non-abelian, soluble, monomial, rational
Aliases: D6≀C2, C62⋊1D4, S32⋊2D4, C32⋊C22≀C2, C22⋊2S3≀C2, Dic3⋊D6⋊1C2, C62⋊C4⋊5C2, C6.D6⋊1C22, S32⋊C4⋊2C2, (C2×C3⋊S3)⋊6D4, (C2×S3≀C2)⋊3C2, (C22×S32)⋊4C2, (C2×S32)⋊1C22, C3⋊S3.6(C2×D4), (C2×C32⋊C4)⋊C22, C2.22(C2×S3≀C2), (C3×C6).22(C2×D4), (C2×C3⋊S3).10C23, (C22×C3⋊S3).52C22, SmallGroup(288,889)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6≀C2
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a3b, dad=a-1, cbc-1=a2b3, bd=db, dcd=c-1 >
Subgroups: 1336 in 224 conjugacy classes, 29 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C22×C6, C22≀C2, C3×Dic3, C32⋊C4, S32, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×D4, S3×C23, C6.D6, C3⋊D12, C3×C3⋊D4, S3≀C2, C2×C32⋊C4, C2×S32, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, S32⋊C4, C62⋊C4, Dic3⋊D6, C2×S3≀C2, C22×S32, D6≀C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, S3≀C2, C2×S3≀C2, D6≀C2
Character table of D6≀C2
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12 | |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 9 | 9 | 12 | 18 | 4 | 4 | 12 | 36 | 36 | 4 | 4 | 4 | 4 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | |
| 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 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | -3 | 3 | 2 | -1 | 0 | 1 | 1 | -1 | -1 | 0 | 0 | orthogonal faithful |
| ρ16 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | -2 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | 4 | -4 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | -1 | -1 | -2 | 1 | 2 | -1 | 1 | -1 | 1 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ18 | 4 | -4 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 3 | -3 | 2 | -1 | 0 | -1 | 1 | 1 | -1 | 0 | 0 | orthogonal faithful |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 1 | -2 | 0 | 0 | 2 | 2 | 1 | -2 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from C2×S3≀C2 |
| ρ20 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 1 | 2 | 0 | 0 | 2 | 2 | 1 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | orthogonal lifted from C2×S3≀C2 |
| ρ21 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 1 | -2 | 0 | 0 | -2 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S3≀C2 |
| ρ22 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 1 | 2 | 0 | 0 | -2 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3≀C2 |
| ρ23 | 4 | 4 | -4 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | -1 | -1 | -2 | 1 | 2 | 1 | -1 | 1 | -1 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ24 | 4 | -4 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 3 | -3 | 2 | -1 | 0 | 1 | -1 | -1 | 1 | 0 | 0 | orthogonal faithful |
| ρ25 | 4 | -4 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | -3 | 3 | 2 | -1 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | orthogonal faithful |
| ρ26 | 4 | 4 | 4 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 12 points - transitive group 12T125
Generators in S12
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)
(1 5 3 2 6 4)(7 12)(8 10)(9 11)
(1 11)(2 9)(3 10 6 12)(4 8 5 7)
(7 8)(10 12)
G:=sub<Sym(12)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12), (1,5,3,2,6,4)(7,12)(8,10)(9,11), (1,11)(2,9)(3,10,6,12)(4,8,5,7), (7,8)(10,12)>;
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12), (1,5,3,2,6,4)(7,12)(8,10)(9,11), (1,11)(2,9)(3,10,6,12)(4,8,5,7), (7,8)(10,12) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12)], [(1,5,3,2,6,4),(7,12),(8,10),(9,11)], [(1,11),(2,9),(3,10,6,12),(4,8,5,7)], [(7,8),(10,12)]])
G:=TransitiveGroup(12,125);
►On 24 points - transitive group 24T594
Generators in S24
(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 9 3 8 2 7)(4 11 5 12 6 10)(13 18 17 16 15 14)(19 20 21 22 23 24)
(1 13 8 16)(2 17 7 18)(3 15 9 14)(4 24 11 19)(5 20 10 23)(6 22 12 21)
(1 5)(2 4)(3 6)(7 11)(8 10)(9 12)(13 23)(14 22)(15 21)(16 20)(17 19)(18 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), (1,9,3,8,2,7)(4,11,5,12,6,10)(13,18,17,16,15,14)(19,20,21,22,23,24), (1,13,8,16)(2,17,7,18)(3,15,9,14)(4,24,11,19)(5,20,10,23)(6,22,12,21), (1,5)(2,4)(3,6)(7,11)(8,10)(9,12)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24)>;
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), (1,9,3,8,2,7)(4,11,5,12,6,10)(13,18,17,16,15,14)(19,20,21,22,23,24), (1,13,8,16)(2,17,7,18)(3,15,9,14)(4,24,11,19)(5,20,10,23)(6,22,12,21), (1,5)(2,4)(3,6)(7,11)(8,10)(9,12)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24) );
