metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1D6, D24⋊2C2, C24⋊1C22, C12.12D4, C4.14D12, D12⋊4C22, M4(2)⋊1S3, C22.5D12, C12.32C23, Dic6⋊4C22, (C2×C6).5D4, C24⋊C2⋊1C2, C4○D12⋊2C2, (C2×D12)⋊7C2, C3⋊1(C8⋊C22), C6.13(C2×D4), (C2×C4).15D6, C2.15(C2×D12), (C3×M4(2))⋊1C2, C4.30(C22×S3), (C2×C12).27C22, SmallGroup(96,115)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D6
G = < a,b,c | a8=b6=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 210 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C24, Dic6, C4×S3, D12, D12, D12, C3⋊D4, C2×C12, C22×S3, C8⋊C22, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, C8⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C8⋊C22, C2×D12, C8⋊D6
Character table of C8⋊D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 12A | 12B | 12C | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 12 | 12 | 2 | 2 | 2 | 12 | 2 | 4 | 4 | 4 | 2 | 2 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | -1 | 1 | -2 | 2 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | -1 | 1 | 2 | -2 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | -1 | 1 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | -1 | 1 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ17 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 24 points - transitive group 24T107
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 13 21)(2 10 22 6 14 18)(3 15 23)(4 12 24 8 16 20)(5 9 17)(7 11 19)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 24)(7 23)(8 22)(10 16)(11 15)(12 14)
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,13,21)(2,10,22,6,14,18)(3,15,23)(4,12,24,8,16,20)(5,9,17)(7,11,19), (1,21)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(10,16)(11,15)(12,14)>;
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), (1,13,21)(2,10,22,6,14,18)(3,15,23)(4,12,24,8,16,20)(5,9,17)(7,11,19), (1,21)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(10,16)(11,15)(12,14) );
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)], [(1,13,21),(2,10,22,6,14,18),(3,15,23),(4,12,24,8,16,20),(5,9,17),(7,11,19)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,24),(7,23),(8,22),(10,16),(11,15),(12,14)]])
G:=TransitiveGroup(24,107);
C8⋊D6 is a maximal subgroup of
D12⋊1D4 D12.3D4 D12.5D4 D12.6D4 Q8⋊5D12 D4.10D12 C24.19D4 C24.42D4 C24.9C23 D4.11D12 D4.12D12 S3×C8⋊C22 D8⋊5D6 D24⋊C22 C24.C23 C8⋊D18 C24⋊1D6 D24⋊S3 D12⋊18D6 D12.28D6 C24⋊3D6 C24⋊D10 D24⋊D5 D60⋊36C22 C60.38D4 C8⋊D30
C8⋊D6 is a maximal quotient of
C8⋊Dic6 C42.16D6 D24⋊C4 C8⋊D12 C42.19D6 C42.20D6 C23.40D12 D12.31D4 D12⋊13D4 D12⋊14D4 C23.43D12 C23.18D12 D12⋊3Q8 C4⋊D24 D12.19D4 D12.3Q8 Dic6⋊8D4 Dic6⋊3Q8 C23.52D12 C23.53D12 C23.54D12 C24⋊2D4 C24⋊3D4 C8⋊D18 C24⋊1D6 D24⋊S3 D12⋊18D6 D12.28D6 C24⋊3D6 C24⋊D10 D24⋊D5 D60⋊36C22 C60.38D4 C8⋊D30
Matrix representation of C8⋊D6 ►in GL4(𝔽73) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 7 |
| 0 | 0 | 14 | 7 |
G:=sub<GL(4,GF(73))| [0,0,7,59,0,0,14,66,1,0,0,0,0,1,0,0],[1,72,0,0,1,0,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,72,1,0,0,0,0,66,14,0,0,7,7] >;
C8⋊D6 in GAP, Magma, Sage, TeX
C_8\rtimes D_6
% in TeX
G:=Group("C8:D6"); // GroupNames label
G:=SmallGroup(96,115);
// by ID
G=gap.SmallGroup(96,115);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,188,50,579,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^8=b^6=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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