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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C24, Dic6, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, 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 | 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 |
(1 40 27 18 7 46 33 24)(2 47 28 13 8 41 34 19)(3 42 29 20 9 48 35 14)(4 37 30 15 10 43 36 21)(5 44 31 22 11 38 25 16)(6 39 32 17 12 45 26 23)
(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 44 19 38)(14 37 20 43)(15 42 21 48)(16 47 22 41)(17 40 23 46)(18 45 24 39)(25 28 31 34)(26 33 32 27)(29 36 35 30)
G:=sub<Sym(48)| (1,40,27,18,7,46,33,24)(2,47,28,13,8,41,34,19)(3,42,29,20,9,48,35,14)(4,37,30,15,10,43,36,21)(5,44,31,22,11,38,25,16)(6,39,32,17,12,45,26,23), (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)(13,44,19,38)(14,37,20,43)(15,42,21,48)(16,47,22,41)(17,40,23,46)(18,45,24,39)(25,28,31,34)(26,33,32,27)(29,36,35,30)>;
G:=Group( (1,40,27,18,7,46,33,24)(2,47,28,13,8,41,34,19)(3,42,29,20,9,48,35,14)(4,37,30,15,10,43,36,21)(5,44,31,22,11,38,25,16)(6,39,32,17,12,45,26,23), (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)(13,44,19,38)(14,37,20,43)(15,42,21,48)(16,47,22,41)(17,40,23,46)(18,45,24,39)(25,28,31,34)(26,33,32,27)(29,36,35,30) );
G=PermutationGroup([[(1,40,27,18,7,46,33,24),(2,47,28,13,8,41,34,19),(3,42,29,20,9,48,35,14),(4,37,30,15,10,43,36,21),(5,44,31,22,11,38,25,16),(6,39,32,17,12,45,26,23)], [(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),(13,44,19,38),(14,37,20,43),(15,42,21,48),(16,47,22,41),(17,40,23,46),(18,45,24,39),(25,28,31,34),(26,33,32,27),(29,36,35,30)]])
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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