metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D6, C23⋊3D6, C6.7C24, D12⋊8C22, D6.3C23, C3⋊12+ 1+4, C12.21C23, Dic6⋊8C22, Dic3.4C23, (C2×C4)⋊3D6, (C2×D4)⋊7S3, (S3×D4)⋊4C2, (C6×D4)⋊7C2, D4○(C3⋊D4), C4○D12⋊5C2, D4⋊2S3⋊4C2, (C4×S3)⋊1C22, (C2×C12)⋊3C22, (C3×D4)⋊7C22, C3⋊D4⋊3C22, (C2×C6).2C23, C2.8(S3×C23), C4.21(C22×S3), (C22×C6)⋊5C22, (C22×S3)⋊3C22, (C2×Dic3)⋊4C22, C22.6(C22×S3), (C2×C3⋊D4)⋊11C2, SmallGroup(96,211)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊6D6
G = < a,b,c,d | a4=b2=c6=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 370 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C2×D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, 2+ 1+4, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, D4⋊6D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S3×C23, D4⋊6D6
Character table of D4⋊6D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 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 | 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 | -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 | -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 | 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 | 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 | -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 | 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 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -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 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | 1 | -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 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -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 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | -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 |
| ρ13 | 1 | 1 | 1 | -1 | -1 | 1 | 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 |
| ρ14 | 1 | 1 | -1 | 1 | 1 | -1 | 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 |
| ρ15 | 1 | 1 | 1 | -1 | -1 | 1 | 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 |
| ρ16 | 1 | 1 | -1 | 1 | 1 | -1 | 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 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ23 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ24 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T95
(1 23 7 20)(2 21 8 24)(3 19 9 22)(4 18 11 15)(5 16 12 13)(6 14 10 17)
(1 23)(2 21)(3 19)(4 15)(5 13)(6 17)(7 20)(8 24)(9 22)(10 14)(11 18)(12 16)
(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 12)(2 11)(3 10)(4 8)(5 7)(6 9)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)
G:=sub<Sym(24)| (1,23,7,20)(2,21,8,24)(3,19,9,22)(4,18,11,15)(5,16,12,13)(6,14,10,17), (1,23)(2,21)(3,19)(4,15)(5,13)(6,17)(7,20)(8,24)(9,22)(10,14)(11,18)(12,16), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,8)(5,7)(6,9)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24)>;
G:=Group( (1,23,7,20)(2,21,8,24)(3,19,9,22)(4,18,11,15)(5,16,12,13)(6,14,10,17), (1,23)(2,21)(3,19)(4,15)(5,13)(6,17)(7,20)(8,24)(9,22)(10,14)(11,18)(12,16), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,8)(5,7)(6,9)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24) );
G=PermutationGroup([[(1,23,7,20),(2,21,8,24),(3,19,9,22),(4,18,11,15),(5,16,12,13),(6,14,10,17)], [(1,23),(2,21),(3,19),(4,15),(5,13),(6,17),(7,20),(8,24),(9,22),(10,14),(11,18),(12,16)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,8),(5,7),(6,9),(13,23),(14,22),(15,21),(16,20),(17,19),(18,24)]])
