COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA. COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
178 Charles C. Sims TABLE 1. The Primitive Groups of Degree not exceeding 20 TABLE 1 (continued) - - - - Dew e No. - < lrder - t N G. Generators t - Comments Degree NO. Irder t N G. Generators f 6 7 8 9 10 1 1 2 1 2 1 2 3 4 5 1 2 3 4 1 2 3 4 i I 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 2 3 6 12 24 5 10 20 60 120 60 120 360 720 7 14 21 42 168 2520 5040 56 168 168 336 1344 , 1 LO160 1 10320 36 72 72 72 144 216 432 504 1512 $9! 9! 60 120 360 720 720 2 3 2P 4 2 3P 5 2P 3 4P 6 2 2 SP 7 2P 2P 2P 3 k 8 2 2 2 2 2 3P 3P 7P 9 2P 3 3 - - 3Gl 2Gl e.a. 3Gl e.a. 3G2 - a5 5Gl a5, b: 5Gl as9 b, - 4Gl 5G4 4G2 - 5G2 a5, b:, a6 6Gl 5G3 a,, bs, a6 - 5G4 6G3 5G5 - a7 7Gl a,, 6; 7Gl a,, b5 7Gl a,, 4 - a,, c7 - 6G3 7G6 e.a. 7Gl a,, a8 e.a. 7G3 a7, b27, a8 - 7G3 a,, b:, 4 8G3 7G4 a,, b7, 4 e.a. 7G5 a,, c7, a6 - 7G6 8G6 7Gl e.a. a,, 4 e.a. a,, 4, dg e.a. ag7 cg e.a. as, d, cgdg e.a. aa, ~9, da e.a. a,, 4, eg e.a. 09, Cg9 eg - 8Gl a7, a,, f, 9G8 8G2 a7, G, a,, fa - 8G6 9GlO 8G7 - do, b lo lOG1 40, b 1~ - 9G1 a,, 4, 60 lOG3 9G2 ag, 4, da, CIO lOG3 9G3 a9, Cg, Go - - - - A5 - k(2 5) - PGL(i, 5) A6 - S6 - - PSL(3,2) A, - s, PSL(2, 7) - PGL(2, 7) 43 - &l - - - - PSL(2, 8) -4 a - sa A.5 - SS PSL(2,9) - - %L(2,9) 11 12 13 14 15 16 8 9 1 2 3 4 5 6 I 8 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 1 2 3 4 1 2 3 4 5 6 1 2 3 4 5 6 7 .lO! lo! 11 22 55 110 660 7920 ..ll! ll! 660 1320 7920 5040 .12! 12! 13 26 39 52 78 156 5616 .-13! 13! 1092 2184 -.14! 14! 360 720 2520 0160 -.15! 15! 80 160 240 288 320 480 576 8~ 10 2 2P 4 9P 11 2P 3 3P 5 1OP 12 2 2 1lP 13 2P 3 12P 14 2 2 13P 15 2 2 - 9GlO A 10 lOG8 9611 - S10 - llG1 - llG1 llG1 - - 1OGl PSL(2,ll) - lOG6 Ml1 - lOG8 A 11 llG7 lOG9 - S - llG3 ally b&, al2 P;L(2, 11) 12Gl llG4 all, b al2 - PGL(2, 11) - llG5 do. ho, cll, b I~ Ml1 - llG6 *a, 4, cads, ~10, Ml2 41, ~12 - llG7 A 12 12G5 llG8 - S12 - al3 13Gl 63, b* l5 13Gl a13, b4l3 13Gl ala, j3 - 89" 13Gl cJ13, 13 13Gl a13, b - l~ - als. cl3 PSL(3, 3) - 12G5 A 13 13GE 12G6 - S - 13G5 a13, bz 13, aI4 PgL(2, 13) 14Gl 13G6 ala, 43, alp - PGL(2, 13) - 13G8 Ald 14G: 13G9 - S14 - als9 b 15 -46 15G1 al5, cl5 S6 - bm 4, 447 - 45, e15 . PSL(4, 2) - 14G3 A 15 15G: 14G4 - S15 e.a. 65, al6 e.a. al,, al6, e?6 e.a. alSe15s al6 e.a. bm %6,fi6, g,, e.a. a15s al69 e16 e.a. alse16, al6, e:6 e.a. b15, hfi6, gl6h6 6 7 720 1440 3 3 lOG3 lOG3 9G4 9G5 09, 4, CA GO ag, Cg, ds, CID - [continued on p. 1801 - ! Computational methods for permutation groups 179
