345678)l925(&12345678)l925j96, 96^, ^1423 • ^ L(«12345678)i28(Ol2345678)l28j t L(^12345678)288(Ol2345678)288J48, 48 ^, CH2.56. Ol2-56' ^ L(Ol2345678)384(Ol2345678)384j32, 32 f, dll-t L(«12345678)384(&12346678)384j32, 32 t, dn-t L(<*1/345678)i923(&12345678)i923J64, 64^, 0135-246 •^, fll3-24.0i3.24-^ L (^12345678) 1344 (Ol2346678) 1344 J8, 8^ Imprimitive Substitution Groups of Degree Sixteen Order No. 18432 1,2 3,4 5,6 7 8 9 10 11 16384 1,2 3,4 5,6 7,8 9, 10 11, 12 13, 14 12288 1 2 3,6 7 8 9216 1,4 5 6 8192 1,4 5 6 7 8, 11 12 L(^12345678)676j(Ol2345C78)676,Jl6. W '» au-b]2.t L(ctl2345678)576,(Ol2345678)576,Jl6. 16 ^ fllZ . &12 • ^ l(^12346678)i92j(Oi2345678)i92jJi8, 48 ai6-26.37-48- ^, Ol6'26.37.48 • t L(fll2345678)l922(Ol2345678)l922_]48, 48 ^ L(Ol2345678)l922(Ol23456789)l92j_l48, 48 i L(^12345678)96io(Ol2345678)96ioJ ^ L(<*12346C78) 96 1,(^12345678) 96,1 J ^ L(^12345678)96, 2(012345678)9612] t L(012345678)i28(Oi2345678)i28J64, 64 L(<*12346678)l28(Ol2345678)l28j64, 64 L(<^12345678)l28(Ol2345678)l28j64, 64 L(^12345678)l28(Ol2345678)l28j64. 64 L(^12345678)l28(Ol2345678)l28j64. 64 [_(<^12345678)l28(Ol2345678)l28_|64. 64 L(^12345678)l28(Ol2345678)l28_|64, 64 L(012W6678)i922("12345678)i922332, 32 L(^12345678)384(Ol2345678)384_|l6, 16 L(^12345678)l92,(&12345678)l92,j32. 3 fll3-24 . Ol3'24 • ^ 01423 •Ol3-24-^ L(ctl2345678)l924(Ol2345678)l9212345678)l28j32, 32 t 49 L(ai2345678)l28(^12345678)l28j32, 32 t 50 |I(ai2345678)643(&12345678)643J t 51 L(ai2345678)645(&12345678)645J t 52 L(Ol2345678)64e(Ol2345678)646J t 53 I_(ai2315678)647(&12345678)647J t 54 Qai2345678)642(&12345678)642J t 55 L(ai2345678)644(&12345678)644J t 6272 1 Qai2345678)56(&12345678)56j t Imprimitive Substitution Groups of Degree Sixteen Order 6144 4608 4096 No. 1,2 3,4 5,7 8,9 10, 11 12, 13 14, 16 1 2 1,2 3,4 5,6 7,8 9, 10 11, 12 13,20 21 22,23 24,25 26,27 28,29 30,31 32,33 34,35 36,37 38,39 40,41 42,43 44,45 46,47 L(fll2345678 L (^12345678 L(<^12345678 an-66-b L(^12345678 L ('^12346678 L (^12346678 L(fll2345678 ai3-24 • b L (^12345678 L(^I2346678 L(^12345678 L(o 12345678 LC** 12345678 l(^12345678 L (^12345678 L (^12345678 L('*12345678 ai5-26-37 fll 7362 845' L('^1234ft678) L(ai2345678) L ('^12345678 L(<*12345678 C (^12345678 L(<^12345678 L(^12345678 L(<^12345678 L(^12345678 L (^12345678 L (^12345678 L (^12346678 L (^12345678 |_(<^12346678 192j(&12345678)l92,Jl6. 1« '» fll5-26-37-48 • ^ I92,(6l234567s)l92,jl6' '6 ^ CH5-26-37-48 • ^ 96io(ol2346678)96ioJ32, 82 ^ fll23-667'^> 2-66 -^ 384(Ol2345678)384J8, 8 ^ 012.^ 384(Ol2345678)384j8. 8 ^, ^12 • ^ 192,(&12345678)l92,j8, 8^ ffll3-24 • &13.24 • ^ 96ij(Ol2345678)96,2j32, 32 ^, fll35'246-^. 3^24 • t 48u(6l2345678)48ieJ t 48i7(6l2345678)48nJ ^ 128(Ol2345678)l28_]l6, 16 ^ fll2.56«^ 128(^1234667s)l28]]l6. 16 t, CH2-66-^ 128(w]2345678)l28jl6. 16^, fll3-24.57-48 • ^ 128(Ol2345678)l28jl6. 16 ^ 013.24.57-68 • ' 128(f'l2346678)l28jl6, 16 ^ fll2-56.^ 128(Ol2345678)l28J16. 16 ^ Ol2.56.^ 28(l'l2346678)l28jl6. 16 ^ Cln-t, dli'U-t, i-t, 01625.37.48.^, Ol736.2846.^ Ol324.^» t 28(Ol2345678)l28jl6, 28(Ol2345678)l28jl6, 128(Ol2346678)l28jl6, 128(Ol2345678)l28jl6, 128(Ol2345678)l28jl6, 128(0l2345678)l28jl6, 128(6l2345678)l28jl6, 128(&12345678)l28jl6, 128(^12345678) 128 Jl6, 128(Ol2345678)l28_]l6, 128(Ol2346678)l28jl6, 128(Ol2346678)l28jl6, 128(Ol2346678)l28jl6, 128(Ol2345678)l28jl6. 6 fll3.24 . i 6 it Oi5.26.37.48 • t 6 i, 01735.2846 • i 6 t, ai2 . t 6 i} Oi2 . t it, a\2.t t, 015.26-37.48.^ t, Oi5.26.37.48.^ 6 t, Oie26.37.48 . ^ t) 015.26-37-48-^ 6 t, Oi626.37.48 • ^ 6 t, ai3.2i.t 6 t, ai324 . t 6 t, Oi3.24.< 10 Grace Marie Bareis Order No. 48,49 [](ai2345678)l28(&12345678)l28jl6, 16 ^, Clisii.t 50,51 • L(ai2345678)l28(&12345678)l28jl6, 16 ^, 0'U-2i-t 52,53 [^(ai2345678)l28(&12345678)l28jl6, 16 ^j ^1324-^ 54,55 [](ai2345678)l28(&12345678)l28jl6, 16 ^, 013-24.^ 56,57 Qai2345678)l28(^12345678)l28jl6, 16 ^j ai324.^ 58,59 Qai2345678)l28(&12345678)l28jl6, 16 ^, ai3.24-^ 60,61 [](ai2346678)]28(&12345678)l28jl6, 16 ^, 01324-^ 62,63 L(ctl2345678)l28(&12345678)l28jl6, 16 ^> ttn-t 64,65 |^(ai2345678)643(&12345678)643j32, 32 ^, ai5-26-37.48. < 66, 67 |I(ai2345678)643(&12345678)643j32, 32 t, Cll3-2A't 68,69 |3(ai2345678)643(&12345678)643j32, 32 ^, ai3-24 • ^ 70,71 [](ai2345678)643 (^12345678)643^32, 32 ^j ai2.56«^ 72,73 Qai2345678)643(&12345678)643j32, 32 ^, ai2-56-^ 74,75 C(^12345C78)646(&12345678)645i]32, 32 ^, ai625.37-48- ^ 76,77 [(ai2345678)645(&12345678)645_]32, 32 ^, «12.56-^ 78,79 [(ai2345678)646(&12345678)64jl32, 32 ^, ai3-24 • ^ 80, 81 Qai2345678)646(fel2345678)64gl32, 32 t, ai5-2G-Z7-AS't 82,83 [](ai2345678)646(&12345678)646j32, 32 ^, «12 • ^ 84,85 Qai2345678)646(&12345678)64ell32, 32 ^, ^12 • < 86,87 [(ai2345678)647(&12345678)647l|32, 32 ^, ai735.2846-< 88,89 [](ai2345678)647(&12345678)647j32, 32 ^j ^12 • ^ 90,91 [(ai2345678)647(fel2345678)647l!32, 32 ^, 012-^ 92,93 [(ai2345678)642(&12345678)642l|32, 32 ^, ai5-26-37-48- ^ 94,95 [](ai2345678)642(fel2345678)642j32, 32 ^, 01324-^ 96,97 C(ai2345678)642(&12345678)642_|32, 32 ^, 01324-^ 98,99 [^(ai2345678)642(&12345678)642j32, 32 ^, ai2.56-^ 100,101 [(ai2345678)642(^12345678)642l|32, 32 ^, ai2.56-^ 102,103 [(012345678)64^(^12345678)644132,32^, ai625-37.48- ^ 104,105 [(ai2345678)644(fel2345678)644j32, 32 ^, 01324-^ 106,107 [(ai2345678)644 (^12345678)6432. 32 ^, ^1324.^ 3072 1 [(ai2345678)l922(^12345678)l922j8, 8 t 2,5 [(ai2345678)l923(fcl2345678)l92jj8, 8 ^, ^12 ^, fll3-24-&13'24.^j ai423'Ol3-24-^ 6 [(ai2345678)l924(&12345678)l924j8, 8 t Imprimitive Substitution Groups of Degree Sixteen 11 Order 2688 2304 2048 No. 7 8 9 10, 14 15 1 2 1 2 3,4 5,6 7,8 1 2 3 4 5 6 7 8 9, 12 13 14 15 16, 19 20 21 22 23,26 27 |_(«12345678)l92^ (Ol2345678 L (^12345678) 1 925 (Ol2345678 L (^12346678) 192 J (^12345678 L (^12345678) 96 „(Ol2345678 L (^12345678) 96 1 , (Ol2345678 L (^12345678) 1344 (Ol2345678 L (^12345678) leSj ("12345678 L (Ctl2345678) 1152 (Ol2345678 L('^12345678)48ig (Ol2345678 L («12345«78)48 ig (Ol2345678 L(^12345678)48,g(Ol2345678 L(^12346678)48, 7(^12345678 L(^12346678)32,o(^12345678 L (^12345678)32,, (^12345678 L (^12345673)32 J 2(012345678 L(^12346678)32„ (Ol2345678 L (^12345678)32 ,4 (Ol2345678 C(^12345678)32,j(Ol2346678 L(fll2345678)32j7(Ol2345678 L(^l2345678)32ig(Ol2345678 L(^1234 5678)64,(^12345678 1924 J8. 8 t 192jj8, 8 t 192jj8. 8 i ' 96,0 J16, 16 t, dUZ'WT-tf ,37.48 •<, ai6-25-37.48'Ol2.66.^ 96i,Jl6. 16 t 1344 Jl, 1 t I682J8. 8 t 1152 Jl . 1 t 48,g_l24, 24 fll6'26-87.48 • ^ 48,gj24, 24 t, fll2.56.^ 48,4 J24. 24 t, ai2-66-^ 48,7J24, 24 ^ CH3-24-78-^ 32,oIl i 32„] t 32, J t 32,,] < 32,4] i 32,3^ 32,7] < S2,Jt 64,Jl6, 16 i, ttlZ-U'tf ai5-26-37-48 • t, «1735.2846 • t \_ (fll2345678) 64, (Ol2345678 L (Ct 12345678)64, (&12345678 L(^12345678)64, (Ol2346678 L (^12345678)64, (Ol2345678 dli.K.t, ai324.6fi./ L('^12345678)64, (Ol2345678 L (^12345678)64, (Ol2345678 C('^12346678)64,(Ol2345678 L(Ol2346678)64, (Ol2345678 012-66 -^ ai324-56-' L (^12346678)64, (Ol2346678)64,Jl6, 16 t 64,Jl6. 16 t 64,Jl6. 16 t 64,Jl6. 16 t 64,Jl6. 16 t, ttlS'U-t, 64,Jl6, 16 i 64,Jl6. 16 t 64,J16. 16 t Mjjie, 16 t, Cm.24''» 12 Grace Marie Bareis Order No. 28 |^(ai2346678)643(Ol2345678)643Jl6, 16^ 29 L(ai2345678)643(&12346678)643l6, 16 t 30,33 [_(ai2345678)643(Ol2345678)643Jl6, 16 ^ 0'\1-0v2..t, ai2-56-^> ai5'26.37'48-^ 34 L(fll2345678)643(wi2345678)643Jl6, 16^ 35,38 C(ai2345678)643(&12345678)643_ll6. 16 ^, Ol2 . 6l2 • <, fll2-56-^j Cti5.26-37'48 • ^ 39 L(ai2345678)643(Ol2345678)643Jl6, 16 ^ 40, 42 [I(ai2345678)645(&12345678)645Jl6. 16 Oi625-37-48- ^, an . bi2 . t, a56 .bn.t 43,46 L(ai2345678)645 (^12345678)645^16, 16 ^> CH3-24-^ fll625-37-48-^j fll7362845'^ 47 [_(ai2345678)645(6l2345678)645J16, 16 t 48 L(ai2345678)645(&1234o678)645Jl6, 16 t 49 L(ai2345678)645(&12345678)646Jl6, 16 t 50,53 |I(fll?345678)645(&12345678)645Jl6, 16 ^, «1625-37-48- ^j 0X2 . 6i2 . t, ai324 .bi2-t 54,57 [](fll2345678)645(fel2345678)64eJl6, 16 ^, fll5-26- &15-26- ^, «15-26-37-48'^> a57-68- t'l3-24 • ^ 58,61 Qai2345678)64j(&12345678)646Jl6, 16 ^, ^12 ^j Ctl5-26-37-48-^) CH625-37-48 • ^ 62 [_(ai2345678)646 (1^12345678)646 Jl6, 16^ 63 L(ai2345678)64e (^12345678)646 Jl6, 16 t 64 L(«12345678)646(0l2345678)646Jl6, 16 i^ 65,68 |I(ai2345678)646(&12345678)64eJ16, 16 ^, 012 • ^ «15;26-37-48-^> 013-24 • O13.24 • ^ 69 L (^12345678) 646(^12345678) 646 J16, 16 t 70,71 C(ai2345678)647(&12345678)647J16, 16 01735-2846 -^j 057-68-013-24-^ 72,75 Qoi2345678)647(fel2345678)647Jl6, 16 ^, 0,12 -t, Ol735-2846-^) Ol8462735-^ 76 l(^12345678)647(&1234.5678)647Ji6, 16 ^ 77 L(Oi2345678)647(&12345678)64tJi6, 16 ^ 78 [_(ai2345678)647 (^12345678)647 J16, 16 ^ Imprimitive Substitution Groups of Degree Sixteen 13 Order No. 79,82 [_(ai2345678)647(Ol234S678)647Jl6, 16 ^ «! 735-2846. ^ fll3-24- 013-24 -^ fll423-Ol3-24-^ 83,86 [_(ai2346678)64,(6i2345678)64jjl6. 16 ^ 01324-^, 015-26-37-48 -i^j ^17362845-^ 87 L(ai2346678)642(Ol2345678)64,J16, 16^ 88 L(Ol2345678)642(Ol2346678)642Jl6, 16 i 89 L(^12346678)642(Ol2345678)642Jl6. 16 t 90,93 L(ai2345678)642(Ol2345678)64jJl6, 16 ^ 012-66-^, 015-26-37-48-^. 012- Ol2-^ 94 L('^12345678)64,(tl2345678)64jjl6. 16 t 95,98 L(ai2345678)64,(Ol2345678)64jJl6, 16 ^, CH2-66-^ Ol5-26-37-48-^> du-bn-t 99 L(O12345678)642(012345678)642Ji6, 16 i 100,103 |_('^12346678)642(^12346678)642_|i6. 16 <, 012-66- ^ 01324 -<, 014.23-56-^ 104 L(Oi2345678)642(012345678)642Ji6, 16 t 105 L(Oi2345678)642(^12345678)64jJi6, 16 t 106 C('^12345678)642 (012345678)642^16, 16 t 107,110 L(^12345678)e42(Ol2345678)642Jl6, 16 ^) Oi2.56-^ Ol324-^ Oi4.23-56-^ 111 l(^12345678)642(012345678)64,Ji6. 16 t 112 L(^12346678)642(Ol2346678)64,Jl6, 16^ 113 L(Ol2345678)642(Ol2345678)64jJl6. 16^ 114,117 L(ai2346678)644 (^12345678)644 Jl6, 16 ^ 0x324.^, ai626-37-48 • ty Oi736.2846 • i 118 L(^12345678)644(Ol2345678)644Jl6. 16^ 119 l(Oi2345678)644(012345678)644Ji6, 16^ 120 1.(^12346678)644 (&12346678)644J16. 16^ 121, 123 L(Oi2345678)644(012345678)644Ji6, 16 Oi626-37-48'^ on.bu.t, a^'buJ 124,127 L(Oi2346678)644(6l2345678)644Jl6. 16 ^ Oi826-37-48 • '> Ol2.0l2-^> 066 -012.^ 128, 131 L(Oi2345678)l28(6l2345678)l28j8, 8 ^ Oi2.66-^ Ol3-24-^ 01423-56 -^ 14 Grace Marie Bareis Order No. 132, 135 L(^12346678)l28(Ol2345678 0'13-24-t, (lU23-56-i 136, 139 L(ai2345678)l28 (^12345678 ais-24-t, ai423-66'^ 140, 143 [_(ai2345678)l28(&12345678 aiz-24-t, CI'U2Z'bG-t 144, 147 |_(ai2345678) 12 8 (^12345678 «13-24-^, ttl423-56'^ 148, 151 L(ai2345678)l28(&12345678 fll3-24-^, Cfi423.56.^ 152 L(«12345678)l28(^12345678 153 C(^12345678)l28(&12345678 154 L(^12345678)l28(Ol2345678 155, 158 L(ai2345678)l28(^12345678 ai2.t, ai32'i-57-es't 159, 162 L(^12345678)l28(&12345678 (ll2't, ai32i-57-GS-t 163, 166 [_(ai2345678)l28(^12345678 fll3-24-57-68-^ 167, 170 L(«12346678)l28(&12345678 (^n-t, ttl324-b7-6S-t 171, 174 L(ai2345678)l28(^12345678 (ll2-t, CH324.57-68''^ 175, 178 L(ai2345678)l28 (^12345678 ttl2-t, tti324.57-68'^ 179 l(^12345678)i28(012345678 180 l(^12345678)i28(&12345678 181 L(^12345678)i28(Oi2345678 182, 185 L(^12345678)i28(012345678 «1324-^> ai4.23.56-*^ 186, 189 L('^12345678)l28(Ol2345678 ai324.^, (Iu.23.b6't 190, 193 L ('^12345678) 128 (Ol2345678 ai324.^, (lU-23-b&-t Ol28j8, 8^, o-n-be-t. Ol28j8, 8^, (Il2'b6-t, Ol28j8, 8^, (ll2-56-t, Ol28_l8, 8^, ai2-56't. Ol28J8, 8^, «12-56.^, )l28j8, 8^ )l28j8, 8/ i)l28l8, 8^ 0128^8, 8^, fll3-24-57-68'^j O128J8, 8^, ai3.24-57.68-^, Ol28j8, st, (112 -t, «1324.57-68-^j )l28j8, st, ttl3.24-57-68-^, )l28j8, st, «13-24-57-68.i^, )l28j8, st, «13-24-57-68'^j )l28j8, st )l28_]8, st )l28j8, st )l28j8, st, 0-12 • 56 -t. )l28J8, st, ttl2-b6-t, )l28j8, st. Cll2-56-t, Imprimitive Substitution Groups of Degree Sixteen 15 Order No. 194, 197 L(ai2346678)l28(&12345678 198, 201 L (^12346678) 128 (Ol2346678 01324 -^ «14-23-6«-^ 202, 205 L(ai2346678)l28(Ol2345678 fll324-^> ai4.23-66'^ 206 [_ (^12345678) 128 (Ol2345678 207 L(cH2345678)l28(Ol2345678 208 |_(fll2315678)l28(Ol2345678 1536 1 L(ai2345678)384(Ol2346678 2, 3 [_(ai2345678)96ij(^1234B678 4, 5 L(ai2345678)96i2(^12345678 6, 7 L(ai2345678)l92j(Ol2345678 8, 9 L(ai2345678)l92j(Ol2345678 10, 11 |_(ai2345678)96io(Ol2345678 1152 1, 2 L(ai2345678)5762(oi2345678 3, 4 L(ai2345678)676,(Ol2345678 5, 8 L(^12345678)48i,(Ol2346678 ai5'26.37'48-^ 016-25 9 |_(^12346678)48,g (012345678 10 L(Ol2345678)48:j(oi2345678 11 L(Oi2345678)48ig(Ol2345678 12 L(ai2346678)24,(Ol2345678 13 I_(^12346678)24„(Ol2346678 14 L(Oi2346678)24io(Ol2345678 1024 1, 2 L(ai2345678)32,o(Ol2345678 3, 4 l_('*12345678)32io(oi2345678 5, 6 L(^12345678)32,o(Ol2345678 7, 8 L (012345678)32 „(Ol2345678 9, 10 L(Oi2345678)32ii(Ol2345678 11, 12 L(Oi2346678)32„(6l2346678 13, 14 [_(ai2345678)32i,(oi2346678 15, 16 L(Oi2345678)32„ (^12345678 17, 18 |_(ai2345678)32ij(Ol2345678 128j8. 8 t, an-K-t, 128 J8, 8 ^ Oi2.66.<, 128 Js, 8 t) Oi2.56.^, U8, 8 t IJ8, 8 ^ U8, 8 t ii, 2 ^ 9612J8, 8^ Ol3-24 • &13-24 . < 9612J8. 8 ^> Oi423 • 01423 • ^ I922J4. 4 ^ Oi5.26.37-48'^ I922J4, 4 t, Ol6-26.37'48 • ^ 96ioj8, 8 ^ Oi2.56-Ol2.66-^ 5762 Jl , 1 t> O12 .bi2't 676jjl , 1 ^ 012 . 612 . < 48igjl2, 12 t, Oi2.66.'> 37-48 -^ 48iJl2. 12 t '48,Jl2. 12 t '48, J12. 12 t '24 J « 24 „] t 24,0] i '32ioJl6. 16 32,oJl6, 16 '32,oJl6. 16 32nJl6, 16 32i,J16, 16 '32iiJl6. 16 '32,2J16, 16 '32,2Jl6, 16 3212J16, 16 t, 015.26-37-48 -^ t, Oi324.^ t, 016.26-37-48-^ tj Oi3.24-< t, 013.24.^ i> Oi6.26-37-48-^ tf Oi2.66'^ t, Oi2.56-^ 16 Grace Marie Bareis Order No. 19,20 21,22 23,24 25,26 27,28 29,30 31, 32 33, 34 35,36 37, 38 39,40 41,42 43, 44 45,46 47,48 49, 50 51,58 59,60 61,64 65,66 67, 68 69, 70 71, 72 73,74 75, 76 77,80 81,82 83,86 87,88 .(b 3 (,"12345678 3213II16, 3213II16, 3214J16, 32]4Jl6, 32iJl6, 321J16. 32nJ16, 3217J16, 3217J16, 32iJl6, 32i6lll6, 32i6J16, 32i6l|l6, Js, 8 ^ 64, J8, 8 L (^12345678)32 L(^12345678)32i3 (Ol2345678 L(C^12345678)32 14(012345678 L(<^12345678)32 14(012345678 L (^12345678)32 15(012345678 L(^12345678)32ib(012346678 L(^12345678)32i7(oi2345678 L(«12345678)32i7 (Ol2345678 L(^12345678)32i7(Ol2345678 L(^12345678)32ig (Ol2345678 L (^12345678)32 ie(oi2345678 L (Cfl2345678) 32 jg (Ol2345678 L(<^12345678)32ig (Ol2345678 L (^12345678)32 ig(oi23i5678 C(^12345678)643(&12345678)64 L(^12345678)643 (Ol2345678 L(^12345678)643(Ol2345678)643j8, 012-56 -^j Oi7.28-35-46-^j Ol423-56 O1324.O12./, 013-24 -^ L (^12345678)643 (Ol2345678 L(Ol2345678)643 (&12345678 ai2.bi2.t, 01526.37-48 C(Ol2345678)643(Ol2345678 C(Ol234£678)643 (Ol2345678 L(Ol2345678)643 (0l2345678 L(Ol2345678) 643(012345678 L(Ol2345678)643 (&12345678 L(Ol2345678)643 (Ol2345678 L(Oi2345678)643 (0l2346678)643 J8 ai2.bi2.t, a5Q.bi2.t L(Ol2345678)643(Ol2345678)643j8, 8 i) L(Ol2345678)643(Ol2345678)643j8, 8 t, O12 . O12 . t, a56 .h\2-t C(Ol2345678)643(&12345678)643j8, 8 ^> ] 643II8 t 3 643II8 643]8 643]8 643II8 64, J8, 8 64, J8, 8 ) 0x735.2846 -^ J Oi2.56-^ , ffl625.37-48 • ' , 013.24-57-68 -^ ) 0x7362845-^ , Oi2.56-^ , Oi2.5768-^ J Ol5-26-37-48-^ , 015.26-37-48 -^ , Oi5.26-37.48-^ J 013.24-56-^ , Oi3.24.56-^ , 0X2.34-^ , Ol2-34«^ Ol3.24-57-68-^ 013.24-57-68-^ 01735-2846 • t, t, (Il2'bi2.t, 0x2.56- ^ 0x735-2846-^} 0x3-24 -^ flX423-56 • t ' 0x735-2846-^ 0x735-2846-^ 0x735-2846.^ Ox3-24 - 1 0x2-56 -^> 0x2-56-^ 0x2-56 -^, 0x2.56-^ Imprimitive Substitution Groups of Degree Sixteen 17 Order No. 89,92 [_(ai2345678)64,(Ol2345678)64jj8. 8 ^ CH2-34-^, fll2 • Ol2 . t, 034 .Oi2.t 93,94 [_ (^12345678)64, (&12346678)64,j8, 8 ^ 012.34 • ^ 95,98 [_(^12345678)64,(&12345678)64,j8, 8 ^ fll2-34'^ ai2 . Oi2 . /, fl34 -012.1 99, 100 L(ai2346678)64,(Ol2345678)64,j8, 8 ^» 012-34 .^ 101, 110 L(ai2345678)645(Ol2345678)64jJ8, 8 ^ O13.24 • <, fll2-56-^> ttl324-66.^ CH625-37-48 • '» ai3-24-56 • &12 • ^ «12-&12-^ 056. &12.^ 017362845 -^ ai324.&12.^ Ill, 115 L(ai2345678)64i(^2345678)64jj8, 8 ^> Ol3.24.^» 01626.37.48 . ^ Ol2 • &12 • <, 01736.2845. ^12 . ^ 116, 119 L(Oi2345678)645(Ol2345678)64jj8. 8 ^ Ol2.66.<, Ol2.6l2.<, 056. 612. < 120,121 L(Oi2345678)64j(Oi2346678)64j_]8. 8 ^> O13.24.67.68 • ^ 122,123 |_(ai2345678)64j(Ol2345678)64jj8. 8 ^> Oi3.24.67.68. ^ 124, 127 [_(Ol2346678)645(Ol2345678)64j_]8. 8 ^ Oi2 . O12 . <, Ol2.56.^> 056.012.^ 128, 129 [_(ai2345678)64j(&12345678)64jj8, 8 ^, Oi2.66.^ 130, 137 L(Oi2345678)64,(6l2345678)64gj8, 8 ^ O12 . <, 013.24.67*68 • ^> 01423.57-68 . ^ O13.24 • 013-24 • ^, O1423 . O13.24 . <, O12.57-68- ^13-24 -^ 057.68-^13-24-^ 138, 141 (_(^12345678)64,(&12345678)64,j8. 8 ^> O13.24.67.68. ^> O13.24 . O13.24 . t, 057.68 • 013-24 . t 142, 145 L(ai2345678)64j(Ol2345678)64,j8, 8 ^ Oi2.<, O1423.57-68 • ^> Oi827-36-46-^ 146,149 [_(ai2345678)64j(&12345678)64jj8, 8 <, 016.26-37.48. '» O18.24 . O13.24 . t, 016482637 . 01324 . t 150, 153 L(Oi2345678)64j(Ol2345678)64gj8. 8 ^, Oi2.66.^> Ol3.24.&13.24.^> O1423.56 . &13-24 - ^ 154, 155 L(^12345678)64,(&12345678)64,j8. 8 ^ Oi2.66-^ 156,159 [_(Oi2345678)64,(6l2345678)64,j8, 8 <, O12.66.', 013.24.013.24.^ 01423.56 . O13.24 . ^ 160,161 L(Oi2345678)64,(&12345678)64gj8, 8 ^, Oi2.66.< 18 Grace Marie Bareis Order No. 162,171 C(ai2345678)647(&12346678)647j8, 8 ^> ^12 • ^, «13-24-57-68- ^» «1423-57-68 • ^) O1735.2846 • ^j 018462735-^, ttl3-24-&13-24-^) fll423 • 013-24 • ^5 Ct57-68- Ol3.24- ^> ftl2-57-68' t>13.24'^ 172,175 [_(ai2345678)647(^12345678)647j8, 8 ^) 012-^, «1 735.2846 • ^) fll8462735-^ 176,183 [[(ai2345678)647(&12345678)647j8, 8 ^, «12 • ^, fll3-24-57-68-^j Cti423. 57-68 • ^> 013.24- Ol3.24- ^, ai423-&13-24-^, «57-68 - &13-24 - ^> «12-57-68- Ol3-24. ^ 184, 185 |I(ai2345678)647(&12345678)647j8, 8 ^j 012-56-^ 186,187 [^(ai2345678)647 (^12345678)647 J8, 8 ^, 012.56-^ 188,191 C(ai2345678)647(&12345678)647j8, 8 ^, ai2-56-^^ Ol3-24- ^13-24 -^j CH423-Ol3-24-^ 192,193 [^ (^12345678)647 (^12345678)647 Js, 8 ^, ai2.56-^ 194,195 [(ai2345678)642(&12345678)642l]8, 8 ^^ 013-24-57-68-^ 196,197 [](ai2345678)642(^12345678)642_l8, 8 ^} 013-24-57-68-^ 198,201 |[(ai2345678)642(^12345678)642l8, 8 ^j 012-56-^, Ol2 . 612 . i, 066 - O12 . i 202,203 |I(Oi2345678)642(^12345678)642J8, 8 ^) Oi2.56-^ 204,211 [ (0123456 78) 64 2(^12345678) 642118, 8 ^, Oi2.56-^, Oi324.<, 014-23-56-^, 0]5.26-37-48-^> Oi7362845-^j O12 . O12 . ^, O13.24 .Oi2.t 212,215 |I(Oi2345678)642(&12345678)642_|8, 8 ^^ Oi2-56-i^j O12 • O12 • t, O56 .hi2.t 216,219 .n(Oi2345678)642(&12345678)642j8, 8 ^j Oi5.26-37-48- ^> O12 - 612 - 1, 01736-2845 - Ol2 - ^ 220,221 l(Oi2345678)642(^12345678)642.J8, 8 ^j Oi2.56-^ 222,223 |I(Oi2346678)642(&12345678)642j8, 8 ^ 012-56-^ 224,225 [(Oi2345678)642(&12345678)642l|8, 8 ^, Oi5.26-37-48- ^ 226,227 [] (Oi2345678)€42 (^12345678)642 J8, 8 i^j 015.26-37-48 • i^ 228,229 [(012345678)642 (&12345678)642ll8, 8 ^, 01324.^ 230,231 C(0]2345678)642(&12345678)642J8, 8 ^j Oi4.23-56-^ 232,233 [_ (Oi2345678)642 (^12345678) 642 J8, 8 ^j 01324-^ 234,235 n(^12345678)642(^12315678)64,]8, 8 ^, Oi4-23-56-^ *» Imprimitive Substitution Groups of Degree Sixteen 19 Order No. 236,237 [_(ai2345678)64,(Ol2345678)64,j8. 8 ^ CH3-24.57.68-' 238,239 L(ai2345678)64,(Ol2345678)64,j8, 8 '. fll3-24-57-68- ' 240,241 [_(ai2345678)64,(6l2345678)64,j8, 8 ^ 012.66-' 242,243 L(fll2345678)64j(Ol2345678)64,j8, 8 ', fll2.56-' 244,245 L(ai2345678)644(6l2345678)644j8, 8 'j fll3.24-57-68- ' 246,247 C(«12346678)644(0l2345678)644J8, 8 '> fll3-24.57-68- ' 248,255 L(CH2345678)644(012345678)644J8. 8 'f 01324-', fll625.37-48-'j fll735-2846 - 'j ^12 - Ol2 - 'j fl56-Ol2.', 013-24 - &12 ', Ol423-56-Ol2-' 256,263 L(ai2345678)644 (012345678)644^8, 8 'j 01324-', 01625.37-48-', Oi736.2846 - ', Oi2 . 6l2 - ', O56.O12.', O13-24 - ^12 • ', O1423.66 - O12 . ' 264,267 C(Oi2345678)644(Ol2345678)644j8, 8 ', Oi626-37-48 - ', O12.612-', 017352846-' 268,271 C(Oi2345678)644(Ol2345678)644j8. 8 ', O12.56-', O12 - O12 - ', O56 . 612 • ' 272,273 l(Oi2345678)644 (012345678)644^8, 8 ', Oi2.66-' 274,277 L(Oi2345678)l28(Ol2345678)l28_l4, 4 ', O12.66-', Oi3.24.57.68 . ', 01423.5867 - ' 278,281 L(^12345678)l28(Ol2345678)l28j4, 4 ', O12.66-', 013-24-67-68 • ', Oi423-5867 - ' 282,285 L(^12345678)l28(Ol2345678)l28j4, 4 ', O12.56.', Oi3.24.57.68 . ', 01423.5867 . ' 286,289 L(Oi2345678)l28(^12345678)l28j4, 4 ', O12.66.', 013-24.57-68 - ', O1423.5867 - ' 290,293 L(Oi2345678)l28(Ol2345678)l28j4, 4 ', O12.66-', Oi3.24.57.68.', 01423.6867 -' 294,297 [_(Oi2346678)l28(&12346678)l28j4, 4 ', O12.66-', 013-24-57.68.', O1423.5867.' 298,301 L(Oi2345678)l28(Ol2345678)l28j4, 4 ', O12.66.', 013.24-57-68- 'j O1423.5867 • ' 302,305 L(Oi2345678)l28(&12346678)l28_]4. 4 ', Oi2.66-', 013-24-57-68-', O1423.5867.' 306 |_(*12346678)l28(Ol2345678)l28j4, 4 ' 20 Geace Marie Bareis Order No. 307 L(ai2345678)l28(&12345678)l28j4, 4 t 308 L('^12345678)l28(Ol2345678)l28j4, 4 t 309 C(«12345678)l28(Ol2345678)l28i, 4 t 310 L(^12345678)l28(Ol2345678)l28j4, 4 ^ 311 L(^12345678)l28(wi2345678)l28j4, 4 t 896 1,7 L (^12345678) 56 (^12345678) 56 J8, 8 ^) fll234567 ^, tti357246-'^, «1473625'^^ «1526374 • ^, CH642753-^j fll 765432 . t 768 1 l(^12345678)48,b(012345678)48i6I]8, 8 t 2 l(*12345678)48i7(&12345678)48i7J8, 8 t 3,6 L(ai2345678)96io(oi2345678)96ioJ4, 4 ^, C[l5-26-37-48 • ^j fll2-56- 012.56 '^5 Cfi6.25-37-48-Ol2-56-i^ 7,8 C(ai2345678)l923(t>12345678)l923j2, 2 ^, 013-24 • &13-24- ^ 9,10 L(^12345678)l924(&12345678)l924j2, 2 ^, 012- &12-^ 11,12 L(^12345678)l925(Ol2345678)l925j2, 2 ^, ^12 • Ol2 • ^ 13,14 L(*12345678)384(Ol2345678)384jl, 1 ^, CH2-34.56-78' ^ 672 1 L(^12345678)336(Ol2345678)336Jl, 1 ^ 576 1,2 C(Ol2345678)249(&12345678)249l2, 12 ^, CH5-26-37-48- ^ 3, 4 L(ai2345678)24u(Ol2345678)24iiJl2, 12 t, ai6-25-37-48- ^ 512 1 L(^12345678)l627(^12345678)l627J t 2 C (^12345678) I625 (^12345678) 162bI] ^ 3 C(ttl2345678)l628(Ol2345678)l628J t 4 C(^12345678)l626(Ol2345678)l626J ^ 5 L(*12345678)l63o(&12345678)l630_] t 6 L(ai2345678) 1629(012345678)1629 J t 7,10 L(^12345678)32io(Ol2345678)32ioJ8, 8 ^j Ctl324.