metabelian, supersoluble, monomial
Aliases: D6.3D6, Dic3.3D6, C62.7C22, C3:D4:3S3, C22.1S32, (C2xC6).4D6, C3:D12:2C2, C3:4(C4oD12), (C6xDic3):6C2, (C2xDic3):3S3, (S3xDic3):5C2, C32:2Q8:4C2, C6.D6:2C2, C32:5(C4oD4), C32:7D4:1C2, C3:3(D4:2S3), (S3xC6).3C22, C6.11(C22xS3), (C3xC6).11C23, C3:Dic3.6C22, (C3xDic3).9C22, C2.12(C2xS32), (C3xC3:D4):1C2, (C2xC3: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, C2xC4, D4, Q8, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C3:D4, C2xC12, C3xD4, C3xDic3, C3:Dic3, S3xC6, C2xC3:S3, C62, C4oD12, D4:2S3, S3xDic3, C6.D6, C3:D12, C32:2Q8, C6xDic3, C3xC3:D4, C32:7D4, D6.3D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, S32, C4oD12, D4:2S3, C2xS32, 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 C4oD4 |
| 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 C4oD4 |
| 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 C4oD12 |
| 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 C4oD12 |
| 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 C4oD12 |
| 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 C4oD12 |
| 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 C2xS32 |
| 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 S3xC4oD12 Dic6.24D6 S3xD4:2S3 Dic6:12D6 D12:13D6 C32:2+ 1+4 D18.3D6 Dic3.D18 C62.8D6 C62.9D6 (S3xC6).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 Dic3xC3:D4 C62.117C23 C62:6D4 D18.3D6 Dic3.D18 C62.8D6 (S3xC6).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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