p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊4D8, C82⋊6C2, C42.659C23, C4.3(C2×D8), C8⋊4D4⋊4C2, (C2×C8).241D4, C2.6(C8⋊4D4), (C4×C8).394C22, C4⋊1D4.45C22, C22.60(C4⋊1D4), (C2×C4).716(C2×D4), SmallGroup(128,444)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊4D8
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 464 in 140 conjugacy classes, 48 normal (5 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, D4, C23, C42, C2×C8, D8, C2×D4, C4×C8, C4⋊1D4, C2×D8, C82, C8⋊4D4, C8⋊4D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4⋊1D4, C2×D8, C8⋊4D4, C8⋊4D8
►On 64 points
Generators in S64
(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 38 23 25 11 45 61 50)(2 39 24 26 12 46 62 51)(3 40 17 27 13 47 63 52)(4 33 18 28 14 48 64 53)(5 34 19 29 15 41 57 54)(6 35 20 30 16 42 58 55)(7 36 21 31 9 43 59 56)(8 37 22 32 10 44 60 49)
(1 61)(2 60)(3 59)(4 58)(5 57)(6 64)(7 63)(8 62)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(26 32)(27 31)(28 30)(33 42)(34 41)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(49 51)(52 56)(53 55)
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), (1,38,23,25,11,45,61,50)(2,39,24,26,12,46,62,51)(3,40,17,27,13,47,63,52)(4,33,18,28,14,48,64,53)(5,34,19,29,15,41,57,54)(6,35,20,30,16,42,58,55)(7,36,21,31,9,43,59,56)(8,37,22,32,10,44,60,49), (1,61)(2,60)(3,59)(4,58)(5,57)(6,64)(7,63)(8,62)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(26,32)(27,31)(28,30)(33,42)(34,41)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(49,51)(52,56)(53,55)>;
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), (1,38,23,25,11,45,61,50)(2,39,24,26,12,46,62,51)(3,40,17,27,13,47,63,52)(4,33,18,28,14,48,64,53)(5,34,19,29,15,41,57,54)(6,35,20,30,16,42,58,55)(7,36,21,31,9,43,59,56)(8,37,22,32,10,44,60,49), (1,61)(2,60)(3,59)(4,58)(5,57)(6,64)(7,63)(8,62)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(26,32)(27,31)(28,30)(33,42)(34,41)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(49,51)(52,56)(53,55) );
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)], [(1,38,23,25,11,45,61,50),(2,39,24,26,12,46,62,51),(3,40,17,27,13,47,63,52),(4,33,18,28,14,48,64,53),(5,34,19,29,15,41,57,54),(6,35,20,30,16,42,58,55),(7,36,21,31,9,43,59,56),(8,37,22,32,10,44,60,49)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,64),(7,63),(8,62),(9,17),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(26,32),(27,31),(28,30),(33,42),(34,41),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(49,51),(52,56),(53,55)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 8A | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D8 |
| kernel | C8⋊4D8 | C82 | C8⋊4D4 | C2×C8 | C8 |
| # reps | 1 | 1 | 6 | 6 | 24 |
Matrix representation of C8⋊4D8 ►in GL4(𝔽17) generated by
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,2,1] >;
C8⋊4D8 in GAP, Magma, Sage, TeX
C_8\rtimes_4D_8
% in TeX
G:=Group("C8:4D8"); // GroupNames label
G:=SmallGroup(128,444);
// by ID
G=gap.SmallGroup(128,444);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,436,1123,136,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations