metabelian, supersoluble, monomial
Aliases: D6.3D6, Dic3.3D6, C62.7C22, C3⋊D4⋊3S3, C22.1S32, (C2×C6).4D6, C3⋊D12⋊2C2, C3⋊4(C4○D12), (C6×Dic3)⋊6C2, (C2×Dic3)⋊3S3, (S3×Dic3)⋊5C2, C32⋊2Q8⋊4C2, C6.D6⋊2C2, C32⋊5(C4○D4), C32⋊7D4⋊1C2, C3⋊3(D4⋊2S3), (S3×C6).3C22, C6.11(C22×S3), (C3×C6).11C23, C3⋊Dic3.6C22, (C3×Dic3).9C22, C2.12(C2×S32), (C3×C3⋊D4)⋊1C2, (C2×C3⋊S3).5C22, SmallGroup(144,147)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.3D6
G = < a,b,c,d | a6=b2=c6=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >
Subgroups: 284 in 88 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C4○D12, D4⋊2S3, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C32⋊7D4, D6.3D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, C2×S32, D6.3D6
Character table of D6.3D6
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | 12E | |
| size | 1 | 1 | 2 | 6 | 18 | 2 | 2 | 4 | 3 | 3 | 6 | 6 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 6 | 6 | 6 | 6 | 12 | |
| 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 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | -2 | 0 | 0 | 1 | -1 | 2 | 1 | -2 | 1 | 1 | -1 | 0 | -1 | 1 | -1 | 1 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | -1 | -1 | -1 | -1 | 0 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 2 | 0 | -2 | 2 | -1 | -2 | 1 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | 2 | 0 | 0 | 1 | -1 | 2 | 1 | -2 | 1 | 1 | -1 | 0 | 1 | -1 | 1 | -1 | 0 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | -2 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | 2 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | -1 | -2 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | 0 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 2i | -2i | 0 | 0 | 0 | -√-3 | 1 | -2 | √-3 | 0 | √-3 | -√-3 | 1 | 0 | -i | -√3 | i | √3 | 0 | complex lifted from C4○D12 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | -2i | 2i | 0 | 0 | 0 | √-3 | 1 | -2 | -√-3 | 0 | -√-3 | √-3 | 1 | 0 | i | -√3 | -i | √3 | 0 | complex lifted from C4○D12 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | -2i | 2i | 0 | 0 | 0 | -√-3 | 1 | -2 | √-3 | 0 | √-3 | -√-3 | 1 | 0 | i | √3 | -i | -√3 | 0 | complex lifted from C4○D12 |
| ρ22 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 2i | -2i | 0 | 0 | 0 | √-3 | 1 | -2 | -√-3 | 0 | -√-3 | √-3 | 1 | 0 | -i | √3 | i | -√3 | 0 | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S32 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 2 | 2 | 2√-3 | 0 | -√-3 | √-3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2√-3 | 2 | 2 | -2√-3 | 0 | √-3 | -√-3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T205
(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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 12 4 9)(2 7 5 10)(3 8 6 11)(13 23 16 20)(14 24 17 21)(15 19 18 22)
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), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,22)(8,21)(9,20)(10,19)(11,24)(12,23), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,12,4,9)(2,7,5,10)(3,8,6,11)(13,23,16,20)(14,24,17,21)(15,19,18,22)>;