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)], [(1,9,3,8,2,7),(4,11,5,12,6,10),(13,18,17,16,15,14),(19,20,21,22,23,24)], [(1,13,8,16),(2,17,7,18),(3,15,9,14),(4,24,11,19),(5,20,10,23),(6,22,12,21)], [(1,5),(2,4),(3,6),(7,11),(8,10),(9,12),(13,23),(14,22),(15,21),(16,20),(17,19),(18,24)]])
G:=TransitiveGroup(24,594);
►On 24 points - transitive group 24T655
Generators in S24
(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 3 18 5 14)(2 17 4 13 6 15)(7 22 11 20 9 24)(8 23 12 21 10 19)
(1 7 18 20)(2 21 17 12)(3 9 16 24)(4 23 15 10)(5 11 14 22)(6 19 13 8)
(1 20)(2 19)(3 24)(4 23)(5 22)(6 21)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
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), (1,16,3,18,5,14)(2,17,4,13,6,15)(7,22,11,20,9,24)(8,23,12,21,10,19), (1,7,18,20)(2,21,17,12)(3,9,16,24)(4,23,15,10)(5,11,14,22)(6,19,13,8), (1,20)(2,19)(3,24)(4,23)(5,22)(6,21)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)>;
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), (1,16,3,18,5,14)(2,17,4,13,6,15)(7,22,11,20,9,24)(8,23,12,21,10,19), (1,7,18,20)(2,21,17,12)(3,9,16,24)(4,23,15,10)(5,11,14,22)(6,19,13,8), (1,20)(2,19)(3,24)(4,23)(5,22)(6,21)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13) );
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)], [(1,16,3,18,5,14),(2,17,4,13,6,15),(7,22,11,20,9,24),(8,23,12,21,10,19)], [(1,7,18,20),(2,21,17,12),(3,9,16,24),(4,23,15,10),(5,11,14,22),(6,19,13,8)], [(1,20),(2,19),(3,24),(4,23),(5,22),(6,21),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)]])
G:=TransitiveGroup(24,655);
►On 24 points - transitive group 24T686
Generators in S24
(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 5 11 10 4 8)(2 6 12 9 3 7)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)
(1 21 5 15)(2 14 6 22)(3 18 9 24)(4 19 10 17)(7 20 12 16)(8 13 11 23)
(1 9)(2 10)(3 5)(4 6)(7 11)(8 12)(13 20)(14 19)(15 24)(16 23)(17 22)(18 21)
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), (1,5,11,10,4,8)(2,6,12,9,3,7)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,21,5,15)(2,14,6,22)(3,18,9,24)(4,19,10,17)(7,20,12,16)(8,13,11,23), (1,9)(2,10)(3,5)(4,6)(7,11)(8,12)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)>;
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), (1,5,11,10,4,8)(2,6,12,9,3,7)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,21,5,15)(2,14,6,22)(3,18,9,24)(4,19,10,17)(7,20,12,16)(8,13,11,23), (1,9)(2,10)(3,5)(4,6)(7,11)(8,12)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21) );
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)], [(1,5,11,10,4,8),(2,6,12,9,3,7),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22)], [(1,21,5,15),(2,14,6,22),(3,18,9,24),(4,19,10,17),(7,20,12,16),(8,13,11,23)], [(1,9),(2,10),(3,5),(4,6),(7,11),(8,12),(13,20),(14,19),(15,24),(16,23),(17,22),(18,21)]])
G:=TransitiveGroup(24,686);
►On 24 points - transitive group 24T687
Generators in S24
(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 5 3 4 2 6)(7 12 8 10 9 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 13 7 24)(2 17 8 20)(3 15 9 22)(4 16 10 21)(5 14 11 23)(6 18 12 19)
(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)
G:=sub<Sym(24)| (13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,12,8,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,13,7,24)(2,17,8,20)(3,15,9,22)(4,16,10,21)(5,14,11,23)(6,18,12,19), (13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
G:=Group( (13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,12,8,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,13,7,24)(2,17,8,20)(3,15,9,22)(4,16,10,21)(5,14,11,23)(6,18,12,19), (13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
G=PermutationGroup([[(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,5,3,4,2,6),(7,12,8,10,9,11),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,13,7,24),(2,17,8,20),(3,15,9,22),(4,16,10,21),(5,14,11,23),(6,18,12,19)], [(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
G:=TransitiveGroup(24,687);
►On 24 points - transitive group 24T688
Generators in S24
(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 9 7 6 11 3)(2 10 8 5 12 4)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)