^j fll5-26-37-48 • "^j ^^17362845 • t 11 L(ai2345678)32io(oi2346678)32ioj8, 8^ 12 L(^12345678)32io(Ol2345678)32ioJ8, 8 t 13 Qtt]2345678)32io(^12345678)32joj8, 8 ^ 14, 17 L(^12345678)32io(&12345678)32]o]]8, 8 ^j 01324-^, «12.56 . O12.66 • t, ^34 -56 • 012-56 • t 18,21 [_(ttl2345678)32io(&12345678)32ioJ8, 8 ^, ^1324 ^, fll2.56-Ol2-56-^} a34-56. 012.56 -i Imprimitive Substitution Groups of Degree Sixteen 21 Order No. 22,25 I_(ai2346678)32n(^12345678)32,ij8, 8 ^ al3•24•^ ai5-26-37-48.^ Ol736-2816.^ 26 L (^12345678)32 i,(Ol2345678)32,,j8. 8 i 27 I_(ai2345678)32,,(Ol2345678)32„j8. 8 f 28 L(ai2345678)32ii(^12345678)32nj8, 8 t 29,32 [_(ai2346678)32ji(&12345678)32iiJ8. 8 ^, 012.34. ^ 013-24. 'j 014.23'' 33,36 L(Ol2345678)32ii(^2346678)32ijJ8, 8 ^ Ol2-34.', 013.24 .^ 012.66- 012.56-' 37 L(^12345678)32,i(&12345678)32j,j8. 8 ' 38,41 L(Oi2345678)32ij(tl2345678)32,j_]8. 8 '> 016-26 • ^6'26 • '> Oi3.24.57.68.') 01753.2846.' 42,47 [_(Oi234567s)32j,(&12345678)32„j8. 8 'j O12 . 612 . ', Ol2.66.'> 066.fel2-'> Oi5.26.37-48' '> 01626-37-48 • ^12 • ' 48,51 |_(Oj2345678)32, 2(012345678)32, jj8. 8 '> O12.56.', Ol5.?C.37.48.'. Oi6.26.37.48.' 52,55 L(^12346678)32, 2(612346678)32, 2J8. 8 '> O12.34.', 035.46 . ^35.46 . '> 012.3645 . O36.46 - ' 56,59 L(^12345678)32,2(6l2346678)32ijj8, 8 ', O13.24-67-68 • '» O12-66-', 01324-6768.' 60 L(Oi2346678)32,2(6l2346678)32„j8, 8 ' 61 L(Oi2345678)32, 2(612345678)32, 2J8, 8 ' 62 [_(ai2345678)32„(6l2345678)32i2j8. 8 ' 63,67 L(^12345678)32,,(6l2345678)32i,j8. 8 'j O12.34-', O34.66.', 012-36-46- 012-36-46 •'» O3646- 012-36-46- ' 68 |_(Oi234667r)32,2(o12346678)32,2J8, 8 ' 69, 73 L(^12345678)32,2(6l2346678)32,2j8. 8 '> Oi2.6«.', 016-26-87-48 •'» O12.612.', Oi626-37-48- 6l2. ' ' 74 1_(^12345678)32, 2(012345678)32, ,J8, 8 ' 75,79 L('^12345678)32„(6l2346678)32„J8, 8 ', Oi2.34-', O12.66.', O13.24 - 613.24 - ', 01423-56 - 013-24 • ' 80 L(Ol2346678)32, 2(012345673)32, 2J8. 8 ' 81,83 L(Ol2348678)32,2(Ol2345678)32i,j8, 8 '» 012-34-', 012-36-46- 012.36-46-' 22 Grace Marie Bareis Order No. 84,87 88, 91 92 93,96 97, 101 102, 104 105 106, 108 109, 113 114, 118 119, 123 124 125, 128 129, 132 133 134 135 136, 139 140, 143 144 145 L(^12345678)32i3 (Ol2345678 ttl3-24.0l3.24.^, ^57. 6 L(fll2345678)32i3(Ol2345678 «1735-2846 . ^, «12 • &12 L (^12345678)32 13(012345678 L(<*12345678)32i3 (012345678 ai2 . O12 • t, fll324. 57.68 L(^12345678)32 14(012345678 fll625-37-48 • ^j ^12 • Ol2 L(^12345678)32i4(Ol2345678 «1625-37.48.^ L(^12345678)32i4 (Ol2346678 L (^12345678)32 14(012345678 fll625-37.48-^ C(ai2345678)32is(& Ql3-24.0i3.24.^, ai3-24 L (^12345678)32 fll7362845.^ [_ (^12345678) 32 fll7362845-^ L (^12345678) 32 L (^12345678)32 fll3-24-Ol3.: L (^12345678)32 Ol2.5768-^j L(ctl2345678)32 L (^12345678)32 L (^12345678)32 L (^12345678) 32 fll3-24-Oi3. L(^12345678)32 Q15-26-37-48 L (0^12345673)32 L (^12345678)32 '12345678 (Ol2345678 fll2 • O12 . (0l2345678 fll2 • O12 . (O] 23456 7 8 7(012345678 ;4 . t, aU'2, 7(012345678 fll736-2845' 7(012345678 7 (&12345678 (0l2345678 7(012345678 •t^.t, ai537 e(Oi2345678 t, Cti7352846 6(Ol2345678 6(Ol2345678 '3213J8, %t, Cfl735.2846 • ^, 13-24 -^ gjs, 8 t, ai2-56-^> '32 ,J8, 8 t, C^l 735.2846 • ^> , a56 . O12 • ^ 14 Js, 8 t, 0'lZ-2A-b7-6S-t> '32 '3214J8, 8 t uJ8, 8 t, fll3.24.57.68.^> '3215J8. 8 ^> ^57-68 .013.24.^^ •57.68 .^j fll7362845.^ '32i6j8, 8 t, «12.56.^, ', a56.0i2./ '3215^8, 8 t, (Il2-h6-t, ', O56 -hii.t '32i6l|8, 8 t '3217J8, 8 t, 12-5768 -t, . &13.24 • t '32i7j8,8^j ^15.26.37.48.^> t '32i7j8, 8 t '3217J8, 8 t '3217J8, 8 t '3217J8, 8 t, Cl\2-b76S-t> •2648.013.24.^ 16 Js. 8 ^' fll3.24.56.^> .t 32i6]8, 8 t '32i6l|8, 8 t Imprimitive Substitution Groups of Degree Sixteen 23 Order No. 146 L(^12346678)32,j(^2345678)32„j8, 8 t 147,150 Qai2345678)32,4(6l2345678)32„j8. 8 ^ 012.34. ^ al6•26•37•48•^ 01625.3847.^ 151 [_(ai2346678)32ig(ftl2345678)32igj8, 8^ 152 [_(ai2348678)32„(Ol2345678)32ijj8, 8 t 153 |_(Ol2346678)32,e(Ol2345678)32,eJ8, 8^ 154,157 L(^12345678)32jg(&12345678)32i,j8. 8 ^, fll2.34.^, «13.24.56 . t, 01324 . ^1324 • t 158 L(ai2346678)32,g(&12345678)32jgj8. 8 t 159, 162 Qai2346678)32ij(&12345678)32,8j8. 8^ 012-34. <, Oi3,24.56.^ Oi324 . 0J324 . ^ 163 |_(Oi2345678)32,8(t'l2345678)32igj8. 8^ 164,167 Qoi2345678)32ij(6l2346678)32,jj8, 8 ^ Oi2.34.^ Ol3.24.56.^ Oi4,23.56.^ 168 I_(^12345678)32,j(&12346678)32„_|8. 8^ 169 [_(Oi2346678)32ij(t'l2345678)32igj8. 8^ 170 L(^12345678)32„(&12346678)32,gj8. 8 t 171, 178 l(Oi2345678)64,(6i2345678)64jJ4. 4 '> Oi2.66.^, Ol3.24.67.68.^ Oi423.6367.^ Oi2 . O12 . <, O66.0i2.<, Oi324.67.68 . &12 • t, Oi4.23.6867 -Ou-t 179,182 |_(Oi2346678)64,(&12346678)64,_l4, 4 ^> 012.56. ^> O13.24. 57.68. ^ 01423.5867.^ 183 C (^12345678)64, (&12345678)64,j4. 4 t 184,191 L(^12345678)64,(&12345678)64,_]4, 4 ^ Oi2.66.^, 013.24.57.68.^, Oi423.5867.^> Oi2.0i2.<, 056.612. <, Oi324.57.68 . 612 . tf Oi4.23.5867 'O^.t 192 L(Oi2345678)64,(6l2345678)64,_l4. 4 t 193,196 [_(Oi2345678)64,(6l2345678)64,_l4, 4 ^ Ol2.34.^, Ol5.26.37.48.^ Oi625.3847.^ 197,200 L (^12345678)64, (6l2345678)64,_l4. 4 ^ Oi2.34.^ Ol6.26.37.48 • '> Oie26.3847 . t 201,204 [(ai2345678)64,(6l2346678)e4,j4, 4 ^ 012-34.^ 016.26.37.48.^ Oi625.3847 . ^ 205,208 C(Oi2346678)64,(6l2345678)64,j4, 4 ^ Ol2.34.<, 016.26.37.48.^ Oie26.8847.^ 24 Grace Marie Bareis Order No. 209,212 L(^12345678)643(Ol2345678)64sJ4, 4 ^^ dlZ-U-t, fl^l5-26-37-48-i^j CH625-3847.^ 213,216 L(^12345678)643 (0X2345678)643^4, 4 ^, fll2-34-^, «15-26-37.48-^j CH625-3847 • ^ 217 L(^12345678)643(&12345678)643J4, 4 t 218 C(ai2345678)643(&12345678)643_]4, 4 t 219 L(^12345678)643(Ol2345678)643j4, 4 ^ 220,227 L(^12345678)645(Ol2345678)646j4, 4 ^j fll3-24.57-68' ^j fll2.56'^j CH324-5768-^) ^12 • Ol2 • ^, CH324.57'68- Ol2 • ^, 056 • Oi2 . /, 013.24-5768 -On-t 228,235 L(^12345678)645(Oi2345678)645j4, 4 ^, ai3.24.57-68 • i^j ai2.56.^, ai324-5768 • ^, dVi-hn-t, 01324-57.68 • ^12 • ^j O56.0l2.i, 013-24-5768-012-^ 236 L(^12345678)646(&12345678)645J4, 4 ^ 237,244 L (^12345678) 646 (0l2345678)646j4, 4 ^j 012-56- ^j Oi2.i, 056-^, Oi3.24.t>i3.24.^, O1423-56 - Ol3-24 • ^, 01423.613.24.^, 013-24-56- 013.24.^ 245,248 L (^12345678)646 (012345678)646^4, 4 ^j Oi2-56-^, O12 . ^, 055.^ 249,252 L(^12345678)646(6l2345678)646j4, 4 ^;. Oi2.56.^, O12 . ^, a^.t 253,254 L('^12345678)646(Oi2345678)646j4, 4 ^, 013.24.013.24.^ 255 L(^12345678)646(Ol2345678)646j4, 4 t 256,263 L(^12345678)647(0l2345678)647j4, 4 ^j Oi2.56.^, O13.24. 57.68 . t, 01423.5867 . t, O13.24 • O13.24 . t, 01423-56 . O13-24 - ^, 057.68.013.24.^, O12. 5867. 013.24.^ 264,271 l^(Oi23456.78)647(6l2345678)647]]4, 4 ^, Oi2.56.^, 013.24.57.68.^, 01423-5867-^, 013.24.013.24.^ Ol423.56.0i3.24-^, 057.68-013.24-^, Oi2-5867 • O13.24. if 272 L(^12345678)647(Ol2345678)647j4, 4 t 273,280 L(^12345678)642(Ol2345678)642j4, 4 i^, 01423.5867.^, Ol2.56.^, 014.23.58.67.^, O12 . O12 . i, 014.23-5867.012.^, O56 . 612 . t, 01423.58. 67 .^12.^ 281,284 [_(Oi2345678)642(6l2345678)642J4, 4 ^, Oi423.5867 . ^, Ol2.56.^, Oi4.23.58.67.^ Imprimitive Substitution Groups of Degree Sixteen 25 Order No. 285, 292 |_(ai2»46678)64,(tl2345678)64,3l. 4 t Ol2-56.<, fll4-23'68.67 • ^ ^12 • 6 dU-bn't) fll423-68'67 • Ol2 . ^ 293 l(<^12345678)642(012345678)64,J4, 4 294 [_ (^12345678)64, (6l2346678)64,j4, 4 295, 298 [_(^12345678)64,(fel2345678)64,_|4. 4 fll6-25-37-48-^ fll526-3847-^ 299, 302 |_(ai2346678)642(6l2345678)642j4, 4 ai6-25'37-48'^ 01626-3847 • ^ 303, 306 [_(ai2346678)642(6l2345678)64 J4, 4 Ol6-26-37-48'^, Ctl526-3847 • ^ 307, 310 [_(ai2345678)64,(Ol2345678)642j4, 4 fll6-26'37-48 • t, fll526-3847 • t 311, 314 |_(Ol2346678)642(Ol2345678)642j4. 4 Ol6-25-37-48.^, 01526-3847 •' 315, 318 [_(^12345678)642(Ol2345678)64jj4, 4 ai6.26«37.48 • t, 01626-3847 • t 319 L(^12345678)64j(Ol234S678)64jj4. 4 320 L(^12346678)642(Ol2345678)642j4. 4 321 L (012345678)64 ,(^12345678)64 2 J4, 4 322, 329 [_(ai2345678)644(Ol2345678)644j4, 4 Ol3-24-57-68-^ O1423.5867 • ^, O 01324-67-68 • t, 014.23-5867 -612- 330, 337 L(^12348678)644(Ol2346678)644j4. 4 Ol3-24-67-68-^ 01423-6867-^ O Ol3»l-57-68-&12-^ 014.23-5867- 338 l(^12346678)644(&12346678)644J4, 4 339, 340 L(ai2345678)l28(6l2346678)l28j2. 2 341, 342 L('^12346678)l28(Ol2345678)l28j2, 2 343, 344 L(ai2345678)l28(Ol2346678)l28j2. 2 345, 346 |_(^12346678)l28(Ol2346678)l28j2, 2 347,348 L(Oi2346678)l28(Ol2346678)l28j2. 2 349, 350 L(Oi2346678)l28(tl2345678)l28j2. 2 351, 352 |_(Oi2345678)l28(Ol2346678)l28j2, 2 01423-6867 '^j .t, Oi4.23.6867 Ol2.34.^, 012-34-^, 0X2.84 .^ Ol2-34-^, Oi2.34.<, 012.34. ^> .6i2.<, ttl2.66.^> !.6l2.^ 056.6l2.<, Ol2.56.^, I'bn.t, a^^.bn.t, n-t , 012-34-^ , 012.34-^ , 012.34-^ , 012.34.^ I 012.34.^ , Oi2.34.< , 012.34.^ 26 Grace Marie Bareis / Order No. 353,354 C(ai2345678)l28(&12345678)l28_l2, 2 ^ dU-Zi-t 384 1 L(^12345678)l922(Ol2345678)l922Jl, 1 ^ 2.5 L(^12345678)l923(Ol2345678)l923Jl, 1 ^, «12.34-56.78 • ^j fll3-24 • 013-24 . t, fll4-23.56-78 • &13-24 • t 6.9 L (^12345678) 1924(012345678) 1924 Jl, 1 ^, ai2-34-56-78- ^j an-bi2.t, «34-56-78-&12-^ 10,13 C(Ol2345678)l924(Ol2345678)l924Jl, 1 ^, «12-34-56.78- ^j ai2.0l2-i^j ^34.56.78.012.^ 14, 17 L(ai2345678)l92B(&12345678)l925Jl, 1 ^, ai2.34-56.78- ^, ai2 . 612 • t, a34.56- 78 • &12 - ^ 18.21 L(ai2345678)l926(&12345678)l926Jl, 1 ^j 012-34-56-78 • ^, ai2 . 612 .t, ^34 -56 - 78 • O12 - ^ 22.25 l(«12345678)96i2(^12345678)96i2J2, 2 ^, «12 - O12 • ^, ttl3.24.0]3.24.^, ^1423 . O1423 . ^ 26,27 L (ai2345678)48i6 (^12345678)48 16 J4, 4 ^, C[l5.26.37-48 - ^ 28,29 L(ai2345678)48ie(^12345678)48i6j4, 4 ^J ^15-26-37-48 • ^ 30,32 L(ai2345678)24io(oi2345678)24ioj8, 8 ^> 0135-246-^, 12 L(ai2345678)l626 (^12345678)1626^8, 8 ^> Ol625-3847-^ 13, 14 |_(^12345678)l625(&12345678)l625_]8, 8 ^> ai3-24.57-68. ^ 15,16 Qctl2345678)l625(^12345678)l625j8, 8 ^. 012.56-^ 17, 18 L(^12345678)l625(Ol2345678)l626j8, 8 ^, 012-56-^ 19,20 Qai2345678)l626(^12345678)l62sl8, 8 ^, 012-56-^ 21.22 L(ttl2345678)l628(^12345678)l628j8, 8 ^) 01735-2846-^ 23,24 L(^12345678)l628(^12345678)l628J8, 8 ^5 Ol2-34-^ 25.26 |_(^12345678)l626 (^12345678) I626J8. 8 ^, Ol7362845-^ Imprimitive Substitution Groups of Degree Sixteen 27 Order No. 27,28 L(^12346678)i6„(^12345678)i6„J8, 8 ^ 012-34.^ 29,30 [_(^12345678)l6,o(^12345678)l6,«j8. 8 ^ ai6-26.37-48' ^ 31,32 [_(ai2345678)l6,o(^12346678)l6,oJ8. 8 ^ fll8-24-66«^ 33,34 l(^12345678)i6,o(^12345678)i6joJ8, 8 ^ ai6-26-37-48- ^ 35,36 L(^12346678)l62,(&12345678)l6jgj8, 8 ^ ai625-3847-^ 37,38 L(ai2345678)l6jg(6l2345678)l6„j8, 8 ^> ai3-24.66-^ 39, 42 |_(ai2346678)32,o(^12345678)32ioj4, 4 t, ttiz-u-t, fll2.66'C>12-66-^j 0[34'56-O]2-56'^ 43,44 L (012345678)32 10 (^12346678)32, oJj. 4 ^ dn-U-t 45,46 C(^12346678)32io(^12345678)32ioj4, 4 ^> «1324.5768.< 47, 52 (_(ai2346678)32io(^12315678)32io!]4. 4 fll5-26-37-48' ^j fll7362345«^ fll2-56- &12-56- ^> 034-56 • Ol2'56« '> ai6-25-37-4S • 6l2'66 • t, 01526-3847 • &12.B6 • t 53,56 L(al2^45678)32lo(^12345678)32,oJ4. 4 ^ 012-34-^, Ol2-56-Ol2-56-^ 034-56- O12.56- ^ 57,62 [_(ai2345678)32ji(&12345€78)32n_|4. 4 ^ 012-34-^, ai5-2'i-37-48-^> Ol625-3847.^> ai2-66. &I2-66. ^, 016-26-37-48 -Ol2-56-^ 63, 65 [_(Oi2J45678)32ii(&12345678)32,iJ4, 4 012-34-^ Oi2.56-&12-66-^ 01846-2735 -012-56-^ 66,67 L (012345678)32 „(Ol2346678)32,,j4, 4 ^ O12.34 - ^ 68,70 |_(^12345678)32,i(6l2345678)32,,j4, 4 ^ 012.34-^ 012-56-^12-56-^ 71,72 [_(Oi2346678)32i,(&12346678)32.ij4, 4 <, Ol3-24-67-68. < 73,74 L(Oi2346678)32„(&12346678)32,,j4, 4 ^> 012-34-^ 75,85 L(Oi2345678)32i,(&12346678)32i,_l4. 4 ^ Oi3.24-67-68- ^> Ol2-56-^ 01324.5768- ^ 015.26-37-48- ^> O12 . O12 . ^, 01428-57-68 • &12 - ^ O13.24 - fel3-24 - 1) 01423-86 • O13.24 • t, O1423 - ^1423 - ^ 034.57.68-t>1423-^ 86, 87 [_(^12345678)32i,(&12345678)32;,j4, 4 tf 015.26-37-48-^ 88,89 L(^12345678)32i,(tl2346678)32i,j4- ■* ^ O18.24.57-68- < 90,92 [_(Oi2346678)32„(6l2346678)32,jj4. 4 ^ Oi324-5768-^ Ol5-26-37-48-^ 93,94 C(ai2345678)32„(6l2845678)32i,j4. 4 ', ai6-26-37.48« ^ 28 Grace Marie Bareis Order No. 95,96 97,98 99, 106 107, 110 111, 112 113, 114 115, 118 119, 120 121, 122 123, 124 125, 126 127, 128 129, 130 131, 138 139, 142 143, 144 145, 148 149, 150 151, 152 153, 154 155, 156 157, 158 159, 160 161, 162 163, 169 L(cH2345678)32i2(Ol2345678)32i2j4, 4 t, an.m-t L(^12345678)32i2(012345678)32i2J4, 4^, «1324.5768-^ L(^12345678)32^2(^12345678)32i2j4, 4 t, dll-U-U 0'\1-h&-t, ai5-26-37.48-^j 0^1625-3847 • ^, C[i6.25-37-48- ^j fll3.24 • 013.24 • t, fll537-2648 • Ol3.24 • ^ L(^12345678)32i2(Ol2345678)32i2j4, 4 t, «12-56-^j «13-24 • &13-24 • t, ai637-2548 • Ol3.24 • t L(^12345678)32j2(^12345678)32i2J4, 4 i, dU-Si-t C(^12345678)32, 2(012345678)32 12 J4, 4^, fll2-34-^ L(ctl2345678)32]2(Ol2345678)32i2j4, 4 t, ttu-Zi't, (lU-2i-Oi3.24.t, ai537.2648-^ L(ctl2345678)32i2 (^12345673)32 12 J4, 4 t, ai5.26-37-48- ^ l(^12345678)32]2(012345678)32i2J4, 4 t, ai625-3847-^ L(<^12345678)32i2(Ol2345678)32i2j4, 4 ^j ai2.56'^ L(ctl2345678)32 12 (012345673)32 12 J4, 4 t, Cl\2-5e-t L(<^12345678)32 12 (^12345673)32 12 J4, 4 t, a34-56 • ^ L(^12345678)32i2(^12345678)32i2j4, 4 t, a34-56-^ L(^12345678)32i2 (012345673)32, 2 J4, 4 t, Clu-Zi-t, fl]2.56-^j Ol5-26.37-43-^> ttl6-25.37-48' ^j «1526-3847 • ^> ai423- 01423-^, CH6432537 • 01423 • ^ L(^12345678)32 12(012345678)3212 J4, 4 t, ttvz .56- ^j «1423 • ^1423 • t, 015432637 • Ol423 • t L(Ol2345678)32i2(oi2345673)32i2j4, 4 t, Cll2-U'i C(Ol2345678)32, 2(012345673)32 12_|4, 4 t, Cll2-U-t, fll423 • 01423 • ^j fll324 • Oi423 • ^ L(012345678)32i2 (^12345673)32 12 J4, 4 i, (ll2-U-t L(ctl2345678)32i2 (^12345673)32 12 J4, 4 t, ai5-26-37-48- ^ l(012345673)32,2(^12345678)32i2J4, 4 t, ai6-25-37'48 • ^ L(ctl2345678)32i2(Ol2345678)32i2j4, 4 t, ttii.^Q.t L(012345678)32i2(^12345678)32i2J4, 4 ^, 034.56.^ L(Ol2345678)32i2(^12345678)32i2j4, 4 t, dn-bG-t L(ctl2345678)32i2(Ol2345678)32i2j4, 4 t, Cll2-b6-t L('^12345678)32i3 (012345673)32 13 J4, 4 Ol735-2846 • ^> dU-b^-t, a^^.hn.t, 057-68 • ^13-24 -^j 012.5367 •Ol3.24-^> 034-57-68 • 01423 • t, O1423.56 • O13.24 • t Imprimitive Substitution Groups of Degree Sixteen 29 Order No. 170 L(^12345678)32,j(6i2345678)32,jJ4, 4 ai736.2846-^ 171,172 |_(ai2345678)32„(&12345678)32,jj4. 