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), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,22)(8,21)(9,20)(10,19)(11,24)(12,23), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,12,4,9)(2,7,5,10)(3,8,6,11)(13,23,16,20)(14,24,17,21)(15,19,18,22) );
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)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,22),(8,21),(9,20),(10,19),(11,24),(12,23)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,14,15,16,17,18),(19,24,23,22,21,20)], [(1,12,4,9),(2,7,5,10),(3,8,6,11),(13,23,16,20),(14,24,17,21),(15,19,18,22)]])
G:=TransitiveGroup(24,205);
►On 24 points - transitive group 24T221
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)
(1 9 5 7 3 11)(2 10 6 8 4 12)(13 23 15 19 17 21)(14 24 16 20 18 22)
(1 24 4 21)(2 19 5 22)(3 20 6 23)(7 18 10 15)(8 13 11 16)(9 14 12 17)
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16), (1,9,5,7,3,11)(2,10,6,8,4,12)(13,23,15,19,17,21)(14,24,16,20,18,22), (1,24,4,21)(2,19,5,22)(3,20,6,23)(7,18,10,15)(8,13,11,16)(9,14,12,17)>;
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16), (1,9,5,7,3,11)(2,10,6,8,4,12)(13,23,15,19,17,21)(14,24,16,20,18,22), (1,24,4,21)(2,19,5,22)(3,20,6,23)(7,18,10,15)(8,13,11,16)(9,14,12,17) );
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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,15),(8,14),(9,13),(10,18),(11,17),(12,16)], [(1,9,5,7,3,11),(2,10,6,8,4,12),(13,23,15,19,17,21),(14,24,16,20,18,22)], [(1,24,4,21),(2,19,5,22),(3,20,6,23),(7,18,10,15),(8,13,11,16),(9,14,12,17)]])
G:=TransitiveGroup(24,221);
D6.3D6 is a maximal subgroup of
D12.33D6 S3×C4○D12 Dic6.24D6 S3×D4⋊2S3 Dic6⋊12D6 D12⋊13D6 C32⋊2+ 1+4 D18.3D6 Dic3.D18 C62.8D6 C62.9D6 (S3×C6).D6 D6.S32 D6.4S32 D6.3S32 C62.90D6 C62.93D6 C62.96D6
D6.3D6 is a maximal quotient of
C62.6C23 Dic3⋊5Dic6 C62.16C23 C62.17C23 C62.18C23 Dic3.D12 C62.24C23 C62.28C23 C62.29C23 C62.37C23 C62.38C23 C62.47C23 Dic3⋊4D12 D6.D12 C62.65C23 D6⋊4Dic6 C62.94C23 C62.95C23 C62.97C23 C62.98C23 C62.100C23 C62.101C23 C62.56D4 C62⋊3Q8 C62.60D4 C62.111C23 C62.113C23 Dic3×C3⋊D4 C62.117C23 C62⋊6D4 D18.3D6 Dic3.D18 C62.8D6 (S3×C6).D6 D6.S32 D6.4S32 D6.3S32 C62.90D6 C62.93D6 C62.96D6
Matrix representation of D6.3D6 ►in GL4(𝔽7) generated by
| 0 | 0 | 3 | 1 |
| 5 | 4 | 6 | 0 |
| 1 | 6 | 5 | 5 |
| 3 | 3 | 2 | 0 |
| 3 | 3 | 1 | 5 |
| 2 | 5 | 2 | 0 |
| 6 | 6 | 1 | 2 |
| 3 | 1 | 5 | 5 |
| 1 | 1 | 1 | 3 |
| 0 | 6 | 6 | 3 |
| 2 | 2 | 6 | 4 |
| 2 | 6 | 2 | 1 |
| 0 | 0 | 1 | 3 |
| 4 | 0 | 1 | 4 |
| 3 | 3 | 5 | 1 |
| 1 | 6 | 3 | 2 |
G:=sub<GL(4,GF(7))| [0,5,1,3,0,4,6,3,3,6,5,2,1,0,5,0],[3,2,6,3,3,5,6,1,1,2,1,5,5,0,2,5],[1,0,2,2,1,6,2,6,1,6,6,2,3,3,4,1],[0,4,3,1,0,0,3,6,1,1,5,3,3,4,1,2] >;
D6.3D6 in GAP, Magma, Sage, TeX
D_6._3D_6
% in TeX
G:=Group("D6.3D6"); // GroupNames label
G:=SmallGroup(144,147);
// by ID
G=gap.SmallGroup(144,147);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^6=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations
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