metabelian, supersoluble, monomial, rational
Aliases: D6⋊3D6, Dic3⋊2D6, C62⋊2C22, C3⋊S3⋊3D4, C22⋊3S32, C3⋊3(S3×D4), (C2×C6)⋊5D6, C3⋊D4⋊2S3, C32⋊7(C2×D4), C3⋊D12⋊6C2, (S3×C6)⋊4C22, C6.D6⋊3C2, C6.18(C22×S3), (C3×C6).18C23, (C3×Dic3)⋊2C22, (C2×S32)⋊4C2, C2.18(C2×S32), (C3×C3⋊D4)⋊4C2, (C22×C3⋊S3)⋊2C2, (C2×C3⋊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, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3⋊D4, C3×D4, C22×S3, C3×Dic3, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×D4, C6.D6, C3⋊D12, C3×C3⋊D4, C2×S32, C22×C3⋊S3, Dic3⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, S32, S3×D4, C2×S32, 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 S3×D4 |
| 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 C2×S32 |
| 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 S3×D4 |
| 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 D6≀C2 C62⋊D4 D12⋊23D6 D12⋊27D6 S32×D4 Dic6⋊12D6 D12⋊13D6 C32⋊2+ 1+4 D18⋊D6 C62⋊D6 C62⋊5D6 D6⋊4S32 (S3×C6)⋊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 (S3×C6)⋊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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