4 ^ (ll2-34-t 173,174 [_(ai2345678)32i,(&12345678)32„j4, 4 ^ dn-U-t 175,182 [_(ai2345678)32,4(&12345678)32,4j4, 4 Ql2-56.^ ai324.6768« ^» Clb6-bl2.t, al3.24.5768•^12•^ 01423-56 • &13'24 • ^ 034.6768«&13'24.^, fll3-24-56- Ol423' ^> ^6768 • ^1423 • ^ 183,185 [_(ctl2345678)32,4(&12346678)32nJ4. 4 ^ dn-bi-ty fll626-37'48 • t 186 L(^12345678)32,4(6l2345678)32i4j4, 4 fll2.34.< 187, 188 L(ai2345678)32,5(&12345678)32i5_]4, 4 ttl73e2846'^ fll2-34.5867- 01423-^ 189,191 L(ai2345678)32 15(612345678)32 15 J4. 4 ^ fll423-6867 • ^> ai7362845>^ 192,195 C(ai2345678)32i5(&12345678)32,5j4. 4 ^ ai2-34.<, 01423 • ^1423 • t, 01324 • ^1423 • t 196,199 ^(012345678)32,5(612345678)32 ,5 J4. 4 ', Oi2.34.<, 0X423 • 6l423 • t> O1324 . O1423 . t 200, 206 [_(Oi2345678)32 ,7(612345678)32 17 J4, 4 Oi2.5768-^> Ol5.26-37-48-^ Oi648-2537 • ^ O13.24 • 013-24 • ^> 014.23 • 613.24 • ^ 01637-2648- 613.24.^, Oi647'2638 • 6x3.24 • ^ 207,210 C(^12345678)32i7(6l2345678)32,7j4, 4 ^ Oi2.5768-^, 013.24 • 6l3-24 • tf 01648-2537 • t 211,214 [_('^12345678)32,7(6l2345678)32i7_l4, 4 ^ Oi2.34.<, 013-24 - 613.24 • t, O14.23 . 613.24 • t 215,218 n(^12345678)32,7(6l2345678)32,734. 4 <, Ox3.24.67-«8 • <> 013-24 - 613-24 - ^ 057.68-613.24.^ 219, 221 [(012345678)32 ,7 (6l2345678)32i7l4. 4 0X3.24-67.68. ^ 0X3.24- 613-24 -^ 057.68.613-24.^ 222,225 [(Oi2345678)32 ,7(612345678)32, 7 J4. 4 ^ O13.24.67.68. <» 013-24 - 613.24 - ^ 057.68-613.24-^ 226,233 [_(^l2345678)32,j(6l2345678)32j,_i4, 4 ^ 012-34 - ^> 013.24-56 -^> 015.26-37-48 -^ 0]625-3847-^ Oj 7362846. ^» Ol324 - 61324 • ^ 015372648- 61324.^ 234, 236 C(Oi2345678)32,j(6l2345678)32,gj4, 4 O13.24.66.'> Ol324 . 6x324 . t, Oi 728.3646 • 6l324 . t 30 Grace Marie Bareis Order 192 128 No. 237, 238 239, 240 241, 244 245 246 247, 248 249, 250 251, 252 253, 254 255, 256 257, 260 261, 264 1,2 3 4,5 6, 13 14 15 16, 17 1 2 3 4 5 6,9 10 11 C (^12345678)32 6(Ol2345678)32i6j4, 4 t, Clu-3A-t L(^12345678)32i6(Ol2345678)32igJ4, 4 t, dU-U-t [_('^12345678)32i6(&12345678)32i6j4, 4 t, «12-34-^> ttl324 • Ol324 • ^, dim-Ouii.t L(Cfl2345678)32j6(&12345678)32i6j4, 4 ai5-26.37-48- ^ C(fll2345678)32i8(Ol2345678)32i6j4, 4 ai625-3847-^ L (^12345678)32 16 (^12345678)32 15 J4, 4 t, CH3-24-56-^ C(ai2345678)32i6(&12345678)32i5J4, 4 t, ^13.24. 56-^ L(^12345678)32ie(&12345678)32i6_l4, 4 t, «13-24-56-^ L (^12345678)32 16(^12345678)32 15 J4, 4 t, ai4.23-56-^ L(«12345678)32 16 (^12345678)32 16 J4, it, «1324-5867-^ L(^12345678)32i6(^12345678)32i6!]4, 4^, CH324.5867 • ^j fll324 • Oi324 . t, ^5768 • Ol324 • t L(^12345678)32i6(^12345678)32i6J4, 4 t, CH324-5867 • ^j fll324 • &1324 • t, Ct5768 • Ol324 • t L(^12345678)96io(^12345678)96ioJ1. 1 ^j ai2-56- 012-56- ^ L (^12345678) 96] (^12345678) 96ioJl, I t C(«]2345678)96ii (^12345678) 96iiJl, it) «12-56- Ol2.56' ^ L(^12345678) 96)2 (^12345678) 9612J1, ^ ^' «12-34-56-78. '> 0'12-bn.t, a34-56-78'Ol2.^, «13-24 • &13-24 • ^j ai4-23-56-78-&13-24-^, 01423 • &1423 • ^j ai324-56-78- Ol423' ^ L(O12345678)48i6(0i2345678)48i6J2, 2 t L(012345678)48i7(^12345678)48i7J2, 2 t E(<^12345678)249(fel2345678)249_l4, 4 ai5.26-37-48- ^j fll6-25-37-48- 012-56 -^ L(<*12345678) 826 (^12345678) 825 J t l(012345678) 826 (^12345678) 826 J t Qai2345678) 82, (1^12345678) 822J t L(012345678)824(^12345678)824J t L(Ol2345678) 823(012345678) 823 J t |_(^12345678)l627(^12345678)l627j4, 4 t, 012.56 '^j 0X324.5768 • t, Ol3-24-57-68 • t L(Ol2345678)l627(&12345678)l627j4, 4 t L(O]2345678)i627(0i2345678)i627Ji, 4 t Imprimitive Substitution Groups of Degree Sixteen 31 Order No. 12 13, 16 17 18 19 20,23 24 25,28 29 30,33 34 35 36 37,40 41 42,45 46 47 48 49,52 53 54,57 58 59 60 61,64 L(^U346678)l6,7 (Ol2345678)l6,7_l4 , L (Cfli345678) 16,7 (^123466 ai5-26-37-48-^ ^16 L(Ol2345678)l6,7(^] L (^12345678) 16,7 (&J L\'-1ZM00/B/10,7V'' \_ (ai2345678) 16,7 (^123456 L (^12346678) 16,7 (^12345678 fll5-26-37-48-^ «16-26- L(^12345678)l6,7 (6l2346678)l6,7j4 , L (Ctl2345678) 16,7 (t'l2345678) 16,7 J4 , fll2-56-^ fll7-28-36-46-^ '123456 *123456 '8)l6, •25-37-48 Ol6,7 j)l6_ Ol6,7_l4. l)l6,7j4. 615.26. L (^12345678) 16,7 (&12345678) L (CH2346678) 16,7 (Ol2346678) 1 fll5.26.37.48.^ 016-25.37 l(012345678) 16,7(0] L(<^12345678)l627(&12345678)l6, L(^12345678)l627 (Ol2345678) L (<'tl2345678) 16,5 (Ol234S fll625 • 01625 . ty ai3.24.57.68 • t L (^12345678) 16, j (Ol2345678) 16, L (^12345678) 16,5 (Ol2345678) 16, j ^13.24.57.68.^ 01423-5867 • L(Ol2345678)l625(6 L (^12345678) '12345678) 16 ^ 16,j(6l2345678)l6,5j4, L(Ol2345678)l6,j(oi2345678)l6,jj4, L(ai2345678)l6,s(6l2345678) . „. )l6,5]4. ai7.28.36.46 • t, ^1626 • 6l625 . t L(O12345678)i6,j(0i2345678)i6,jJ4, \_ (012346678) 16, j (Ol2345678) 16, , J4 . 01626.3847. ^ 0]626.; ,J. 01626.3847 . t> 0]626.3847.i L(Ol2346678)l6,s(Ol2345678)l6,5J4 L(012346678)i6,j(6i2345678)i6^ L(Ol2345678)l6,4(Ol2346678)l6,jj4, L(Ol2345678)l6„(frl2345678)l6,j_J4, aifl26'3S47.^ 01626.3847-^ „1. l8-.< 'I2345678)l6,7j4. „J. ,1. ,1. ,1. ,1. ,1. , 013.24-67-68 -^ 01324-6768 • t , ai6.26«&16.26.<, , Oi2.66.<, ', ai3.w.t, , 012.66 .^j , Oi2.66.<, , ai3.24.67.«g.^, , 0]2.5fi.<, , Oi2.66.^) 32 Grace Marie Bareis Order No. 65 L(^12345678)l625(^12345678)l625J4, 4 t 66 L(^12345678)l626(^12345678)l625l]4, 4 ^ 67 L(^12345678)l6j5(&12345678)l625j4, 4 t 68,69 L(^12345678)l628(^12345678)l628ll4. 4 01735-2846.^, «57-68-&13-24-^ 70,73 L(^12345678)l628(Ol2345678)l628j4, 4 ^ ai2-34-^, Ctl2-56. Ol2-56-^j CH735-2846.^ 74 L(Ol2345678)l628 (^12345678) I628J4, 4 t 75,78 L(^12345678)l628(^12345678)l628j4, 4 ^, a]2.3-l-^, ttl3.24-&13.24.^, Ol4.23-Ol3-24.^ 79,82 L(«12345678)i626(^12345678)i626J4, 4 ^, ai2-34.^, ft] 324 • Oi324 . ^, a\ 7302 845 t 83 L(Ol2345678)l626(^12345678)l626_l4. 4 ^ 84, 85 L('^12345678)l626 (^12345678) I626J4, 4 01324 • &1324 • t, Ol7362845-^ 86,89 L(^12345678)l626(^12345678)l626j4, 4 ^, Ol2-56- &12-56- ^, 034.56 • 012-56 .^j 017362845-1^ 90,93 L(Ol2345678)l63o(Ol2345678)l63oJ4, 4 i^, 013.24-56-^, Ol5-26-37-48-^j Oi7352846-^ 94 L(Ol2345678)l63o(^12345678)l63oJ4, 4 t 95 L(Ol2345678)l63o(^12345678)l63oj4, 4 t 96 L(Ol2345678)l63o(oi2345678)l63oj4, 4 1^ 97, 100 L (^12345678) 1629(0l2345678)l629j4, 4 ^, Oi3.24.56-^j 01625-3847 - ^, Oi8452736-^ 101 L(Ol2345678)l629(^12345678)l629j4, 4 t 102 L(Ol2345678)l629(^12345678)l629j4, 4 ^ 103 L(Ol2345678)l629 (^12345678) I629J4, 4 ^ 104, 105 [_(Ol2345678)32io(Ol2345678)32ioj2, 2 ^j O12.56. 6l6-25-37-48- ^ 106,109 L(Ol2345678)32u(^12345678)32iij2, 2 ^, O13.24-57-68 • ^, 012.34-^, Oi4.23-57-68-^ 110 L(Ol2345678)32ii(&12345678)32nJ2, 2 t 111 L(Ol2345678)32ii(Ol2345678)32iiJ2, 2 ^ 112, 118 L(Ol2345678)32i2(Ol2345678)32i2j2, 2 ^, 013.24-57-68-^5 Ol324-5768-^> Oi2.34-^, dvi.hvi.t, O13.24- Ol3-24- ^j Ol423 - 01423 - t Imprimitive Substitution Groups of Degree Sixteen 33 Order No. 119,120 Qai2345678)32 „ (^12345678)32 „J2. 2 ^ 013-24 • fel3.24. < 121 L(ai2346678)32„(Ol2345678)32i2j2. 2 ^ 122 [.('^12345678)32 i,(Ol2345678)32i,32, 2 t 123,124 L (ai2345678)32, 2 (^2345678)32 12 JJ, 2 ^> fll423 • ^1324 • ^ 125 L(ai2346678)32„(i'l2345678)32i2j2, 2 t 126,128 L(ai2345678)32i2(&12345678)32„j2, 2 ^ CH35-246 • Ol63-264 • ^ fll53-264 • 0136-246 • ' 129 L(^12345678)32,j(Ol2345678)32i,J2, 2 t 130,135 L(^12346678)32,j(6i2345678)32,2J2, 2 ^j ai2-34-^ fll2-56-^ 013-24 • 6l3-24 . <, 057-68 - Ol3-24 - ^ 0X637.2648 •t'13-24-^ 136,138 L(ai2346678)32„(&12345678)32i,j2, 2 ^ Oi2.34-^, 013-24.67-68 • t 139, 142 [_(Oi2345678)32|2(Ol2345678)32i,J2. 2 ^ Oi2.34-^ Ol3-24-57-68.^ Oi4.23-57-68 - ^ 143,146 1^(^12345678)32 1,(^12345678)32 12 J2. 2 ^ 012-86. <, 01626.8847 • t, Oi526-3847 • t 147, 150 I_(Oi2345678)32, 2 (^12345678)32 12 J2, 2 ^ Oj2.56-^ Ol5-26-37-48-^ 016-25-37-48-^ 151,154 [_(ai2345678)32„(6l2345678)32ijj2, 2 '. O13.24-57-68- ^ 016-26.87.48.^ Oi7.28-35-46 • ' 155, 158 [_(Oi2345678)32 1.(^12345678)32 12 J2, 2 ^> 013-24-57-68- ^ Ol625-3847-^ Ol827-3645-^ 159 L(^12346678)32„(&12346678)32„j2. 2 ^ 160 L('^12345678)32i,(Ol2346678)32,2J2. 2 t 161 L(Ol2345678)32„(Ol2345678)32i2j2. 2 ^ 162 L(^12345678)32i2(^12346678)32i2j2, 2 ^ 163,164 |_(Oi2345678)32 ,2 (^12345678)32 ijJZ, 2 ^ 013-24-013.24-^ 165,167 [_("l2345678)32,2(^12345678)32i2_l2, 2 ^ 0135.246 • &163-264 •/, 01324 - Ol423 - 1 168,175 L(Oi2346678)32i2(Ol2346678)32i,j2. 2 ^> O12.34./, 016-26-37-48- ^ Oi626-3847 - ^> 0x2 . O12 - <, 034.612-^, Ol5-26- 0)6-26 -^ Oi626 - Oi625 - ^ 176 L (^12346678)32 „(&12346678)32|,j2, 2 i 34 Grace Marie Bareis Order No. 177,179 L(ai2345678)32i,(&12345678)32i2J2, 2 i^, ttli-Zi-t, fll5-26-37-48-^ 180 L(^12345678)32i2(^12346678)32i2j2, 2 t 181,185 L(ai2345678)32 12(^12345678)32121, 2 ^j Cll2-Zi't, ttl7-28-35.46'^, ttl827-3645 • ^j fll423 • Ol423 • ^ 186,190 |_(^12345678)32i2(^12345678)32i2J2, 2 ^j ai2-34-^, «15-26-37-48'^7 '^13-24 • O13.24 • ^, tti625-3847 • ^ 191,193 L(ai2345678)32i2(Ol2345678)32ijj2, 2 if> ai2-34.i^, Cfl6-25-37-48-^ 194 L(ai2345678)32 12(012346678)32121, 2 ^ 195, 196 L(0^12345678)32i3 (^12345678)3213!, 2 C[l3.24-57-68'^> tt57-68-Oj3.24.^ 197,200 L(^1234o678)32i3(&12345678)32i3J2, 2 ^, Ctl2-34-^, fll3-24-57-68-^j «14-23-57-68 • ^ 201,202 L(ai2345678)32i4(&12345678)32i4j2, 2 ai2-56-^j ^56 . O12 . < 203,204 |_ (012345678)3214(012345678)32 14 J2, 2 ttl2 -34 .^j 034.56.^ 205,206 Qoi2345678)32i5(^]2345678)32i5j2, 2 CH423-5867-^ fli2-34.5867- 01423 -^ 207,210 L(ai2345678)32 15 (^12345678)32)5 J2, 2 i^, ai2-34-^, Oi423-5867-^j ai324-5867 • i^ 211,212 L (^12345673)32 17 (^12315678)32 17 J2, 2 ^j ai3-24- &13-24. ^ 213,216 [_(^12345678)32i6 (^12345678)32 ieJ2, 2 ^, ai2-34-^, fll324-5867 • t, 01423-5867 • i 217 L(012345678)32 15(012345678)32 16 J2, 2 t 218 L(^12345678)32ie(Ol2345678)32i6J2, 2 t 112 1 L(^12345678)56(Ol2345678)56jl, 1 t 96 1,2 L(^12345678)48i6 (^12345678)4816 Jl, 1 ^> ai5.26-37-48 • ^ 3,4 L(Ol2345678)48i6(^12345678)48i6Jl, 1 i^j ai5-26-37-48- ^ 5,6 |_(fll2345678)48i7(^12345678)48i7Jl, 1 ^j ai2-34-56-78 • ^ 7,8 L(cH2345678)48i7 (^12345673)4817 Jl, 1 ^j ai2-34-56-78 • ^ 9,10 n(^12345678)249 (^12345678)249 J2, 2 ^, ^12.56 • O12.56 • ^ 11,12 L (^12345678)24 10 (^12345678)24 10 J2, 2 ^j ai3-24-78- &13-24-78- ^ 64 1,2 L (0H2345678) 826 (^12345678) 825 J4, 4 ^, tti5.26.37-48 • ^ 3 C(^12345678) 825 (^12345678) 826 J4, 4 ai324-5768- ^ Imprimitive Substitution Groups of Degree Sixteen 35 Order C(<*12346678)8„(tl2345678)8Mj4, 4 t, 01526-3847 • < L(Ol2345678)8,,(^12348«78)8„_]4. 4 t, 017362845- < E(ai2345678)8,4(6l2345678)8,4_]4. 4 015-26-37-48 • ^ C(Ol2346678) 824(612345678)8,434. 4 t, Ol324-5768-< L(ai2345678)8,3(&12345678)82jj4. 4 t, 015-26-37-48-^ [_(Ol2345678)l627(&12345678)l6,7j2. 2 t, 01324-6768.^, Ol2-56-^, 013.24-57-68 • ^ Oi6.26.37-48- ^> Oi6.26-37-48' ^ 015-26- ^6-26-^, 016-25-015-26-' L(Ol2345678)l6,7(&12345678)l6j7j2, 2 ^ 01324-5768. ^ 012-56 -^ 018.24-67-68 • ' L(Ol2345678)l6,7(&12346678)l6j7j2, 2 ty 01324-6768.', 012-56 -'> O13.24-57-68-' L(Ol2346678)l6,7(&12345678)l6,7j2. 2 01324-6768.', O12.66.', O13.24.57.68 . ' L(Oi2345678)i6,7(&12345678)i6j7J2, 2 ', Oi5.26-37-48. ' [_(Ol2345678)l6,7(&12345678)l6j7j2. 2 ', Oi6.26-37-48- ' L(Ol2345678)l6,7(6l2345678)l6j7j2, 2 01324-5768.' L(Ol2346678)l6j7(&12345678)l6i7j2. 2 ', 01324-5768-' L(012345678)i627(&12345678)i627J2, 2 ', Ol324-5768-' l(012345678) 1627(012345673) 1627^2, 2 015.26-37-48-' L(Ol2346678)l6,7(&12346678)l627j2, 2 ', Oi6.26-37-48- ', 015-26 -&15-26.'> 016-25 - ^15-26 - ' E(Ol2345678)l627(6l2345678)l627j2, 2 ', 012.34-56-78.' L(Ol2345678)l625(6l2345678)l626j2. 2 ', O12.56*. ', O13-24-57-68.', 01423-5867.', 01626-3847-', 01526-3847-', Oi625 - Ol625 • ', 01526-01625-' L(Ol2345678)l624(6l2346678)l62sJ2. 2 '» Oi3.24-57-68. ' L(Ol2346678)l6,j(tl2345678)l6,jJ2- 2 '> Oi2.56-' L(Ol2346678)l6,j(&12345678)l6,5j2, 2 ', O12.56-' L(Ol2345678)l6j5(6l2345678)l6jj_]2, 2 Oi625-3847 - ', 012.66-3847 . Ol625 - ', Oi625 - Ol625 - ' C(Ol2346678)l6,s(6l2346678)l6,sJ2. 2 '> Oi626-3847.' C(Ol2345678)l6,s(6l2345678)l625j2. 2 ', Oi526.3847.' L(Oi2346678)l6„(6l2345678)l6,jj2, 2 ', Oi625-3847.' ^ ^0. 4, 5 6, ,7 1 8 9, 10 11, 12 13, 20 21, 24 25, 28 29, 31 32, 33 34, 35 36 37, 38 39, 40 41 42, 45 46, 47 48, 55 56, 57 58, 59 60, 61 62, 64 65, 66 67, 68 69, 70 36 Grace Marie Bareis Order L (^12345678) I626 (^12345678) I625J2, 2 t, ai526-3847. i^ L(ai2345678)l625(&12345678)l625j2, 2 t, a13.04.57.68 J, ^1526 • Oi625 • i, CH3682457 • ^1625 • t L(c'l2345678)l625(Ol2345678)l625J2, 2 t, ai3-24-57-68 • ^ L(ctl2345678)l628(^12345678)l628j2, 2 01735-2846 • ^, a57-68-Ol3-24-^, Ct58.67 • ^13-24 • ^ L(cfl2345678)l628 (^12345678) I628J2, 2 t, ai735.2846' ^ L(Ol2345678)l628(^12345678)l628j2, 2 ai2-34-56-78 • ^ l(012345678)i628(^12345678)i628J2, 2 t, ai2-34-56-78 • ^ |_(*12345678)l6,6(&12345678)l626j2, 2 017362845 •^. 012.57-68 • 013-24.56 -i^ L(Ol2345678)l626(Ol2345678)l626j2, 2 ^, Ol7362845-^ C(Ol2345678)l63o(^12345678)l63oJ2, 2 t, 01324.5867-^ L(Ol2345678)l63o(oi2345678)l63oJ2, 2 t, 01324-5867-^ L(Ol2345678)l629(oi2345678)l629j2, 2 ^, 01324-5867-^ L(Oi2345678)i629(012345678)i629J2, 2 ^, Oi324-5867-^ L(Ol2345678)32ii(^12345678)32nJl, lU O12.34.56-78 • ^j Oi3.24-57-68-^j Oi4.23-58-67 - ^ 103,106 L(^12345678)32i2(Ol2345678)32i2Jl, 1 ^, Oi2.34.56.78 - ^j O13.24 - 013-24 . t; O14.23. 56-78 • ^13-24 - 1 107, 108 L(^12345678)32i7(^12345678)32i7Jl. 1 ^' 012.34.56-78.^ 48 1,4 L (^12345678)249 (012345678)249 Jl, 1 ^, O15.26.37.48 . ^, Ol6.25.37.48 . O12.56 • t, O12.56 . O12.56 • t 5,6 l(^12345678)24ii(&12345678)24iiJi, 1 ^, O12.56 . &12.56 . ^ 7,10 L(^12345678)24io(Ol2345678)24ioJl, 1 ^j Ol2-34-56-78 - ^) 013-24-78 - 013.24-78 - 1, O14.23-56 - &13-24-78 • t 32 1 L('^12345678)826(0l2345678)825j2, 2 Oi324-5768-^ 2 L (^12345678) 822(012345678)822^2, 2 Oi6482637-^ 3,10 L (^12345678) I627 (Ol2345678) I627 Jl, 1 ^j 012.34-56-78-^) Ol5-26-37-48-^j- O12.34 - Oi2-34 - ^, O15.26 • O15.26 . ^, 037.48- 015-26 -^) 01526-34 - 01526-34 - ^, Oi625-78- Ol526-34 • ^ 11,18 L(^12345678)l627(^12345678)l627Jl, 1 ^) 012.34.56-78 • i^j 0x5.26-37-48 -^j Oi6-26-38-47 - ^j O15.26 • O15.26 - ^j Oi6.25-34-78-&15-26-^j 037.48.615-26-^, Ol2-56-38-47-Oi5.26-^ No. 71, 72 73, 76 77, 78 79, 81 82, 83 84 85, 86 87, 88 89, 90 91, 92 93, 94 95, 96 97, 98 99, 102 iMPRmmvE Substitution Groups of Degree Sixteen 37 Order No. 19, 22 [_(^12346678)l6j7(6l2346678)l6,jjl, 1 t, On-3i'X-7S't, ai5-26-37-48«^j Oi2.34- Ol2-34- ^ 23, 29 L(ai2345678)l6,4(^2345678)l6,5Jl, 1 ai628-3847 • ^ ttb6-7S-bl2.3i't, aie2b'bl6ih-t, fll526-34.78-Ol626.