180 Degrc - No. Order t N (-63 Generators f Comments - - --- 16 8 576 e.a. 17 18 19 20 9 960 e.a. 10 960 2 e.a. 11 960 2 e.a. 12 1152 e.a 13 1920 e.a. 14 1920 2 e.a. 15 2880 2 e.a. 16 5760 2 e.a. 17 5760 2p e.a. 18 11520 2p e.a. 19 40320 3 e.a. 20 ‘22560 3 e.a. 21 $16! 14~ - 22 16! 16 6G21 1 17 - 2 34 17Gl 3 68 17Gl 4 136 17Gl 5 272 2 17Gl 6 4080 3 - 7 8160 3 17G6 8 16320 3 17G6 9 $17! 15p - 10 17! 17 17G9 1 2448 2p - 2 4896 3 18Gl 3 i-18! 16p - 4 18! 18 18G3 1 19 - 2 38 19Gl 3 57 19Gl 4 114 19Gl 5 171 19Gl 6 342 2 19Gl 7 ;. 19! 171 - 8 19! 19 19G7 1 3420 2p - 2 6840 3 20Gl 3 +.20! 18~ - 4 20! 20 20G3 - Charles C. Sims TABLE 1 (continued) 15Gl 15G2 15G3 15G4 15G5 15G6 16G3 16G6 6GlC 6G21 6622 17G4 17G5 17G9 7GlC 18G3 18G4 19G5 19G6 19G7 19G8 bl5, %6.&. g1m k,Lh,hl, a153 a16, h6 wa5, a16, cl6 a15, alBr f 16 b157 a16y fiS, b-16, h 1 6 a15, al,, ~1~ al51 al67 k16 a16e15, %6, f 16 a15e15p %Y k 16 a,,, hi, %6 %5, c15, %6 b15, d15, al6 d16, e15; aI6 al7 a17, b?, a17, b4l7 a17, bf7 a179 b17 wk a16, cl7 wh a16, d6, cl7 a15e15y %6r e16Y '17 a17, %, a18 aI,, b17, al8 al8 alsl % a197 b:, b al,, 19 a19, bf, agO al8, b9, %O - - - - - - - - - - - A 16 S 16 PSL(2, 16) A 17 S I&(2, 17) PGL(2, 17) A 18 S 18 A 19 s P&(2,19) PGL(2, 19) 40 S 20 5 6 7 8 Computational methods for permutation groups a = (1,2, 3,4, 5) b = (1) (2, 3, 534) a = (1,6) (2) (3,4) (5) a = (1,2, 3,4,5, 6,7) b = (1) (2,4,3,7,5,6) c = (1) (2,3) (4,7) (5) (6) a = Cl,81 (234) (3, 7) (5, 6) b = Cl,41 (2,8) (395) (67) c = (1, 7) (2, 5) (3, 8) (4, 6) d = (1, 8) (2, 7) (394) (5, 6) TABLE 2. Generating Permutations ~=(1,2,3)(4,5,6)(7,8,9) b = (1, 4, 7) (2, 5, 8) (3, 6, 9) c = (1) (2, 6, 4, 9, 3, 8, 7, 5) d = (1) (2) (3) (4, 7) (5, 8) (6, 9) e = (1) (2, 4, 9) (3, 7, 5) (6) (8) f = (1) (2, 7) (3, 6) (4, 5) (8, 9) a = (1, 8) (2, 5, 6, 3) (4, 9, 7, 10) b = (1, 5, 7) (2, 9, 4) (3, 8, 10) (6) c = (1, 10) (2) (3) (4, 7) (5, 6) (8, 9) Permutations a = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) b = (1) (2, 3, 5, 9, 6, 11, 10, 8, 4, 7) c = (1, 11) (2, 7) (3, 5) (4, 6) (8) (9) (10) 3 = (1) (2) (3) (4, 8) (5, 9) 6 7) (10, 11) z = (1, 12) (2, 11) (3, 6) (4, 8) (5, 9) (7, 10) 5 = (1) (2, 5) (3, 6) (4, 7) (8) (9) (10) (11, 12) : = (1) (2) (3) (4, 7) (5, 8) (6, 9) (10) (11, 12) z = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) i = (1) (2, 3, 5, 9, 4, 7, 13, 12, 10, 6, 11, 8) : = (1) (2, 3) (4) (5, 10) (6) (7, 11) (8) (9, 12) (13) I = (1, 14) (2, 13) (3, 7) (4, 5) (6) (8, 12) (9) (10, 11) I = (1, 15, 7, 5, 12) (2, 9, 13, 14, 8) (3, 6, 10, 11, 4) 1 = (1, 4, 5) (2, 8, 10) (3, 12, 15) (6, 13, 11) (7, 9, 14) : = (1, 7) (2, 11) (3, 12) (4, 13) (5, 10) (6) (8, 14) (9) (15) ! = (1, 9, 5, 14, 13, 2, 6) (3, 15, 4, 7, 8, 12, 11) (10) t = (1, 3,2) (4, 8, 12) (5, 11, 14) (6, 9, 15) (7, 10, 13) 181 [continued on p. I82/
