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, (C2xC6).5D4, C24:C2:1C2, C4oD12:2C2, (C2xD12):7C2, C3:1(C8:C22), C6.13(C2xD4), (C2xC4).15D6, C2.15(C2xD12), (C3xM4(2)):1C2, C4.30(C22xS3), (C2xC12).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, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C24, Dic6, C4xS3, D12, D12, D12, C3:D4, C2xC12, C22xS3, C8:C22, C24:C2, D24, C3xM4(2), C2xD12, C4oD12, C8:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C8:C22, C2xD12, 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 S3xC8: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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