p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.1D8, C42.240C23, C4⋊C4.62D4, C8⋊2C8⋊16C2, C4.63(C2×D8), (C2×C8).92D4, (C2×D4).59D4, C8⋊6D4.1C2, C4⋊Q16⋊14C2, C4⋊C8.30C22, D4⋊Q8.7C2, C4⋊Q8.61C22, C4.10D8⋊15C2, C2.11(C4⋊D8), (C4×C8).143C22, C2.6(C8.D4), (C4×D4).45C22, C4.122(C8⋊C22), C4.42(C8.C22), C2.12(D4.5D4), C22.201(C4⋊D4), (C2×C4).25(C4○D4), (C2×C4).1275(C2×D4), SmallGroup(128,421)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.D8
G = < a,b,c | a8=b8=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 192 in 83 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C2×M4(2), C2×Q16, C4.10D8, C8⋊2C8, C8⋊6D4, D4⋊Q8, C4⋊Q16, C8.D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C8⋊C22, C8.C22, C4⋊D8, C8.D4, D4.5D4, C8.D8
Character table of C8.D8
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 16 | 16 | 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 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -√2 | -√2 | √2 | 0 | 0 | √2 | orthogonal lifted from D8 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | √2 | -√2 | -√2 | 0 | 0 | √2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | √2 | √2 | -√2 | 0 | 0 | -√2 | orthogonal lifted from D8 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -√2 | √2 | √2 | 0 | 0 | -√2 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | complex lifted from C4○D4 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 4 | 0 | -4 | 0 | 0 | 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 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| 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 | symplectic lifted from D4.5D4, Schur index 2 |
| 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 | symplectic lifted from D4.5D4, Schur index 2 |
►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 22 38 64 55 27 43 16)(2 17 39 59 56 30 44 11)(3 20 40 62 49 25 45 14)(4 23 33 57 50 28 46 9)(5 18 34 60 51 31 47 12)(6 21 35 63 52 26 48 15)(7 24 36 58 53 29 41 10)(8 19 37 61 54 32 42 13)
(1 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 48 21 44)(18 47 22 43)(19 46 23 42)(20 45 24 41)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(49 58 53 62)(50 57 54 61)(51 64 55 60)(52 63 56 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), (1,22,38,64,55,27,43,16)(2,17,39,59,56,30,44,11)(3,20,40,62,49,25,45,14)(4,23,33,57,50,28,46,9)(5,18,34,60,51,31,47,12)(6,21,35,63,52,26,48,15)(7,24,36,58,53,29,41,10)(8,19,37,61,54,32,42,13), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,48,21,44)(18,47,22,43)(19,46,23,42)(20,45,24,41)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(49,58,53,62)(50,57,54,61)(51,64,55,60)(52,63,56,59)>;
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), (1,22,38,64,55,27,43,16)(2,17,39,59,56,30,44,11)(3,20,40,62,49,25,45,14)(4,23,33,57,50,28,46,9)(5,18,34,60,51,31,47,12)(6,21,35,63,52,26,48,15)(7,24,36,58,53,29,41,10)(8,19,37,61,54,32,42,13), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,48,21,44)(18,47,22,43)(19,46,23,42)(20,45,24,41)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(49,58,53,62)(50,57,54,61)(51,64,55,60)(52,63,56,59) );
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)], [(1,22,38,64,55,27,43,16),(2,17,39,59,56,30,44,11),(3,20,40,62,49,25,45,14),(4,23,33,57,50,28,46,9),(5,18,34,60,51,31,47,12),(6,21,35,63,52,26,48,15),(7,24,36,58,53,29,41,10),(8,19,37,61,54,32,42,13)], [(1,12,5,16),(2,11,6,15),(3,10,7,14),(4,9,8,13),(17,48,21,44),(18,47,22,43),(19,46,23,42),(20,45,24,41),(25,40,29,36),(26,39,30,35),(27,38,31,34),(28,37,32,33),(49,58,53,62),(50,57,54,61),(51,64,55,60),(52,63,56,59)]])
Matrix representation of C8.D8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,16,0,0],[0,3,0,0,0,0,6,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
C8.D8 in GAP, Magma, Sage, TeX
C_8.D_8
% in TeX
G:=Group("C8.D8"); // GroupNames label
G:=SmallGroup(128,421);
// by ID
G=gap.SmallGroup(128,421);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,387,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations
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