- Page 43 and 44: 76 W. Lindenberg and L. Gerhards Se
- Page 45 and 46: 80 W. Lindenberg and L. Gerhards Se
- Page 47 and 48: 84 K. Ferber and H. Jiirgensen The
- Page 49 and 50: 7 The construction of the character
- Page 51 and 52: 92 John McKay defining multiplicati
- Page 53 and 54: 96 John McKay Construction of chara
- Page 55 and 56: 100 John McKay In the table, c indi
- Page 57 and 58: 104 C. Brott and J. Neubiiser irred
- Page 59 and 60: 108 C. Brott and J. Neubiiser eleme
- Page 61 and 62: 112 J. S. Frame The characters of t
- Page 63 and 64: 116 J. S. Frame 3. The decompositio
- Page 65 and 66: 120 J. S. Frame The characters of t
- Page 67 and 68: 124 J. S. Frame The characters of t
- Page 69 and 70: TABLE 5. The Z, character block for
- Page 71 and 72: 132 R. Biilow and J. Neubiiser Deri
- Page 73 and 74: A search for simple groups of order
- Page 75 and 76: 140 Marshall Hall Jr. Simple groups
- Page 77 and 78: 144 Marshall Hall Jr. Simple groups
- Page 79 and 80: 148 Marshall Hall Jr. otherwise no
- Page 81 and 82: 152 Marshall Hall Jr. easily found
- Page 83 and 84: 156 Marshall Hall Jr. Simple groups
- Page 85 and 86: 160 Marshall Hail Jr. This leads to
- Page 87 and 88: 164 Marshall Hall Jr. b= (OO)(Ol, 0
- Page 89 and 90: 168 Marshall Hall Jr. 11. R. BRAUER
- Page 91 and 92: 172 Charles C. Sims THEOREM 2.3. Th
- Page 93: 176 Charles C. Sims has a set of im
- Page 97 and 98: An algorithm related to the restric
- Page 99 and 100: A module-theoretic computation rela
- Page 101 and 102: 192 A. L. Tritter expressed in the
- Page 103 and 104: 196 A. L. Tritter a question is mos
- Page 105 and 106: 200 John J. Cannon stack. The Cayle
- Page 107 and 108: , I The computation of irreducible
- Page 109 and 110: 208 P. G. Ruud and R. Keown product
- Page 111 and 112: 212 P. G. Ruud and R. Keown TX wher
- Page 113 and 114: 216 P. G. Ruud and R. Keown 4. L. G
- Page 115 and 116: 220 N. S. Mendelsohn where it is kn
- Page 117 and 118: 224 Robert J. Plemmons such class [
- Page 119 and 120: 228 Robert J. Plemmons ! TOTALS Sem
- Page 121 and 122: 232 Takayuki Tamura 8” = 0B if an
- Page 123 and 124: 236 Takayuki Tamura Case ,Q = {e).
- Page 125 and 126: 240 Takayuki Tamura The calculation
- Page 127 and 128: 244 Takayuki Tamura We calculate46
- Page 129 and 130: 248 Takayuki Tamura Immediately we
- Page 131 and 132: 252 Takayuki Tamura Let U be a subs
- Page 133 and 134: 256 Takayuki Tamura TABLE 8. AI1 Se
- Page 135 and 136: I The author and R. Dickinson have
- Page 137 and 138: 264 Donald E. Knuth and Peter B. Be
- Page 139 and 140: 268 Donald E. Knuth and Peter B. Be
- Page 141 and 142: 272 Donald E. Knuth and Peter B. Be
- Page 143 and 144: 276 Donald E. Knuth and Peter B. Be
178<br />
Charles C. Sims<br />
TABLE 1. The Primitive Groups of Degree not exceeding 20<br />
TABLE 1 (continued)<br />
- -<br />
-<br />
-<br />
Dew e No.<br />
-<br />
< lrder<br />
-<br />
t N G. Generators t<br />
-<br />
Comments Degree NO. Irder t N G. Generators f<br />
6<br />
7<br />
8<br />
9<br />
10<br />
1<br />
1<br />
2<br />
1<br />
2<br />
1<br />
2<br />
3<br />
4<br />
5<br />
1<br />
2<br />
3<br />
4<br />
1<br />
2<br />
3<br />
4<br />
i<br />
I<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
11<br />
1<br />
2<br />
3<br />
4<br />
5<br />
2<br />
3<br />
6<br />
12<br />
24<br />
5<br />
10<br />
20<br />
60<br />
120<br />
60<br />
120<br />
360<br />
720<br />
7<br />
14<br />
21<br />
42<br />
168<br />
2520<br />
5040<br />
56<br />
168<br />
168<br />
336<br />
1344<br />
,<br />
1 LO160<br />
1 10320<br />
36<br />
72<br />
72<br />
72<br />
144<br />
216<br />
432<br />
504<br />
1512<br />
$9!<br />
9!<br />
60<br />
120<br />
360<br />
720<br />
720<br />
2<br />
3<br />
2P<br />
4<br />
2<br />
3P<br />
5<br />
2P<br />
3<br />
4P<br />
6<br />
2<br />
2<br />
SP<br />
7<br />
2P<br />
2P<br />
2P<br />
3<br />
k<br />
8<br />
2<br />
2<br />
2<br />
2<br />
2<br />
3P<br />
3P<br />
7P<br />
9<br />
2P<br />
3<br />
3<br />
-<br />
-<br />
3Gl 2Gl<br />
e.a. 3Gl<br />
e.a. 3G2<br />
-<br />
a5<br />
5Gl<br />
a5, b:<br />
5Gl<br />
as9 b,<br />
- 4Gl<br />
5G4 4G2<br />
- 5G2 a5, b:, a6<br />
6Gl 5G3 a,, bs, a6<br />
- 5G4<br />
6G3 5G5<br />
-<br />
a7<br />
7Gl a,, 6;<br />
7Gl a,, b5<br />
7Gl a,, 4<br />
-<br />
a,, c7<br />
- 6G3<br />
7G6<br />
e.a. 7Gl a,, a8<br />
e.a. 7G3 a7, b27, a8<br />
- 7G3 a,, b:, 4<br />
8G3 7G4 a,, b7, 4<br />
e.a. 7G5 a,, c7, a6<br />
- 7G6<br />
8G6 7Gl<br />
e.a.<br />
a,, 4<br />
e.a.<br />
a,, 4, dg<br />
e.a.<br />
ag7 cg<br />
e.a.<br />
as, d, cgdg<br />
e.a.<br />
aa, ~9, da<br />
e.a.<br />
a,, 4, eg<br />
e.a.<br />
09, Cg9 eg<br />
- 8Gl a7, a,, f,<br />
9G8 8G2 a7, G, a,, fa<br />
- 8G6<br />
9GlO 8G7<br />
- do, b lo<br />
lOG1 40, b 1~<br />
- 9G1 a,, 4, 60<br />
lOG3 9G2 ag, 4, da, CIO<br />
lOG3 9G3 a9, Cg, Go<br />
-<br />
-<br />
-<br />
-<br />
A5<br />
-<br />
k(2 5)<br />
- PGL(i, 5)<br />
A6<br />
- S6<br />
-<br />
-<br />
PSL(3,2)<br />
A,<br />
- s,<br />
PSL(2, 7)<br />
- PGL(2, 7)<br />
43<br />
- &l<br />
-<br />
-<br />
-<br />
-<br />
PSL(2, 8)<br />
-4 a<br />
- sa<br />
A.5<br />
- SS<br />
PSL(2,9)<br />
-<br />
- %L(2,9)<br />
11<br />
12<br />
13<br />
14<br />
15<br />
16<br />
8<br />
9<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
I<br />
8<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
1<br />
2<br />
3<br />
4<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
.lO!<br />
lo!<br />
11<br />
22<br />
55<br />
110<br />
660<br />
7920<br />
..ll!<br />
ll!<br />
660<br />
1320<br />
7920<br />
5040<br />
.12!<br />
12!<br />
13<br />
26<br />
39<br />
52<br />
78<br />
156<br />
5616<br />
.-13!<br />
13!<br />
1092<br />
2184<br />
-.14!<br />
14!<br />
360<br />
720<br />
2520<br />
0160<br />
-.15!<br />
15!<br />
80<br />
160<br />
240<br />
288<br />
320<br />
480<br />
576<br />
8~<br />
10<br />
2<br />
2P<br />
4<br />
9P<br />
11<br />
2P<br />
3<br />
3P<br />
5<br />
1OP<br />
12<br />
2<br />
2<br />
1lP<br />
13<br />
2P<br />
3<br />
12P<br />
14<br />
2<br />
2<br />
13P<br />
15<br />
2<br />
2<br />
- 9GlO<br />
A 10<br />
lOG8 9611 - S10 -<br />
llG1 -<br />
llG1<br />
llG1 -<br />
- 1OGl<br />
PSL(2,ll)<br />
- lOG6<br />
Ml1<br />
- lOG8<br />
A 11<br />
llG7 lOG9 - S<br />
- llG3 ally b&, al2<br />
P;L(2, 11)<br />
12Gl llG4 all, b al2 - PGL(2, 11)<br />
- llG5 do. ho, cll, b I~ Ml1<br />
- llG6 *a, 4, cads, ~10, Ml2<br />
41, ~12<br />
- llG7<br />
A 12<br />
12G5 llG8 - S12 -<br />
al3<br />
13Gl 63, b* l5<br />
13Gl<br />
a13, b4l3<br />
13Gl<br />
ala, j3 -<br />
89"<br />
13Gl<br />
cJ13, 13<br />
13Gl a13, b -<br />
l~<br />
- als. cl3<br />
PSL(3, 3)<br />
- 12G5<br />
A 13<br />
13GE 12G6 - S<br />
- 13G5 a13, bz 13, aI4<br />
PgL(2, 13)<br />
14Gl 13G6 ala, 43, alp - PGL(2, 13)<br />
- 13G8<br />
Ald<br />
14G: 13G9 - S14 -<br />
als9 b 15<br />
-46<br />
15G1 al5, cl5<br />
S6<br />
- bm 4,<br />
447<br />
- 45, e15<br />
. PSL(4, 2)<br />
- 14G3<br />
A 15<br />
15G: 14G4 - S15 e.a. 65, al6<br />
e.a. al,, al6, e?6<br />
e.a. alSe15s al6<br />
e.a. bm %6,fi6, g,,<br />
e.a. a15s al69 e16<br />
e.a. alse16, al6, e:6<br />
e.a. b15, hfi6, gl6h6<br />
6<br />
7<br />
720<br />
1440<br />
3<br />
3<br />
lOG3<br />
lOG3<br />
9G4<br />
9G5<br />
09, 4, CA GO<br />
ag, Cg, ds, CID -<br />
[continued on p. 1801<br />
-<br />
!<br />
Computational methods for permutation groups<br />
179
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