p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8:5D8, C82:7C2, C8:7SD16, C42.653C23, C4.1(C2xD8), C8:2Q8:6C2, C8:5D4:17C2, C4.1(C4oD8), C8:4D4.5C2, C4.4D8:7C2, (C2xC8).220D4, C4.2(C2xSD16), C2.6(C8:5D4), C2.4(C8:4D4), C4:Q8.78C22, (C4xC8).427C22, C2.6(C8.12D4), C4:1D4.41C22, C22.54(C4:1D4), (C2xC4).710(C2xD4), SmallGroup(128,438)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8:5D8
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a3, cbc=b-1 >
Subgroups: 320 in 112 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, Q8, C23, C42, C4:C4, C2xC8, D8, SD16, C2xD4, C2xQ8, C4xC8, D4:C4, C2.D8, C4:1D4, C4:Q8, C2xD8, C2xSD16, C82, C4.4D8, C8:5D4, C8:4D4, C8:2Q8, C8:5D8
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2xD4, C4:1D4, C2xD8, C2xSD16, C4oD8, C8:5D4, C8:4D4, C8.12D4, C8:5D8
►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 63 23 29 47 38 16 54)(2 64 24 30 48 39 9 55)(3 57 17 31 41 40 10 56)(4 58 18 32 42 33 11 49)(5 59 19 25 43 34 12 50)(6 60 20 26 44 35 13 51)(7 61 21 27 45 36 14 52)(8 62 22 28 46 37 15 53)
(1 54)(2 49)(3 52)(4 55)(5 50)(6 53)(7 56)(8 51)(9 58)(10 61)(11 64)(12 59)(13 62)(14 57)(15 60)(16 63)(17 36)(18 39)(19 34)(20 37)(21 40)(22 35)(23 38)(24 33)(25 43)(26 46)(27 41)(28 44)(29 47)(30 42)(31 45)(32 48)
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), (1,63,23,29,47,38,16,54)(2,64,24,30,48,39,9,55)(3,57,17,31,41,40,10,56)(4,58,18,32,42,33,11,49)(5,59,19,25,43,34,12,50)(6,60,20,26,44,35,13,51)(7,61,21,27,45,36,14,52)(8,62,22,28,46,37,15,53), (1,54)(2,49)(3,52)(4,55)(5,50)(6,53)(7,56)(8,51)(9,58)(10,61)(11,64)(12,59)(13,62)(14,57)(15,60)(16,63)(17,36)(18,39)(19,34)(20,37)(21,40)(22,35)(23,38)(24,33)(25,43)(26,46)(27,41)(28,44)(29,47)(30,42)(31,45)(32,48)>;
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), (1,63,23,29,47,38,16,54)(2,64,24,30,48,39,9,55)(3,57,17,31,41,40,10,56)(4,58,18,32,42,33,11,49)(5,59,19,25,43,34,12,50)(6,60,20,26,44,35,13,51)(7,61,21,27,45,36,14,52)(8,62,22,28,46,37,15,53), (1,54)(2,49)(3,52)(4,55)(5,50)(6,53)(7,56)(8,51)(9,58)(10,61)(11,64)(12,59)(13,62)(14,57)(15,60)(16,63)(17,36)(18,39)(19,34)(20,37)(21,40)(22,35)(23,38)(24,33)(25,43)(26,46)(27,41)(28,44)(29,47)(30,42)(31,45)(32,48) );
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)], [(1,63,23,29,47,38,16,54),(2,64,24,30,48,39,9,55),(3,57,17,31,41,40,10,56),(4,58,18,32,42,33,11,49),(5,59,19,25,43,34,12,50),(6,60,20,26,44,35,13,51),(7,61,21,27,45,36,14,52),(8,62,22,28,46,37,15,53)], [(1,54),(2,49),(3,52),(4,55),(5,50),(6,53),(7,56),(8,51),(9,58),(10,61),(11,64),(12,59),(13,62),(14,57),(15,60),(16,63),(17,36),(18,39),(19,34),(20,37),(21,40),(22,35),(23,38),(24,33),(25,43),(26,46),(27,41),(28,44),(29,47),(30,42),(31,45),(32,48)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 8A | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 2 | ··· | 2 | 16 | 16 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D8 | SD16 | C4oD8 |
| kernel | C8:5D8 | C82 | C4.4D8 | C8:5D4 | C8:4D4 | C8:2Q8 | C2xC8 | C8 | C8 | C4 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 8 | 8 | 8 |
Matrix representation of C8:5D8 ►in GL4(F17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 12 | 5 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 14 | 3 |
| 14 | 3 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 3 | 14 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,5,12,0,0,5,5],[3,3,0,0,14,3,0,0,0,0,3,14,0,0,3,3],[14,3,0,0,3,3,0,0,0,0,3,3,0,0,3,14] >;
C8:5D8 in GAP, Magma, Sage, TeX
C_8\rtimes_5D_8
% in TeX
G:=Group("C8:5D8"); // GroupNames label
G:=SmallGroup(128,438);
// by ID
G=gap.SmallGroup(128,438);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,100,1123,136,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations