metabelian, supersoluble, monomial, rational
Aliases: D6:3D6, Dic3:2D6, C62:2C22, C3:S3:3D4, C22:3S32, C3:3(S3xD4), (C2xC6):5D6, C3:D4:2S3, C32:7(C2xD4), C3:D12:6C2, (S3xC6):4C22, C6.D6:3C2, C6.18(C22xS3), (C3xC6).18C23, (C3xDic3):2C22, (C2xS32):4C2, C2.18(C2xS32), (C3xC3:D4):4C2, (C22xC3:S3):2C2, (C2xC3:S3):4C22, Hol(Dic3), SmallGroup(144,154)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3:D6
G = < a,b,c,d | a6=c6=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, cbc-1=a3b, bd=db, dcd=c-1 >
Subgroups: 496 in 124 conjugacy classes, 34 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C4xS3, D12, C3:D4, C3:D4, C3xD4, C22xS3, C3xDic3, S32, S3xC6, C2xC3:S3, C2xC3:S3, C62, S3xD4, C6.D6, C3:D12, C3xC3:D4, C2xS32, C22xC3:S3, Dic3:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, S32, S3xD4, C2xS32, Dic3:D6
Character table of Dic3:D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | |
| size | 1 | 1 | 2 | 6 | 6 | 9 | 9 | 18 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 1 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 2 | 2 | -1 | -1 | 1 | -2 | 1 | 1 | 0 | 1 | -1 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | -1 | 2 | -1 | 1 | 1 | -2 | 1 | 1 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | -2 | 2 | -1 | -1 | 1 | -2 | 1 | 1 | 0 | -1 | 1 | 0 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | -2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 1 | 1 | 0 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -2 | 0 | -1 | 2 | -1 | 1 | 1 | -2 | 1 | -1 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | -1 | -1 | 0 | orthogonal lifted from S3 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2 | 2 | -1 | -3 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | -2 | -2 | 1 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS32 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2 | 2 | -1 | 3 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | -2 | -2 | 1 | 1 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
►On 12 points - transitive group 12T81
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 8 4 11)(2 7 5 10)(3 12 6 9)
(1 5 3)(2 6 4)(7 12 11 10 9 8)
(1 3)(4 6)(8 12)(9 11)
G:=sub<Sym(12)| (1,2,3,4,5,6)(7,8,9,10,11,12), (1,8,4,11)(2,7,5,10)(3,12,6,9), (1,5,3)(2,6,4)(7,12,11,10,9,8), (1,3)(4,6)(8,12)(9,11)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12), (1,8,4,11)(2,7,5,10)(3,12,6,9), (1,5,3)(2,6,4)(7,12,11,10,9,8), (1,3)(4,6)(8,12)(9,11) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,8,4,11),(2,7,5,10),(3,12,6,9)], [(1,5,3),(2,6,4),(7,12,11,10,9,8)], [(1,3),(4,6),(8,12),(9,11)]])
G:=TransitiveGroup(12,81);
►On 24 points - transitive group 24T269
(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 20 4 23)(2 19 5 22)(3 24 6 21)(7 14 10 17)(8 13 11 16)(9 18 12 15)
(1 11 3 7 5 9)(2 12 4 8 6 10)(13 24 17 22 15 20)(14 19 18 23 16 21)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 16)(14 15)(17 18)(19 22)(20 21)(23 24)
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), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,14,10,17)(8,13,11,16)(9,18,12,15), (1,11,3,7,5,9)(2,12,4,8,6,10)(13,24,17,22,15,20)(14,19,18,23,16,21), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)>;
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), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,14,10,17)(8,13,11,16)(9,18,12,15), (1,11,3,7,5,9)(2,12,4,8,6,10)(13,24,17,22,15,20)(14,19,18,23,16,21), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24) );
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)], [(1,20,4,23),(2,19,5,22),(3,24,6,21),(7,14,10,17),(8,13,11,16),(9,18,12,15)], [(1,11,3,7,5,9),(2,12,4,8,6,10),(13,24,17,22,15,20),(14,19,18,23,16,21)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,16),(14,15),(17,18),(19,22),(20,21),(23,24)]])
G:=TransitiveGroup(24,269);
►On 24 points - transitive group 24T270
(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 4 11)(2 7 5 10)(3 12 6 9)(13 23 16 20)(14 22 17 19)(15 21 18 24)
(1 5 3)(2 6 4)(7 12 11 10 9 8)(13 18 17 16 15 14)(19 23 21)(20 24 22)
(1 19)(2 24)(3 23)(4 22)(5 21)(6 20)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)
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), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,23,16,20)(14,22,17,19)(15,21,18,24), (1,5,3)(2,6,4)(7,12,11,10,9,8)(13,18,17,16,15,14)(19,23,21)(20,24,22), (1,19)(2,24)(3,23)(4,22)(5,21)(6,20)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)>;
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), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,23,16,20)(14,22,17,19)(15,21,18,24), (1,5,3)(2,6,4)(7,12,11,10,9,8)(13,18,17,16,15,14)(19,23,21)(20,24,22), (1,19)(2,24)(3,23)(4,22)(5,21)(6,20)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16) );
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)], [(1,8,4,11),(2,7,5,10),(3,12,6,9),(13,23,16,20),(14,22,17,19),(15,21,18,24)], [(1,5,3),(2,6,4),(7,12,11,10,9,8),(13,18,17,16,15,14),(19,23,21),(20,24,22)], [(1,19),(2,24),(3,23),(4,22),(5,21),(6,20),(7,15),(8,14),(9,13),(10,18),(11,17),(12,16)]])
G:=TransitiveGroup(24,270);
►On 24 points - transitive group 24T271
(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 20 4 23)(2 19 5 22)(3 24 6 21)(7 14 10 17)(8 13 11 16)(9 18 12 15)
(1 11 3 7 5 9)(2 12 4 8 6 10)(13 24 17 22 15 20)(14 19 18 23 16 21)
(1 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)
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), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,14,10,17)(8,13,11,16)(9,18,12,15), (1,11,3,7,5,9)(2,12,4,8,6,10)(13,24,17,22,15,20)(14,19,18,23,16,21), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20)>;
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), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,14,10,17)(8,13,11,16)(9,18,12,15), (1,11,3,7,5,9)(2,12,4,8,6,10)(13,24,17,22,15,20)(14,19,18,23,16,21), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20) );
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)], [(1,20,4,23),(2,19,5,22),(3,24,6,21),(7,14,10,17),(8,13,11,16),(9,18,12,15)], [(1,11,3,7,5,9),(2,12,4,8,6,10),(13,24,17,22,15,20),(14,19,18,23,16,21)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20)]])
G:=TransitiveGroup(24,271);
Dic3:D6 is a maximal subgroup of
C62.2D4 C62.9D4 D6wrC2 C62:D4 D12:23D6 D12:27D6 S32xD4 Dic6:12D6 D12:13D6 C32:2+ 1+4 D18:D6 C62:D6 C62:5D6 D6:4S32 (S3xC6):D6 C3:S3:4D12 C62:23D6 C62:24D6 C62:10D6
Dic3:D6 is a maximal quotient of
C62.10C23 C62.23C23 C62.35C23 C62.51C23 C62.53C23 D6:3Dic6 C62.67C23 Dic3:3D12 C62.82C23 C62.83C23 C62.91C23 D6:5D12 D12:D6 D12.D6 Dic6:D6 Dic6.D6 D12.8D6 D12:5D6 D12.9D6 D12.10D6 Dic6.9D6 Dic6.10D6 D12.14D6 D12.15D6 C62.95C23 C62.100C23 C62.113C23 C62.115C23 C62.116C23 C62.117C23 C62.121C23 C62:7D4 C62:8D4 C62:4Q8 C62.125C23 D18:D6 C62:2D6 D6:4S32 (S3xC6):D6 C3:S3:4D12 C62:23D6 C62:24D6
| action | f(x) | Disc(f) |
|---|---|---|
| 12T81 | x12-24x10+216x8-902x6+1752x4-1368x2+200 | 239·312·56·76·236 |
Matrix representation of Dic3:D6 ►in GL4(Z) generated by
| 1 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| -1 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | -1 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | -1 |
G:=sub<GL(4,Integers())| [1,-1,0,0,1,0,0,0,0,0,0,1,0,0,-1,1],[0,0,-1,0,0,0,0,-1,1,0,0,0,0,1,0,0],[-1,1,0,0,-1,0,0,0,0,0,1,-1,0,0,1,0],[1,0,0,0,1,-1,0,0,0,0,1,0,0,0,1,-1] >;
Dic3:D6 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes D_6 % in TeX
G:=Group("Dic3:D6"); // GroupNames label
G:=SmallGroup(144,154);
// by ID
G=gap.SmallGroup(144,154);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,116,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^6=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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