metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.7D8, D12⋊3C4, C4.9D12, C12.1D4, C6.7SD16, C4⋊C4⋊1S3, C4.1(C4×S3), C12.3(C2×C4), (C2×C4).35D6, (C2×C6).30D4, C3⋊1(D4⋊C4), C2.5(D6⋊C4), C2.2(D4⋊S3), (C2×D12).5C2, C6.3(C22⋊C4), (C2×C12).10C22, C2.2(Q8⋊2S3), C22.14(C3⋊D4), (C2×C3⋊C8)⋊1C2, (C3×C4⋊C4)⋊1C2, SmallGroup(96,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.D8
G = < a,b,c | a6=b8=c2=1, bab-1=cac=a-1, cbc=a3b-1 >
Character table of C6.D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 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 | -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 | 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 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | 1 | -1 | -1 | -i | i | i | -i | i | -i | 1 | i | -1 | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | i | 1 | -1 | -1 | i | -i | -i | i | i | -i | 1 | i | -1 | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | -i | 1 | -1 | -1 | -i | i | i | -i | -i | i | 1 | -i | -1 | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | 1 | -1 | -1 | i | -i | -i | i | -i | i | 1 | -i | -1 | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | √3 | -√3 | 1 | -√3 | -1 | √3 | orthogonal lifted from D12 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -√3 | √3 | 1 | √3 | -1 | -√3 | orthogonal lifted from D12 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 2i | -2i | -1 | 1 | 1 | 0 | 0 | 0 | 0 | i | -i | -1 | i | 1 | -i | complex lifted from C4×S3 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -√-3 | -√-3 | 1 | √-3 | 1 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | -2i | 2i | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -i | i | -1 | -i | 1 | i | complex lifted from C4×S3 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | √-3 | √-3 | 1 | -√-3 | 1 | -√-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊2S3 |
►On 48 points
(1 34 20 15 26 41)(2 42 27 16 21 35)(3 36 22 9 28 43)(4 44 29 10 23 37)(5 38 24 11 30 45)(6 46 31 12 17 39)(7 40 18 13 32 47)(8 48 25 14 19 33)
(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)
(2 14)(3 7)(4 12)(6 10)(8 16)(9 13)(17 37)(18 28)(19 35)(20 26)(21 33)(22 32)(23 39)(24 30)(25 42)(27 48)(29 46)(31 44)(34 41)(36 47)(38 45)(40 43)
G:=sub<Sym(48)| (1,34,20,15,26,41)(2,42,27,16,21,35)(3,36,22,9,28,43)(4,44,29,10,23,37)(5,38,24,11,30,45)(6,46,31,12,17,39)(7,40,18,13,32,47)(8,48,25,14,19,33), (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), (2,14)(3,7)(4,12)(6,10)(8,16)(9,13)(17,37)(18,28)(19,35)(20,26)(21,33)(22,32)(23,39)(24,30)(25,42)(27,48)(29,46)(31,44)(34,41)(36,47)(38,45)(40,43)>;
G:=Group( (1,34,20,15,26,41)(2,42,27,16,21,35)(3,36,22,9,28,43)(4,44,29,10,23,37)(5,38,24,11,30,45)(6,46,31,12,17,39)(7,40,18,13,32,47)(8,48,25,14,19,33), (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), (2,14)(3,7)(4,12)(6,10)(8,16)(9,13)(17,37)(18,28)(19,35)(20,26)(21,33)(22,32)(23,39)(24,30)(25,42)(27,48)(29,46)(31,44)(34,41)(36,47)(38,45)(40,43) );
G=PermutationGroup([[(1,34,20,15,26,41),(2,42,27,16,21,35),(3,36,22,9,28,43),(4,44,29,10,23,37),(5,38,24,11,30,45),(6,46,31,12,17,39),(7,40,18,13,32,47),(8,48,25,14,19,33)], [(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)], [(2,14),(3,7),(4,12),(6,10),(8,16),(9,13),(17,37),(18,28),(19,35),(20,26),(21,33),(22,32),(23,39),(24,30),(25,42),(27,48),(29,46),(31,44),(34,41),(36,47),(38,45),(40,43)]])
C6.D8 is a maximal subgroup of
Dic3.SD16 Dic6⋊2D4 C4⋊C4.D6 S3×D4⋊C4 C4⋊C4⋊19D6 D4⋊D12 D6⋊5SD16 D12⋊3D4 (C2×C8).D6 Dic6.11D4 Q8⋊C4⋊S3 Q8⋊7(C4×S3) C4⋊C4.150D6 Q8.11D12 Q8⋊4D12 D12.12D4 Dic3⋊8SD16 D6.4SD16 C8⋊8D12 C24⋊7D4 C4.Q8⋊S3 D24⋊9C4 D12⋊Q8 D12.Q8 Dic3⋊5D8 D6.5D8 D6⋊2D8 C2.D8⋊S3 C8⋊3D12 C24⋊C2⋊C4 D12⋊2Q8 D12.2Q8 C4○D12⋊C4 (C2×C6).40D8 C4⋊C4.228D6 C4⋊C4⋊36D6 C4.(C2×D12) C4⋊C4.236D6 C4×D4⋊S3 C42.48D6 C12⋊7D8 D4.1D12 C4×Q8⋊2S3 C42.56D6 Q8⋊2D12 Q8.6D12 D12⋊16D4 D12⋊17D4 C3⋊C8⋊22D4 C4⋊D4⋊S3 D12.36D4 D12.37D4 C3⋊C8⋊24D4 C3⋊C8⋊6D4 D12.4Q8 C42.70D6 C42.216D6 D12⋊5Q8 D12⋊6Q8 C12.D8 C42.82D6 C18.D8 D12⋊3Dic3 C6.17D24 C62.113D4 D12⋊Dic5 D60⋊12C4 D60⋊9C4 D60⋊C4
C6.D8 is a maximal quotient of
C6.C4≀C2 C12.47D8 D12⋊2C8 C4.D24 D24⋊8C4 C6.D16 C6.Q32 D24.C4 C24.8D4 Dic12.C4 C12.C42 C18.D8 D12⋊3Dic3 C6.17D24 C62.113D4 D12⋊Dic5 D60⋊12C4 D60⋊9C4 D60⋊C4
Matrix representation of C6.D8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 57 | 16 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,1],[57,57,0,0,16,57,0,0,0,0,0,46,0,0,46,0],[1,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;
C6.D8 in GAP, Magma, Sage, TeX
C_6.D_8
% in TeX
G:=Group("C6.D8"); // GroupNames label
G:=SmallGroup(96,16);
// by ID
G=gap.SmallGroup(96,16);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,579,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^3*b^-1>;
// generators/relations
Export
Subgroup lattice of C6.D8 in TeX
Character table of C6.D8 in TeX