p-group, metabelian, nilpotent (class 2), monomial
Aliases: C82⋊16C2, C8⋊6M4(2), C23.14C42, C42.747C23, C8⋊C8⋊23C2, C8⋊C4.13C4, (C2×C4).18C42, C42.50(C2×C4), C2.7(C4×M4(2)), (C4×C8).308C22, C4.60(C2×M4(2)), (C2×M4(2)).23C4, (C4×M4(2)).15C2, C22.44(C2×C42), (C2×C42).144C22, (C2×C8).121(C2×C4), (C22×C4).174(C2×C4), (C2×C4).584(C22×C4), SmallGroup(128,187)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊6M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=cac=a5, cbc=b5 >
Subgroups: 140 in 107 conjugacy classes, 76 normal (7 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C82, C8⋊C8, C4×M4(2), C8⋊6M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C2×C42, C2×M4(2), C4×M4(2), C8⋊6M4(2)
►On 64 points
(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)
(1 10 57 55 20 42 40 25)(2 15 58 52 21 47 33 30)(3 12 59 49 22 44 34 27)(4 9 60 54 23 41 35 32)(5 14 61 51 24 46 36 29)(6 11 62 56 17 43 37 26)(7 16 63 53 18 48 38 31)(8 13 64 50 19 45 39 28)
(2 6)(4 8)(9 45)(10 42)(11 47)(12 44)(13 41)(14 46)(15 43)(16 48)(17 21)(19 23)(25 55)(26 52)(27 49)(28 54)(29 51)(30 56)(31 53)(32 50)(33 37)(35 39)(58 62)(60 64)
G:=sub<Sym(64)| (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), (1,10,57,55,20,42,40,25)(2,15,58,52,21,47,33,30)(3,12,59,49,22,44,34,27)(4,9,60,54,23,41,35,32)(5,14,61,51,24,46,36,29)(6,11,62,56,17,43,37,26)(7,16,63,53,18,48,38,31)(8,13,64,50,19,45,39,28), (2,6)(4,8)(9,45)(10,42)(11,47)(12,44)(13,41)(14,46)(15,43)(16,48)(17,21)(19,23)(25,55)(26,52)(27,49)(28,54)(29,51)(30,56)(31,53)(32,50)(33,37)(35,39)(58,62)(60,64)>;
G:=Group( (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), (1,10,57,55,20,42,40,25)(2,15,58,52,21,47,33,30)(3,12,59,49,22,44,34,27)(4,9,60,54,23,41,35,32)(5,14,61,51,24,46,36,29)(6,11,62,56,17,43,37,26)(7,16,63,53,18,48,38,31)(8,13,64,50,19,45,39,28), (2,6)(4,8)(9,45)(10,42)(11,47)(12,44)(13,41)(14,46)(15,43)(16,48)(17,21)(19,23)(25,55)(26,52)(27,49)(28,54)(29,51)(30,56)(31,53)(32,50)(33,37)(35,39)(58,62)(60,64) );
G=PermutationGroup([[(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)], [(1,10,57,55,20,42,40,25),(2,15,58,52,21,47,33,30),(3,12,59,49,22,44,34,27),(4,9,60,54,23,41,35,32),(5,14,61,51,24,46,36,29),(6,11,62,56,17,43,37,26),(7,16,63,53,18,48,38,31),(8,13,64,50,19,45,39,28)], [(2,6),(4,8),(9,45),(10,42),(11,47),(12,44),(13,41),(14,46),(15,43),(16,48),(17,21),(19,23),(25,55),(26,52),(27,49),(28,54),(29,51),(30,56),(31,53),(32,50),(33,37),(35,39),(58,62),(60,64)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4L | 4M | 4N | 4O | 8A | ··· | 8X | 8Y | ··· | 8AJ |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) |
| kernel | C8⋊6M4(2) | C82 | C8⋊C8 | C4×M4(2) | C8⋊C4 | C2×M4(2) | C8 |
| # reps | 1 | 1 | 3 | 3 | 12 | 12 | 24 |
Matrix representation of C8⋊6M4(2) ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 6 |
| 0 | 0 | 9 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 13 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,1,9,0,0,6,16],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,13,16] >;
C8⋊6M4(2) in GAP, Magma, Sage, TeX
C_8\rtimes_6M_4(2)
% in TeX
G:=Group("C8:6M4(2)"); // GroupNames label
G:=SmallGroup(128,187);
// by ID
G=gap.SmallGroup(128,187);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,758,723,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=b^5>;
// generators/relations