p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊1D8, C42.236C23, C8⋊5D4⋊1C2, C8⋊6D4⋊1C2, C4⋊C4.58D4, C8⋊1C8⋊19C2, (C2×C8).88D4, C4.61(C2×D8), D4⋊Q8⋊7C2, (C2×D4).57D4, C4⋊D8.7C2, C4.D8⋊9C2, C2.8(C8⋊D4), C2.9(C4⋊D8), C4⋊C8.28C22, C4⋊Q8.59C22, C4.10D8⋊14C2, (C4×C8).139C22, (C4×D4).43C22, C4.120(C8⋊C22), C4⋊1D4.32C22, C4.70(C8.C22), C2.16(D4.3D4), C22.197(C4⋊D4), (C2×C4).21(C4○D4), (C2×C4).1271(C2×D4), SmallGroup(128,417)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊D8
G = < a,b,c | a8=b8=c2=1, bab-1=a-1, cac=a3, cbc=b-1 >
Subgroups: 248 in 91 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, 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), D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×D8, C2×SD16, C4.D8, C4.10D8, C8⋊1C8, C8⋊6D4, C4⋊D8, D4⋊Q8, C8⋊5D4, 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, D4.3D4, C8⋊D8
Character table of C8⋊D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 1 | 1 | 8 | 16 | 2 | 2 | 2 | 2 | 4 | 8 | 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 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | √2 | √2 | -√2 | -√2 | 0 | 0 | orthogonal lifted from D8 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -√2 | √2 | √2 | -√2 | 0 | 0 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -√2 | -√2 | √2 | √2 | 0 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | √2 | -√2 | -√2 | √2 | 0 | 0 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | complex lifted from C4○D4 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 4 | 0 | -4 | 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 | 0 | 4 | 0 | -4 | 0 | 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 | -4 | 0 | 4 | 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 | complex lifted from D4.3D4 |
| 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 | complex lifted from D4.3D4 |
►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 28 10 37 51 24 57 43)(2 27 11 36 52 23 58 42)(3 26 12 35 53 22 59 41)(4 25 13 34 54 21 60 48)(5 32 14 33 55 20 61 47)(6 31 15 40 56 19 62 46)(7 30 16 39 49 18 63 45)(8 29 9 38 50 17 64 44)
(2 4)(3 7)(6 8)(9 62)(10 57)(11 60)(12 63)(13 58)(14 61)(15 64)(16 59)(17 40)(18 35)(19 38)(20 33)(21 36)(22 39)(23 34)(24 37)(25 42)(26 45)(27 48)(28 43)(29 46)(30 41)(31 44)(32 47)(49 53)(50 56)(52 54)
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), (1,28,10,37,51,24,57,43)(2,27,11,36,52,23,58,42)(3,26,12,35,53,22,59,41)(4,25,13,34,54,21,60,48)(5,32,14,33,55,20,61,47)(6,31,15,40,56,19,62,46)(7,30,16,39,49,18,63,45)(8,29,9,38,50,17,64,44), (2,4)(3,7)(6,8)(9,62)(10,57)(11,60)(12,63)(13,58)(14,61)(15,64)(16,59)(17,40)(18,35)(19,38)(20,33)(21,36)(22,39)(23,34)(24,37)(25,42)(26,45)(27,48)(28,43)(29,46)(30,41)(31,44)(32,47)(49,53)(50,56)(52,54)>;
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), (1,28,10,37,51,24,57,43)(2,27,11,36,52,23,58,42)(3,26,12,35,53,22,59,41)(4,25,13,34,54,21,60,48)(5,32,14,33,55,20,61,47)(6,31,15,40,56,19,62,46)(7,30,16,39,49,18,63,45)(8,29,9,38,50,17,64,44), (2,4)(3,7)(6,8)(9,62)(10,57)(11,60)(12,63)(13,58)(14,61)(15,64)(16,59)(17,40)(18,35)(19,38)(20,33)(21,36)(22,39)(23,34)(24,37)(25,42)(26,45)(27,48)(28,43)(29,46)(30,41)(31,44)(32,47)(49,53)(50,56)(52,54) );
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)], [(1,28,10,37,51,24,57,43),(2,27,11,36,52,23,58,42),(3,26,12,35,53,22,59,41),(4,25,13,34,54,21,60,48),(5,32,14,33,55,20,61,47),(6,31,15,40,56,19,62,46),(7,30,16,39,49,18,63,45),(8,29,9,38,50,17,64,44)], [(2,4),(3,7),(6,8),(9,62),(10,57),(11,60),(12,63),(13,58),(14,61),(15,64),(16,59),(17,40),(18,35),(19,38),(20,33),(21,36),(22,39),(23,34),(24,37),(25,42),(26,45),(27,48),(28,43),(29,46),(30,41),(31,44),(32,47),(49,53),(50,56),(52,54)]])
Matrix representation of C8⋊D8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 11 | 15 | 15 |
| 0 | 0 | 5 | 11 | 16 | 15 |
| 0 | 0 | 11 | 2 | 6 | 6 |
| 0 | 0 | 1 | 11 | 12 | 15 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 14 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 0 | 16 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,2,5,11,1,0,0,11,11,2,11,0,0,15,16,6,12,0,0,15,15,6,15],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,1,0,0,1,0,0,0,16,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,2,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;
C8⋊D8 in GAP, Magma, Sage, TeX
C_8\rtimes D_8
% in TeX
G:=Group("C8:D8"); // GroupNames label
G:=SmallGroup(128,417);
// by ID
G=gap.SmallGroup(128,417);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,387,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations
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