^» ^3748 • ^^1625 • t, fll5'26.34 • Ol5'26-34 • ^ ai6'26.78'Ol6.2«-34.' 30,34 L(^12345678)l6.j(&12345678)l6,5Jl- 1 '' aj2-34-56-78' ^ 01625-3847 •^ ai2-34 • &12-34- ^ 056-78 • Ol2.34- ^ 35, 40 [_(ai2346678)l628(&12345678)l6,gjl. 1 012-34-56-78-^, Ol3-24-&U-24-^, Ol4-23-66-78'Ol3-24-^, 067-68 - 013-24 • ^ Olo.66-6i2.56-^. 01324-66 -Oj324-56-^ 41, 48 L (^12346678) 16,8 (&12345678)l6j8Jl, 1 ^' Ol2-34-56-78- ^ 013-24-57-68 •', Oi4.23.58-67- ^ O13.24- O13.24. <, 014-23-56-78 • ^3-24 - 1, 057.68 • &13-24 - 1, Ol2.34-58-67-Ol3-24-' 49, 55 C('^12345678)l6,j(6l2345678)l6jj_]l, 1 Oi324-6768. ^, 034-78-^2-66-^ 014.23-78- &13-24-66- ^ O1324 • 0X324 • ^, Ol2-34-6768-&1324-^ 012-56- 012-66 • ^» 013.24-56- 013-24.56-^ .56,59 11(012346678) 16,0 (^12345678)l6,o Jl. 1 ^ 012.84-66-78 • ^ 012-34-^2-34-^ 066-78 -012-34-^ 16 1,7 C(Ol2345678)8,4(&12345678)8jsJl. 1 ^ O12.34 - 6l2-34 - ^ Oi2.66-6l2-56-^ 015-26 - ^15-26 - ^> O34.78- O12.56. <, 016-26-37-48 - 1, Oi6-26-87-48 - Ol2.6« • t 8,13 C(Oi2345678)8„(&12345678)8,jJl. 1 ^ O12.34 - &12-34- <, 013.24-56 -&13 •24- 56 -^ O154 82637 - ^ O14.23.78- ^13-24-66. ^ 014-23.66.014.23.56-^ 14 L(012346678)8„(&12346678)8,,Ji, 1 t Order No. 7962624 1 3981312 1 2,3 1990656 1 995328 1,2 497664 1,2 Systems of Intransitivity of Heads 4, 4, 4, 4. [(ai234) all (&1234) all (C1234) all (^1234) all] t h tz [(01234) all (61234) all (C1234) all (^^1234) all] t h [{(01234) all (61234) all (C1234) all} (^^1234) all]69i2, 12 1 1\ tz, t ti ai2 . ts [{(«i234) all (61234) all (C1234) all} (dnu) all J6912 , ntti [(01234) all (61234) all (C1234) all (^1234) all J12 , 12 , 12 , 12 ^ ^1 h, «i2 .012.^ h di2 . tz [(01234) all (61234) all (C1234) all (^1234) all J12 , 12 , 12 , 12 ^ ti, 012 .Ci2.t ti 331776 1 [{(a:234) all (61234) all (C1234) all}i2, 12, i2((^i234) all]576 , ittitz 248832 1 [(01234) pos (61234) pos (C1234) pos (^1234) pos] t ti 165888 1 [{(01234) all (61234) all (C1234) all} 12, 12, 12 ((^1234) all]576, 4 ^ ^1 2, 4 [{(01234) pos (61234) pos (C1234) pos} (6^1234) P0S]576 ,4ttltz, ttittn. 612 . C12 . di2 . ^3, t 0123 • ^1 <^12 • 612 . C12 . 012 . tz 98304 1 [(O1034) 8(61234) 8(Cl234) 8((^1234) s] t tl tz 82944 1, 2 [(01234) pos (61234) pos (C1234) pos (^1234) P0S]576, 4 t tl, tdnz.ti 49152 1 [(Oi234)8(6i234)8(Cl234)8(c?1234)8]^^l 2, 3 [{ (^^1234) 8 (61234) 8(^1234)8} (C?1234)8]256, i t ti tz, t tl O12 . tz 4, 5 [{ (01234)8(61234)3(^1234)8} (<^1234)8]256, 4 t tl tz, t tl 013.24 . tz 6, 7 [{ (01234)8(61234)3(^1234)8} {dnzi)sjfi66, 4th U} t tl 012 . ^3 36864 1 [{ (01234) all (61234) all (C1234) all} 4, 4, 4 (^^1234) all J64 , it titz 38 Imprimitive Substitution Groups of Degree Sixteen 39 Order No. 24576 1 [{ (01234)8(^1234) 8(^1234) 8 }(rfl234)8]]256. 4 <<1 2 [_ { (01234) 8 (^234) 8 (C1234) 8 } (<^1234) 8 J266 , 4 t ti 3 [| I (01234) 8 (&I234) 8 (C1234) 8 } (^^1234) 8_|266 . 4 ^ 'l 4, 7 [{ (01234) 8(^1234) 8(^1234)3} (<^1234)8jl28, 2ttl U, t t\ di2 . U, t h U13-24 • ts, t i\ "1324 • h 18432 1 [{ (01234) all (61234) all (C1234) all} 4. 4. 4 ((^1234) all]64. ^th 2, 3 [{ (01234) pos (61234) pos (C1234) pos}4. 4. 4 {dvm) pos|]64 ,4tt\h, t ti an . bn . Cu • c?i2 . <3 12288 1 [{ (01234)8(61234)8(^1234)8} (C?1234)8]l28. 2 t h 2 L{ (^1234)8(61234)8(^1234)8}?. 4, 4(oi234)8j64, 4 ^ ^1 ^3 3 {_{ (01234)8(61234)8(^1234)814. 4. 4(C^1234)8J64, 4 t h U 4 {_{ (01234)8(61234)8(^1234)814, 4. 4(<^1234)8_l64, 4 t h U 9216 1 [{ (01234) all (61234) all (C1234) all} 4. 4. 4 (^^1234) allele, it tits 2 [{(01234) pos (61234) pos (C1234) pos} 4. 4. 4 idi23i) P0S]64. 4 tti 6144 1 [{ (01234)8(61234)8(^1234)8} 4. 4. 4(<^1234)8l|64, 4 t tl 2 Ll (Ol234)8(6l234)8(<^1234)8}4, 4, 4(c?1234)8j64, 4 ^ ^1 3 Ll (^1234)8(61234)8(^1234)814, 4. 4(ai234)8j64, 4 ^ ^1 4, 7 L{ (ai234)8(6l234)8(Cl234)8}4, 4, 4 (c?1234) 8 J32 . 2 ^ ^1 tz, t tl C?1324 • ts, O12 .d\i.t t\ 012 • ti, O12 .di2.t tl O13.24 • ^3 8, 11 U (^^1234)8 (61234) 3(^1234)8 }4. 4, 4(oi234)8J32. 2 ^ ^1 ^3, t tl di2 . ts, O13.24 . 013-24 . t tl 013.24 • tz, 013-24 . <^13-24 . t tl 01324 • ^3 12, 15 C{ (01234)8(61234) 8(^1234)8}4, 4, 4 (^1234) 8j32 , 2 t ti tz, t tl diz.24 . ^3. O12 .di2.t tl di2 . tz, O12 .di2.t tl 01324 . ^3 16 Lf ('^1234)8(6l234)8(Cl234)8}l6, 2(^1234)8j32. 2 t ti tz 17, 18 [(01234) eye (61234) eye (C1234) eye (^1234) eye] tti tz, tti ai2.bi2,Ci2.di2.tz 19, 21 [(01234)4 (61234)4 (Cl234)4((/l234)4] t tl tz, t tl ai2 . 612 . C12 . di2 . ^3, t O123 . 6132 . dnz . 24 • ^3 12 [[{ (01234)4(61234)4(^1234)4 } (C?1234)4]32. 2 t Ol2-34-^ 0123 • 6123 . C123 • O123 . ^1 17, 19 [{ (01234)4(61234)4(^1234)4 } 2. 2. 2 (<^1234)4l. 2 t tl 012 • 612 - C12 . 012 . tz, 0)3.24 - C13.24 - 1 tl "13-24 • tz, O13-24 - C13.24 - 1 tl O12 - 612 - C12 - "1324 - tz 576 1 [(0J234) all (61234) all (C1234) all (^1234) all[]i. 1. 1, itti tz 2 [{ { (O1234) pos (61234) pos} 4. 4 (<^1234) pos} 4. 1 (^^1234) pOs34. 1 ^ ^1 42 Grace Marie Bareis Order No. 384 1,6 [{ (ai234) 8(^1234) 8 (C1234) si 2, 2, 2(^^1234) si,! • ai3.24 . 013.24 • t h, fll3-24 • C?13-24 • t h, Ctl324 • «1423 • t t\, ai324 • 61324 . d\2 .Zi.t ti, bl32A • du23 ., t ti, 613.24 • ttl3.24 • t h 7 [{ (01234) eye (61234) eye (C1234) eye} 2, 2. 2 {dnu) Cycjs, 2 ai324.Ci324.^^1 8, 9 [{ (01234) eye (61234) eye (C1234) eye} 2. 2. 2 (<^i234) eye J4 , 1 01324 . duiz • t h O12 . 612 . C12 . c?i4.23 • tz, 01324 • 01423 • t t\ O12 . 612 . C12 . 013.24 • ^3 10 l{ (^1234)4(61234)4(^1234)4 }2, 2, 2(c?1234)4js, 2 Ol3.24. t fll23 • i>123 • ^123 • ttl23 • ^1 Ol324-Ol423-<^12-"34./3» fll4.23. O13.24 . «12-34' ^ 0123 • t'US • <^123 . "123 • ^1 Ol2 • Ol2 • Cl2 • 012 • U, 014.23 • 613 -24 • C?12-34 • t O123 . 6l23 • ^123 • "123 • ^1 01324 • Ol423 . C12 . 034 . <3 48 1,2 Q (01234)4 (61234)4 (01234)4 } 1 . 1 . 1 («?1234)4l . 1 t O123 . Ol23 • C123 . "123 . h, O14.23 • 013-24 • Ol2-34 • ^ 0123 • Ol23 • Cl23 • "123 • ^1 Systems Order 10321920 5160960 2580480 344064 172032 86016 43008 OF Intransitivity of Heads 2, 2, 2, 2, 2, 2, 2, 2. No. [(ai2) (&12) (C12) (dn) (en) (/12) (^'12) (^12)] ^1 h 21504 672 336 1 1 2,3 1 1 1,2 1 1 2,3 1,2 4336 1 7168 1 2688 1 1,3 [(ai2) (bn) (C12) (dn) (en) (/12) (5^12) (^12)] h U [(ai2) (612) (C12) {di2) {en) (fn) ign) (A12)] pos ^1 h, t\ a\2 . h \^{an){bn){cn)(dn)(ei2)(fn)(gi2)(hi2)'] pos ^1 Iq, [(ai2) (bn) (C12) (dn) M (fn) ign) (A12)] h U tz^ h [(ai2) (&12) (C12) ((^12) (^12) (/12) (S'12) (^12)] pos h U k^ h, k U tz^ an .U ' L(an) (bn) (cn) {dn) {en) {fn) {gn) {hn)2 ^1 ^ ^3 [(ai2) (612) {cn) {dn) {en) {fn) {gn) {hn)2 k k^ h^ [(ai2) {bn) {cn) {dn) {en) {fn) {gn) (^12)] pos k k tz, k CLn • k CLn • tz \_{an){bn){cn){dn){en){fn){gn){hn)~\ pos k U tz\ k k tz [_{an) {bn) {cn) {dn) {en) {fn) {gn) {h2)~\ k ti l{an) {bn) {cn) {dn) {en) {fn) {gn) {hn)2 pos ^1 4 [ai2 . 612 . Cn • dn en -fn gn • hi} ti ti tz^ an • bn • Cn -fn • 4 [_an • bn • cn • dn en -fn • 5^12 • ^12] ti an • bn . ^12 • ^ hn • ts, t\ fn • dn • gn • ^2 hn . ts, ti an . bn -fn • gn • ^2 tz [1] k t2 tz [ai2 . bn . Cn • dn . en -fn • gn hn2 ti tz^ an bn • hn . 4' 2188 44