p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8:9M4(2), C23.26C42, C42.743C23, C4:C8.18C4, (C4xC8).35C4, C8:C8:19C2, C8:C4.11C4, C4.9(C8:C4), C4.53(C8oD4), C22:C8.17C4, (C2xC4).60C42, (C22xC8).44C4, C2.5(C4xM4(2)), C42.242(C2xC4), (C4xC8).360C22, C4.57(C2xM4(2)), (C2xC4).41M4(2), C22.8(C8:C4), (C4xM4(2)).13C2, (C2xM4(2)).21C4, C22.40(C2xC42), C2.6(C8o2M4(2)), C42.12C4.41C2, (C2xC42).1030C22, (C2xC4xC8).59C2, C2.5(C2xC8:C4), (C2xC8).117(C2xC4), (C2xC4).580(C22xC4), (C22xC4).392(C2xC4), SmallGroup(128,183)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8:9M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a5, ac=ca, cbc=b5 >
Subgroups: 132 in 104 conjugacy classes, 76 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C2xC8, C2xC8, M4(2), C22xC4, C4xC8, C4xC8, C8:C4, C22:C8, C4:C8, C2xC42, C22xC8, C2xM4(2), C8:C8, C2xC4xC8, C4xM4(2), C42.12C4, C8:9M4(2)
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, M4(2), C22xC4, C8:C4, C2xC42, C2xM4(2), C8oD4, C2xC8:C4, C4xM4(2), C8o2M4(2), C8:9M4(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 19 36 54 27 43 12 59)(2 24 37 51 28 48 13 64)(3 21 38 56 29 45 14 61)(4 18 39 53 30 42 15 58)(5 23 40 50 31 47 16 63)(6 20 33 55 32 44 9 60)(7 17 34 52 25 41 10 57)(8 22 35 49 26 46 11 62)
(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
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), (1,19,36,54,27,43,12,59)(2,24,37,51,28,48,13,64)(3,21,38,56,29,45,14,61)(4,18,39,53,30,42,15,58)(5,23,40,50,31,47,16,63)(6,20,33,55,32,44,9,60)(7,17,34,52,25,41,10,57)(8,22,35,49,26,46,11,62), (17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61)>;
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), (1,19,36,54,27,43,12,59)(2,24,37,51,28,48,13,64)(3,21,38,56,29,45,14,61)(4,18,39,53,30,42,15,58)(5,23,40,50,31,47,16,63)(6,20,33,55,32,44,9,60)(7,17,34,52,25,41,10,57)(8,22,35,49,26,46,11,62), (17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61) );
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)], [(1,19,36,54,27,43,12,59),(2,24,37,51,28,48,13,64),(3,21,38,56,29,45,14,61),(4,18,39,53,30,42,15,58),(5,23,40,50,31,47,16,63),(6,20,33,55,32,44,9,60),(7,17,34,52,25,41,10,57),(8,22,35,49,26,46,11,62)], [(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | M4(2) | M4(2) | C8oD4 |
| kernel | C8:9M4(2) | C8:C8 | C2xC4xC8 | C4xM4(2) | C42.12C4 | C4xC8 | C8:C4 | C22:C8 | C4:C8 | C22xC8 | C2xM4(2) | C8 | C2xC4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C8:9M4(2) ►in GL4(F17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 5 | 15 | 0 | 0 |
| 6 | 12 | 0 | 0 |
| 0 | 0 | 7 | 2 |
| 0 | 0 | 9 | 10 |
| 1 | 0 | 0 | 0 |
| 5 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,4,0,0,1,0],[5,6,0,0,15,12,0,0,0,0,7,9,0,0,2,10],[1,5,0,0,0,16,0,0,0,0,1,0,0,0,0,1] >;
C8:9M4(2) in GAP, Magma, Sage, TeX
C_8\rtimes_9M_4(2)
% in TeX
G:=Group("C8:9M4(2)"); // GroupNames label
G:=SmallGroup(128,183);
// by ID
G=gap.SmallGroup(128,183);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,925,120,758,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^5,a*c=c*a,c*b*c=b^5>;
// generators/relations