p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊2D8, C42.238C23, C4⋊D8⋊6C2, C8⋊6D4⋊2C2, C4⋊C4.60D4, C8⋊2C8⋊14C2, (C2×C8).90D4, C4.62(C2×D8), C8⋊4D4⋊14C2, (C2×D4).58D4, C4.D8⋊10C2, C2.6(C8⋊2D4), C4⋊C8.29C22, C2.10(C4⋊D8), C4.69(C8⋊C22), (C4×C8).141C22, (C4×D4).44C22, C4⋊1D4.34C22, C2.12(D4.4D4), C22.199(C4⋊D4), (C2×C4).23(C4○D4), (C2×C4).1273(C2×D4), SmallGroup(128,419)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊2D8
G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a-1, cbc=b-1 >
Subgroups: 304 in 99 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C4⋊C8, C4×D4, C4⋊1D4, C2×M4(2), C2×D8, C4.D8, C8⋊2C8, C8⋊6D4, C4⋊D8, C8⋊4D4, C8⋊2D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C8⋊2D4, D4.4D4, C8⋊2D8
Character table of C8⋊2D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 1 | 1 | 8 | 16 | 16 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | √2 | 0 | √2 | -√2 | 0 | -√2 | orthogonal lifted from D8 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -√2 | 0 | √2 | -√2 | 0 | √2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | √2 | 0 | -√2 | √2 | 0 | -√2 | orthogonal lifted from D8 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -√2 | 0 | -√2 | √2 | 0 | √2 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | complex lifted from C4○D4 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ20 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 0 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
►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 23 37 12 55 30 57 45)(2 18 38 15 56 25 58 48)(3 21 39 10 49 28 59 43)(4 24 40 13 50 31 60 46)(5 19 33 16 51 26 61 41)(6 22 34 11 52 29 62 44)(7 17 35 14 53 32 63 47)(8 20 36 9 54 27 64 42)
(2 8)(3 7)(4 6)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 43)(18 42)(19 41)(20 48)(21 47)(22 46)(23 45)(24 44)(33 61)(34 60)(35 59)(36 58)(37 57)(38 64)(39 63)(40 62)(49 53)(50 52)(54 56)
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), (1,23,37,12,55,30,57,45)(2,18,38,15,56,25,58,48)(3,21,39,10,49,28,59,43)(4,24,40,13,50,31,60,46)(5,19,33,16,51,26,61,41)(6,22,34,11,52,29,62,44)(7,17,35,14,53,32,63,47)(8,20,36,9,54,27,64,42), (2,8)(3,7)(4,6)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,43)(18,42)(19,41)(20,48)(21,47)(22,46)(23,45)(24,44)(33,61)(34,60)(35,59)(36,58)(37,57)(38,64)(39,63)(40,62)(49,53)(50,52)(54,56)>;
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), (1,23,37,12,55,30,57,45)(2,18,38,15,56,25,58,48)(3,21,39,10,49,28,59,43)(4,24,40,13,50,31,60,46)(5,19,33,16,51,26,61,41)(6,22,34,11,52,29,62,44)(7,17,35,14,53,32,63,47)(8,20,36,9,54,27,64,42), (2,8)(3,7)(4,6)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,43)(18,42)(19,41)(20,48)(21,47)(22,46)(23,45)(24,44)(33,61)(34,60)(35,59)(36,58)(37,57)(38,64)(39,63)(40,62)(49,53)(50,52)(54,56) );
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)], [(1,23,37,12,55,30,57,45),(2,18,38,15,56,25,58,48),(3,21,39,10,49,28,59,43),(4,24,40,13,50,31,60,46),(5,19,33,16,51,26,61,41),(6,22,34,11,52,29,62,44),(7,17,35,14,53,32,63,47),(8,20,36,9,54,27,64,42)], [(2,8),(3,7),(4,6),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,43),(18,42),(19,41),(20,48),(21,47),(22,46),(23,45),(24,44),(33,61),(34,60),(35,59),(36,58),(37,57),(38,64),(39,63),(40,62),(49,53),(50,52),(54,56)]])
Matrix representation of C8⋊2D8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 14 | 3 |
| 0 | 0 | 16 | 16 | 14 | 14 |
| 0 | 0 | 3 | 14 | 1 | 16 |
| 0 | 0 | 3 | 3 | 1 | 1 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 14 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,3,3,0,0,1,16,14,3,0,0,14,14,1,1,0,0,3,14,16,1],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
C8⋊2D8 in GAP, Magma, Sage, TeX
C_8\rtimes_2D_8
% in TeX
G:=Group("C8:2D8"); // GroupNames label
G:=SmallGroup(128,419);
// by ID
G=gap.SmallGroup(128,419);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,387,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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