metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.1D6, C4.15D12, C12.13D4, Dic12⋊2C2, M4(2)⋊2S3, C24.1C22, C22.6D12, C12.33C23, D12.8C22, Dic6.8C22, (C2×C6).6D4, C24⋊C2⋊2C2, C6.14(C2×D4), (C2×C4).16D6, (C2×Dic6)⋊8C2, C4○D12.4C2, C2.16(C2×D12), C3⋊1(C8.C22), (C3×M4(2))⋊2C2, C4.31(C22×S3), (C2×C12).28C22, SmallGroup(96,116)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D6
G = < a,b,c | a8=1, b6=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b5 >
Subgroups: 146 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], Dic3 [×3], C12 [×2], D6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C24 [×2], Dic6, Dic6 [×2], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, C24⋊C2 [×2], Dic12 [×2], C3×M4(2), C2×Dic6, C4○D12, C8.D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D12 [×2], C22×S3, C8.C22, C2×D12, C8.D6
Character table of C8.D6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 12A | 12B | 12C | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 12 | 12 | 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 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | 2 | -2 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | -2 | 2 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ16 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ17 | 2 | 2 | -2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | 2 | -2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
(1 44 31 19 7 38 25 13)(2 39 32 14 8 45 26 20)(3 46 33 21 9 40 27 15)(4 41 34 16 10 47 28 22)(5 48 35 23 11 42 29 17)(6 43 36 18 12 37 30 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)(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 43 19 37)(14 48 20 42)(15 41 21 47)(16 46 22 40)(17 39 23 45)(18 44 24 38)(25 36 31 30)(26 29 32 35)(27 34 33 28)
G:=sub<Sym(48)| (1,44,31,19,7,38,25,13)(2,39,32,14,8,45,26,20)(3,46,33,21,9,40,27,15)(4,41,34,16,10,47,28,22)(5,48,35,23,11,42,29,17)(6,43,36,18,12,37,30,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,43,19,37)(14,48,20,42)(15,41,21,47)(16,46,22,40)(17,39,23,45)(18,44,24,38)(25,36,31,30)(26,29,32,35)(27,34,33,28)>;
G:=Group( (1,44,31,19,7,38,25,13)(2,39,32,14,8,45,26,20)(3,46,33,21,9,40,27,15)(4,41,34,16,10,47,28,22)(5,48,35,23,11,42,29,17)(6,43,36,18,12,37,30,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,43,19,37)(14,48,20,42)(15,41,21,47)(16,46,22,40)(17,39,23,45)(18,44,24,38)(25,36,31,30)(26,29,32,35)(27,34,33,28) );
G=PermutationGroup([(1,44,31,19,7,38,25,13),(2,39,32,14,8,45,26,20),(3,46,33,21,9,40,27,15),(4,41,34,16,10,47,28,22),(5,48,35,23,11,42,29,17),(6,43,36,18,12,37,30,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,43,19,37),(14,48,20,42),(15,41,21,47),(16,46,22,40),(17,39,23,45),(18,44,24,38),(25,36,31,30),(26,29,32,35),(27,34,33,28)])
C8.D6 is a maximal subgroup of
M4(2)⋊D6 D12.2D4 D12.4D4 D12.7D4 C42⋊5D6 Q8.14D12 C24.18D4 C24.42D4 C24.9C23 D4.11D12 D4.13D12 D8⋊4D6 D8⋊6D6 S3×C8.C22 SD16.D6 C8.D18 C24.3D6 Dic12⋊S3 D12.29D6 Dic6.29D6 C24.5D6 Dic60⋊C2 C24.2D10 C20.D12 D12.33D10 C8.D30
C8.D6 is a maximal quotient of
C8⋊Dic6 C42.14D6 C42.16D6 C42.20D6 C8.D12 Dic12⋊C4 C23.39D12 C23.15D12 D12.32D4 C22.D24 Dic6⋊14D4 Dic6.32D4 Dic6.3Q8 C12⋊SD16 C42.36D6 D12⋊4Q8 C4⋊Dic12 Dic6⋊4Q8 C23.51D12 C23.52D12 C23.54D12 C24⋊2D4 C24.4D4 C8.D18 C24.3D6 Dic12⋊S3 D12.29D6 Dic6.29D6 C24.5D6 Dic60⋊C2 C24.2D10 C20.D12 D12.33D10 C8.D30
Matrix representation of C8.D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 72 | 71 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 46 | 46 | 0 | 27 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 27 | 46 | 19 |
| 0 | 0 | 46 | 0 | 0 | 27 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,1,1,0,0,0,1,72,0,0,0,0,0,71,0,72],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,46,0,0,46,0,0,0,46,0,46,0,0,0,0,27,0,0,0,0,0,0,27],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,46,0,27,46,0,0,0,27,27,0,0,0,0,0,46,0,0,0,0,0,19,27] >;
C8.D6 in GAP, Magma, Sage, TeX
C_8.D_6
% in TeX
G:=Group("C8.D6"); // GroupNames label
G:=SmallGroup(96,116);
// by ID
G=gap.SmallGroup(96,116);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,50,579,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^6=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations
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