p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8:9D8, D4:1M4(2), C42.638C23, D4:C8:6C2, C2.4(C4xD8), (C8xD4):34C2, (C4xD8).2C2, (C2xD8).6C4, C8:1C8:29C2, C8:C8:16C2, C4.86(C2xD8), C2.D8.9C4, C8:6D4:28C2, (C2xC8).377D4, D4:C4.3C4, C4.30(C8oD4), C2.8(C8:9D4), (C4xC8).38C22, (C4xD4).9C22, C4.128(C4oD8), C2.6(C8.26D4), C4:C8.220C22, C22.129(C4xD4), C4.24(C2xM4(2)), (C2xC8).30(C2xC4), C4:C4.132(C2xC4), (C2xD4).152(C2xC4), (C2xC4).1474(C2xD4), (C2xC4).499(C4oD4), (C2xC4).330(C22xC4), SmallGroup(128,313)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8:9D8
G = < a,b,c | a8=b8=c2=1, bab-1=cac=a5, cbc=b-1 >
Subgroups: 184 in 91 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), D8, C22xC4, C2xD4, C4xC8, C22:C8, D4:C4, C4:C8, C2.D8, C4xD4, C22xC8, C2xM4(2), C2xD8, C8:C8, D4:C8, C8:1C8, C8xD4, C8:6D4, C4xD8, C8:9D8
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, M4(2), D8, C22xC4, C2xD4, C4oD4, C4xD4, C2xM4(2), C8oD4, C2xD8, C4oD8, C8:9D4, C4xD8, C8.26D4, C8:9D8
►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 54 37 23 15 63 29 44)(2 51 38 20 16 60 30 41)(3 56 39 17 9 57 31 46)(4 53 40 22 10 62 32 43)(5 50 33 19 11 59 25 48)(6 55 34 24 12 64 26 45)(7 52 35 21 13 61 27 42)(8 49 36 18 14 58 28 47)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 56)(18 53)(19 50)(20 55)(21 52)(22 49)(23 54)(24 51)(26 30)(28 32)(34 38)(36 40)(41 64)(42 61)(43 58)(44 63)(45 60)(46 57)(47 62)(48 59)
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), (1,54,37,23,15,63,29,44)(2,51,38,20,16,60,30,41)(3,56,39,17,9,57,31,46)(4,53,40,22,10,62,32,43)(5,50,33,19,11,59,25,48)(6,55,34,24,12,64,26,45)(7,52,35,21,13,61,27,42)(8,49,36,18,14,58,28,47), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,56)(18,53)(19,50)(20,55)(21,52)(22,49)(23,54)(24,51)(26,30)(28,32)(34,38)(36,40)(41,64)(42,61)(43,58)(44,63)(45,60)(46,57)(47,62)(48,59)>;
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), (1,54,37,23,15,63,29,44)(2,51,38,20,16,60,30,41)(3,56,39,17,9,57,31,46)(4,53,40,22,10,62,32,43)(5,50,33,19,11,59,25,48)(6,55,34,24,12,64,26,45)(7,52,35,21,13,61,27,42)(8,49,36,18,14,58,28,47), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,56)(18,53)(19,50)(20,55)(21,52)(22,49)(23,54)(24,51)(26,30)(28,32)(34,38)(36,40)(41,64)(42,61)(43,58)(44,63)(45,60)(46,57)(47,62)(48,59) );
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)], [(1,54,37,23,15,63,29,44),(2,51,38,20,16,60,30,41),(3,56,39,17,9,57,31,46),(4,53,40,22,10,62,32,43),(5,50,33,19,11,59,25,48),(6,55,34,24,12,64,26,45),(7,52,35,21,13,61,27,42),(8,49,36,18,14,58,28,47)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10),(17,56),(18,53),(19,50),(20,55),(21,52),(22,49),(23,54),(24,51),(26,30),(28,32),(34,38),(36,40),(41,64),(42,61),(43,58),(44,63),(45,60),(46,57),(47,62),(48,59)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | ··· | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D8 | C4oD4 | M4(2) | C8oD4 | C4oD8 | C8.26D4 |
| kernel | C8:9D8 | C8:C8 | D4:C8 | C8:1C8 | C8xD4 | C8:6D4 | C4xD8 | D4:C4 | C2.D8 | C2xD8 | C2xC8 | C8 | C2xC4 | D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C8:9D8 ►in GL4(F17) generated by
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 14 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(17))| [0,4,0,0,1,0,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,1,0,0,0,0,0,14,0,0,6,11],[1,0,0,0,0,16,0,0,0,0,16,1,0,0,0,1] >;
C8:9D8 in GAP, Magma, Sage, TeX
C_8\rtimes_9D_8
% in TeX
G:=Group("C8:9D8"); // GroupNames label
G:=SmallGroup(128,313);
// by ID
G=gap.SmallGroup(128,313);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,100,1123,570,136,172]);
// 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^-1>;
// generators/relations