G:=TransitiveGroup(24,95);
►On 24 points - transitive group 24T100
(1 22 14 9)(2 10 15 23)(3 24 16 11)(4 12 17 19)(5 20 18 7)(6 8 13 21)
(1 9)(2 23)(3 11)(4 19)(5 7)(6 21)(8 13)(10 15)(12 17)(14 22)(16 24)(18 20)
(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 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)
G:=sub<Sym(24)| (1,22,14,9)(2,10,15,23)(3,24,16,11)(4,12,17,19)(5,20,18,7)(6,8,13,21), (1,9)(2,23)(3,11)(4,19)(5,7)(6,21)(8,13)(10,15)(12,17)(14,22)(16,24)(18,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23)>;
G:=Group( (1,22,14,9)(2,10,15,23)(3,24,16,11)(4,12,17,19)(5,20,18,7)(6,8,13,21), (1,9)(2,23)(3,11)(4,19)(5,7)(6,21)(8,13)(10,15)(12,17)(14,22)(16,24)(18,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23) );
G=PermutationGroup([[(1,22,14,9),(2,10,15,23),(3,24,16,11),(4,12,17,19),(5,20,18,7),(6,8,13,21)], [(1,9),(2,23),(3,11),(4,19),(5,7),(6,21),(8,13),(10,15),(12,17),(14,22),(16,24),(18,20)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,8),(2,7),(3,12),(4,11),(5,10),(6,9),(13,22),(14,21),(15,20),(16,19),(17,24),(18,23)]])
G:=TransitiveGroup(24,100);
D4⋊6D6 is a maximal subgroup of
C23⋊D12 C23.5D12 M4(2)⋊D6 D12⋊1D4 C24⋊6D6 C22⋊C4⋊D6 C42⋊7D6 C42⋊8D6 D8⋊13D6 SD16⋊13D6 D8⋊5D6 D8⋊6D6 C6.C25 S3×2+ 1+4 D6.C24 D4⋊6D18 D12⋊24D6 D12⋊12D6 D12⋊13D6 C32⋊2+ 1+4 C32⋊82+ 1+4 D20⋊25D6 D20⋊13D6 D20⋊14D6 C15⋊2+ 1+4 D4⋊6D30
D4⋊6D6 is a maximal quotient of
C23⋊3Dic6 C24.35D6 C24.38D6 C23⋊4D12 C24.41D6 C24.42D6 C6.72+ 1+4 C6.82+ 1+4 C6.2+ 1+4 C6.102+ 1+4 C6.112+ 1+4 C6.62- 1+4 D4⋊5Dic6 C42.104D6 C42⋊13D6 C42.108D6 D4⋊5D12 C42⋊18D6 C42.113D6 C42.114D6 C42⋊19D6 C42.115D6 C42.116D6 C42.118D6 C24.43D6 C24⋊7D6 C24⋊8D6 C24.44D6 C24.45D6 C24.46D6 C24⋊9D6 C24.47D6 C6.322+ 1+4 Dic6⋊20D4 C6.342+ 1+4 C6.702- 1+4 C6.712- 1+4 C6.372+ 1+4 C6.382+ 1+4 C6.722- 1+4 C6.402+ 1+4 D12⋊20D4 C6.422+ 1+4 C6.432+ 1+4 C6.442+ 1+4 C6.452+ 1+4 C6.462+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C6.492+ 1+4 C6.752- 1+4 C6.512+ 1+4 C6.522+ 1+4 C6.532+ 1+4 C6.202- 1+4 C6.222- 1+4 C6.562+ 1+4 C6.782- 1+4 C6.252- 1+4 C6.592+ 1+4 C6.812- 1+4 C6.612+ 1+4 C6.622+ 1+4 C6.632+ 1+4 C6.642+ 1+4 C6.652+ 1+4 C6.662+ 1+4 C6.672+ 1+4 C6.682+ 1+4 C6.692+ 1+4 C42.137D6 C42.138D6 C42.140D6 C42⋊22D6 C42⋊23D6 C42⋊24D6 C42.145D6 C42.166D6 C42⋊28D6 D12⋊11D4 Dic6⋊11D4 C42.168D6 C42⋊30D6 Dic6⋊9Q8 C42.174D6 D12⋊9Q8 C42.178D6 C42.179D6 C42.180D6 C24.49D6 D4×C3⋊D4 C24⋊12D6 C24.52D6 C24.53D6 D4⋊6D18 D12⋊24D6 D12⋊12D6 D12⋊13D6 C32⋊2+ 1+4 C32⋊82+ 1+4 D20⋊25D6 D20⋊13D6 D20⋊14D6 C15⋊2+ 1+4 D4⋊6D30
Matrix representation of D4⋊6D6 ►in GL4(𝔽7) generated by
| 3 | 1 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 4 | 5 | 6 | 3 |
| 0 | 3 | 4 | 1 |
| 6 | 6 | 5 | 6 |
| 4 | 0 | 1 | 3 |
| 1 | 1 | 6 | 5 |
| 1 | 6 | 3 | 2 |
| 4 | 4 | 1 | 5 |
| 5 | 4 | 6 | 5 |
| 6 | 4 | 6 | 3 |
| 4 | 4 | 5 | 0 |
| 6 | 6 | 1 | 1 |
| 0 | 6 | 0 | 1 |
| 0 | 5 | 1 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(7))| [3,4,4,0,1,4,5,3,0,0,6,4,0,0,3,1],[6,4,1,1,6,0,1,6,5,1,6,3,6,3,5,2],[4,5,6,4,4,4,4,4,1,6,6,5,5,5,3,0],[6,0,0,0,6,6,5,0,1,0,1,0,1,1,1,1] >;
D4⋊6D6 in GAP, Magma, Sage, TeX
D_4\rtimes_6D_6
% in TeX
G:=Group("D4:6D6"); // GroupNames label
G:=SmallGroup(96,211);
// by ID
G=gap.SmallGroup(96,211);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,579,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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