metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D6, Q8⋊3D6, D24⋊6C2, D4.3D6, D6.7D4, C24⋊3C22, SD16⋊1S3, D12⋊2C22, C12.5C23, Dic3.9D4, D4⋊S3⋊3C2, (S3×D4)⋊3C2, C3⋊C8⋊2C22, C8⋊S3⋊1C2, C3⋊3(C8⋊C22), C2.19(S3×D4), C6.31(C2×D4), Q8⋊3S3⋊1C2, Q8⋊2S3⋊2C2, (C3×SD16)⋊1C2, C4.5(C22×S3), (C3×Q8)⋊2C22, (C4×S3).2C22, (C3×D4).3C22, SmallGroup(96,121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3D6
G = < a,b,c,d | a4=c6=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 202 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, M4(2), D8, SD16, SD16, C2×D4, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, Q8⋊3D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8⋊C22, S3×D4, Q8⋊3D6
Character table of Q8⋊3D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 12A | 12B | 24A | 24B | |
| size | 1 | 1 | 4 | 6 | 12 | 12 | 2 | 2 | 4 | 6 | 2 | 8 | 4 | 12 | 4 | 8 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | -1 | 1 | 2 | 0 | -1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | -1 | -1 | -2 | 0 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | 1 | -2 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | orthogonal faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | orthogonal faithful |
►On 24 points - transitive group 24T140
(1 10 4 7)(2 8 5 11)(3 12 6 9)(13 16 20 23)(14 24 21 17)(15 18 22 19)
(1 20 4 13)(2 24 5 17)(3 22 6 15)(7 23 10 16)(8 14 11 21)(9 19 12 18)
(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 3)(4 6)(7 12)(8 11)(9 10)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)
G:=sub<Sym(24)| (1,10,4,7)(2,8,5,11)(3,12,6,9)(13,16,20,23)(14,24,21,17)(15,18,22,19), (1,20,4,13)(2,24,5,17)(3,22,6,15)(7,23,10,16)(8,14,11,21)(9,19,12,18), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3)(4,6)(7,12)(8,11)(9,10)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20)>;
G:=Group( (1,10,4,7)(2,8,5,11)(3,12,6,9)(13,16,20,23)(14,24,21,17)(15,18,22,19), (1,20,4,13)(2,24,5,17)(3,22,6,15)(7,23,10,16)(8,14,11,21)(9,19,12,18), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3)(4,6)(7,12)(8,11)(9,10)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20) );
G=PermutationGroup([[(1,10,4,7),(2,8,5,11),(3,12,6,9),(13,16,20,23),(14,24,21,17),(15,18,22,19)], [(1,20,4,13),(2,24,5,17),(3,22,6,15),(7,23,10,16),(8,14,11,21),(9,19,12,18)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,3),(4,6),(7,12),(8,11),(9,10),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20)]])
G:=TransitiveGroup(24,140);
Q8⋊3D6 is a maximal subgroup of
SD16⋊13D6 SD16⋊D6 D8⋊15D6 S3×C8⋊C22 D8⋊5D6 D24⋊C22 C24.C23 D72⋊C2 C24⋊1D6 C24⋊6D6 D12.7D6 D12⋊5D6 D12⋊6D6 D12.10D6 C24⋊7D6 GL2(𝔽3)⋊S3 C40⋊1D6 D24⋊6D5 D12.9D10 D12⋊5D10 D12⋊D10 D60⋊C22 Q8⋊3D30
Q8⋊3D6 is a maximal quotient of
C4⋊C4.D6 D4.2Dic6 C4⋊C4⋊19D6 D4⋊D12 C3⋊C8⋊D4 C24⋊1C4⋊C2 D4⋊S3⋊C4 D12⋊3D4 Q8⋊3Dic6 (C2×C8).D6 Q8⋊7(C4×S3) Q8⋊4D12 D6.Q16 C3⋊(C8⋊D4) Q8⋊3(C4×S3) D12.12D4 C24⋊3Q8 C8⋊(C4×S3) D6.4SD16 C24⋊7D4 C4.Q8⋊S3 D24⋊9C4 D12⋊Q8 D12.Q8 Dic3⋊5SD16 SD16⋊Dic3 (C3×D4).D4 D6⋊6SD16 D12⋊7D4 C24⋊8D4 C24⋊9D4 D72⋊C2 C24⋊1D6 C24⋊6D6 D12.7D6 D12⋊5D6 D12⋊6D6 D12.10D6 C24⋊7D6 C40⋊1D6 D24⋊6D5 D12.9D10 D12⋊5D10 D12⋊D10 D60⋊C22 Q8⋊3D30
Matrix representation of Q8⋊3D6 ►in GL4(𝔽5) generated by
| 0 | 0 | 2 | 4 |
| 0 | 1 | 1 | 4 |
| 4 | 2 | 1 | 4 |
| 4 | 4 | 2 | 3 |
| 3 | 3 | 0 | 3 |
| 2 | 4 | 1 | 1 |
| 3 | 4 | 2 | 4 |
| 3 | 3 | 4 | 1 |
| 2 | 2 | 0 | 0 |
| 1 | 4 | 0 | 0 |
| 1 | 1 | 0 | 4 |
| 2 | 4 | 1 | 4 |
| 4 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 |
| 4 | 2 | 1 | 4 |
| 3 | 4 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,0,4,4,0,1,2,4,2,1,1,2,4,4,4,3],[3,2,3,3,3,4,4,3,0,1,2,4,3,1,4,1],[2,1,1,2,2,4,1,4,0,0,0,1,0,0,4,4],[4,4,4,3,0,1,2,4,0,0,1,0,0,0,4,4] >;
Q8⋊3D6 in GAP, Magma, Sage, TeX
Q_8\rtimes_3D_6
% in TeX
G:=Group("Q8:3D6"); // GroupNames label
G:=SmallGroup(96,121);
// by ID
G=gap.SmallGroup(96,121);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,116,86,297,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^6=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
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