direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C6×D8, C12.41D4, C24⋊10C22, C12.44C23, C8⋊2(C2×C6), (C2×C8)⋊3C6, (C2×C24)⋊8C2, (C2×D4)⋊4C6, D4⋊1(C2×C6), C4.6(C3×D4), (C6×D4)⋊13C2, C6.74(C2×D4), C2.11(C6×D4), (C2×C6).52D4, C4.1(C22×C6), (C3×D4)⋊10C22, C22.14(C3×D4), (C2×C12).129C22, (C2×C4).25(C2×C6), SmallGroup(96,179)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D8
G = < a,b,c | a6=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, D4, D4, C23, C12, C2×C6, C2×C6, C2×C8, D8, C2×D4, C24, C2×C12, C3×D4, C3×D4, C22×C6, C2×D8, C2×C24, C3×D8, C6×D4, C6×D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, C2×D8, C3×D8, C6×D4, C6×D8
►On 48 points
(1 14 39 22 29 45)(2 15 40 23 30 46)(3 16 33 24 31 47)(4 9 34 17 32 48)(5 10 35 18 25 41)(6 11 36 19 26 42)(7 12 37 20 27 43)(8 13 38 21 28 44)
(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)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)
G:=sub<Sym(48)| (1,14,39,22,29,45)(2,15,40,23,30,46)(3,16,33,24,31,47)(4,9,34,17,32,48)(5,10,35,18,25,41)(6,11,36,19,26,42)(7,12,37,20,27,43)(8,13,38,21,28,44), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)>;
G:=Group( (1,14,39,22,29,45)(2,15,40,23,30,46)(3,16,33,24,31,47)(4,9,34,17,32,48)(5,10,35,18,25,41)(6,11,36,19,26,42)(7,12,37,20,27,43)(8,13,38,21,28,44), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45) );
G=PermutationGroup([[(1,14,39,22,29,45),(2,15,40,23,30,46),(3,16,33,24,31,47),(4,9,34,17,32,48),(5,10,35,18,25,41),(6,11,36,19,26,42),(7,12,37,20,27,43),(8,13,38,21,28,44)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,18),(19,24),(20,23),(21,22),(25,32),(26,31),(27,30),(28,29),(33,36),(34,35),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45)]])
C6×D8 is a maximal subgroup of
D8⋊1Dic3 D8.Dic3 D8.D6 Dic3⋊D8 C24⋊5D4 D8⋊Dic3 (C6×D8).C2 C24⋊11D4 C24.22D4 D12⋊D4 D6⋊3D8 Dic6⋊D4 C24⋊12D4 C24.23D4 D8⋊13D6
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6N | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | D8 | C3×D4 | C3×D4 | C3×D8 |
| kernel | C6×D8 | C2×C24 | C3×D8 | C6×D4 | C2×D8 | C2×C8 | D8 | C2×D4 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 1 | 1 | 4 | 2 | 2 | 8 |
Matrix representation of C6×D8 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 72 | 0 | 0 |
| 0 | 41 | 57 |
| 0 | 32 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 57 |
| 0 | 41 | 0 |
G:=sub<GL(3,GF(73))| [72,0,0,0,64,0,0,0,64],[72,0,0,0,41,32,0,57,0],[1,0,0,0,0,41,0,57,0] >;
C6×D8 in GAP, Magma, Sage, TeX
C_6\times D_8
% in TeX
G:=Group("C6xD8"); // GroupNames label
G:=SmallGroup(96,179);
// by ID
G=gap.SmallGroup(96,179);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,2164,1090,88]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations