Aliases: C8.7S4, SL2(𝔽3).C8, Q8.(C3⋊C8), C8○D4.2S3, C4.A4.1C4, C8.A4.2C2, C2.3(A4⋊C8), C4.5(A4⋊C4), C4○D4.1Dic3, SmallGroup(192,187)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C8.7S4 |
Generators and relations for C8.7S4
G = < a,b,c,d,e | a8=d3=1, b2=c2=a4, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a4b, dbd-1=a4bc, ebe-1=bc, dcd-1=b, ece-1=a4c, ede-1=d-1 >
Smallest permutation representation of C8.7S4
►On 64 points
Generators in S64
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)
(1 45 9 37)(2 20 10 28)(3 47 11 39)(4 22 12 30)(5 33 13 41)(6 24 14 32)(7 35 15 43)(8 26 16 18)(17 64 25 56)(19 50 27 58)(21 52 29 60)(23 54 31 62)(34 63 42 55)(36 49 44 57)(38 51 46 59)(40 53 48 61)
(1 50 9 58)(2 59 10 51)(3 52 11 60)(4 61 12 53)(5 54 13 62)(6 63 14 55)(7 56 15 64)(8 49 16 57)(17 35 25 43)(18 44 26 36)(19 37 27 45)(20 46 28 38)(21 39 29 47)(22 48 30 40)(23 41 31 33)(24 34 32 42)
(17 35 56)(18 57 36)(19 37 58)(20 59 38)(21 39 60)(22 61 40)(23 41 62)(24 63 42)(25 43 64)(26 49 44)(27 45 50)(28 51 46)(29 47 52)(30 53 48)(31 33 54)(32 55 34)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 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 52 53 54 55 56 57 58 59 60 61 62 63 64)
G:=sub<Sym(64)| (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64), (1,45,9,37)(2,20,10,28)(3,47,11,39)(4,22,12,30)(5,33,13,41)(6,24,14,32)(7,35,15,43)(8,26,16,18)(17,64,25,56)(19,50,27,58)(21,52,29,60)(23,54,31,62)(34,63,42,55)(36,49,44,57)(38,51,46,59)(40,53,48,61), (1,50,9,58)(2,59,10,51)(3,52,11,60)(4,61,12,53)(5,54,13,62)(6,63,14,55)(7,56,15,64)(8,49,16,57)(17,35,25,43)(18,44,26,36)(19,37,27,45)(20,46,28,38)(21,39,29,47)(22,48,30,40)(23,41,31,33)(24,34,32,42), (17,35,56)(18,57,36)(19,37,58)(20,59,38)(21,39,60)(22,61,40)(23,41,62)(24,63,42)(25,43,64)(26,49,44)(27,45,50)(28,51,46)(29,47,52)(30,53,48)(31,33,54)(32,55,34), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)>;
G:=Group( (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64), (1,45,9,37)(2,20,10,28)(3,47,11,39)(4,22,12,30)(5,33,13,41)(6,24,14,32)(7,35,15,43)(8,26,16,18)(17,64,25,56)(19,50,27,58)(21,52,29,60)(23,54,31,62)(34,63,42,55)(36,49,44,57)(38,51,46,59)(40,53,48,61), (1,50,9,58)(2,59,10,51)(3,52,11,60)(4,61,12,53)(5,54,13,62)(6,63,14,55)(7,56,15,64)(8,49,16,57)(17,35,25,43)(18,44,26,36)(19,37,27,45)(20,46,28,38)(21,39,29,47)(22,48,30,40)(23,41,31,33)(24,34,32,42), (17,35,56)(18,57,36)(19,37,58)(20,59,38)(21,39,60)(22,61,40)(23,41,62)(24,63,42)(25,43,64)(26,49,44)(27,45,50)(28,51,46)(29,47,52)(30,53,48)(31,33,54)(32,55,34), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16),(17,19,21,23,25,27,29,31),(18,20,22,24,26,28,30,32),(33,35,37,39,41,43,45,47),(34,36,38,40,42,44,46,48),(49,51,53,55,57,59,61,63),(50,52,54,56,58,60,62,64)], [(1,45,9,37),(2,20,10,28),(3,47,11,39),(4,22,12,30),(5,33,13,41),(6,24,14,32),(7,35,15,43),(8,26,16,18),(17,64,25,56),(19,50,27,58),(21,52,29,60),(23,54,31,62),(34,63,42,55),(36,49,44,57),(38,51,46,59),(40,53,48,61)], [(1,50,9,58),(2,59,10,51),(3,52,11,60),(4,61,12,53),(5,54,13,62),(6,63,14,55),(7,56,15,64),(8,49,16,57),(17,35,25,43),(18,44,26,36),(19,37,27,45),(20,46,28,38),(21,39,29,47),(22,48,30,40),(23,41,31,33),(24,34,32,42)], [(17,35,56),(18,57,36),(19,37,58),(20,59,38),(21,39,60),(22,61,40),(23,41,62),(24,63,42),(25,43,64),(26,49,44),(27,45,50),(28,51,46),(29,47,52),(30,53,48),(31,33,54),(32,55,34)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 8 | 1 | 1 | 6 | 8 | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 8 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 |
| type | + | + | + | - | + | |||||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | C3⋊C8 | C8.7S4 | S4 | A4⋊C4 | A4⋊C8 | C8.7S4 |
| kernel | C8.7S4 | C8.A4 | C4.A4 | SL2(𝔽3) | C8○D4 | C4○D4 | Q8 | C1 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C8.7S4 ►in GL2(𝔽17) generated by
| 8 | 0 |
| 0 | 8 |
| 16 | 1 |
| 15 | 1 |
| 4 | 13 |
| 0 | 13 |
| 0 | 10 |
| 5 | 16 |
| 7 | 8 |
| 14 | 10 |
G:=sub<GL(2,GF(17))| [8,0,0,8],[16,15,1,1],[4,0,13,13],[0,5,10,16],[7,14,8,10] >;
C8.7S4 in GAP, Magma, Sage, TeX
C_8._7S_4
% in TeX
G:=Group("C8.7S4"); // GroupNames label
G:=SmallGroup(192,187);
// by ID
G=gap.SmallGroup(192,187);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,14,36,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=d^3=1,b^2=c^2=a^4,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^4*c,e*d*e^-1=d^-1>;
// generators/relations
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