(1 14 4 13)(2 22 3 21)(5 17 7 16)(6 19 8 24)(9 23 12 20)(10 15 11 18)
(13 14)(15 18)(16 17)(19 24)(20 23)(21 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), (1,9,7,6,11,3)(2,10,8,5,12,4)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,14,4,13)(2,22,3,21)(5,17,7,16)(6,19,8,24)(9,23,12,20)(10,15,11,18), (13,14)(15,18)(16,17)(19,24)(20,23)(21,22)>;
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), (1,9,7,6,11,3)(2,10,8,5,12,4)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,14,4,13)(2,22,3,21)(5,17,7,16)(6,19,8,24)(9,23,12,20)(10,15,11,18), (13,14)(15,18)(16,17)(19,24)(20,23)(21,22) );
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)], [(1,9,7,6,11,3),(2,10,8,5,12,4),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23)], [(1,14,4,13),(2,22,3,21),(5,17,7,16),(6,19,8,24),(9,23,12,20),(10,15,11,18)], [(13,14),(15,18),(16,17),(19,24),(20,23),(21,22)]])
G:=TransitiveGroup(24,688);
►On 24 points - transitive group 24T689
Generators in S24
(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 6 4 2 5 3)(7 11 10 8 12 9)(13 17)(14 18)(15 16)(19 22)(20 23)(21 24)
(1 19 12 17)(2 22 11 13)(3 24 9 14)(4 21 10 18)(5 20 7 16)(6 23 8 15)
(1 12)(2 11)(3 8)(4 7)(5 10)(6 9)(14 15)(16 18)(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), (1,6,4,2,5,3)(7,11,10,8,12,9)(13,17)(14,18)(15,16)(19,22)(20,23)(21,24), (1,19,12,17)(2,22,11,13)(3,24,9,14)(4,21,10,18)(5,20,7,16)(6,23,8,15), (1,12)(2,11)(3,8)(4,7)(5,10)(6,9)(14,15)(16,18)(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), (1,6,4,2,5,3)(7,11,10,8,12,9)(13,17)(14,18)(15,16)(19,22)(20,23)(21,24), (1,19,12,17)(2,22,11,13)(3,24,9,14)(4,21,10,18)(5,20,7,16)(6,23,8,15), (1,12)(2,11)(3,8)(4,7)(5,10)(6,9)(14,15)(16,18)(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)], [(1,6,4,2,5,3),(7,11,10,8,12,9),(13,17),(14,18),(15,16),(19,22),(20,23),(21,24)], [(1,19,12,17),(2,22,11,13),(3,24,9,14),(4,21,10,18),(5,20,7,16),(6,23,8,15)], [(1,12),(2,11),(3,8),(4,7),(5,10),(6,9),(14,15),(16,18),(20,21),(23,24)]])
G:=TransitiveGroup(24,689);
►On 24 points - transitive group 24T690
Generators in S24
(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 11 7 10 8 9)(2 6 12 5 3 4)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 16)(2 21 3 19)(4 20)(5 24 6 22)(7 18 8 14)(9 15 11 17)(10 13)(12 23)
(1 12)(2 8)(3 7)(4 10)(5 11)(6 9)(13 20)(14 19)(15 24)(16 23)(17 22)(18 21)
G:=sub<Sym(24)| (13,14,15,16,17,18)(19,20,21,22,23,24), (1,11,7,10,8,9)(2,6,12,5,3,4)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,16)(2,21,3,19)(4,20)(5,24,6,22)(7,18,8,14)(9,15,11,17)(10,13)(12,23), (1,12)(2,8)(3,7)(4,10)(5,11)(6,9)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)>;
G:=Group( (13,14,15,16,17,18)(19,20,21,22,23,24), (1,11,7,10,8,9)(2,6,12,5,3,4)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,16)(2,21,3,19)(4,20)(5,24,6,22)(7,18,8,14)(9,15,11,17)(10,13)(12,23), (1,12)(2,8)(3,7)(4,10)(5,11)(6,9)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21) );
G=PermutationGroup([[(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,11,7,10,8,9),(2,6,12,5,3,4),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,16),(2,21,3,19),(4,20),(5,24,6,22),(7,18,8,14),(9,15,11,17),(10,13),(12,23)], [(1,12),(2,8),(3,7),(4,10),(5,11),(6,9),(13,20),(14,19),(15,24),(16,23),(17,22),(18,21)]])
G:=TransitiveGroup(24,690);
►On 24 points - transitive group 24T691
Generators in S24
(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 7 12 6 3 9)(2 8 11 5 4 10)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)
(1 18 6 22)(2 19 5 15)(3 16 9 24)(4 23 10 17)(7 20 12 14)(8 13 11 21)
(13 21)(14 20)(15 19)(16 24)(17 23)(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), (1,7,12,6,3,9)(2,8,11,5,4,10)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,18,6,22)(2,19,5,15)(3,16,9,24)(4,23,10,17)(7,20,12,14)(8,13,11,21), (13,21)(14,20)(15,19)(16,24)(17,23)(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), (1,7,12,6,3,9)(2,8,11,5,4,10)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,18,6,22)(2,19,5,15)(3,16,9,24)(4,23,10,17)(7,20,12,14)(8,13,11,21), (13,21)(14,20)(15,19)(16,24)(17,23)(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)], [(1,7,12,6,3,9),(2,8,11,5,4,10),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22)], [(1,18,6,22),(2,19,5,15),(3,16,9,24),(4,23,10,17),(7,20,12,14),(8,13,11,21)], [(13,21),(14,20),(15,19),(16,24),(17,23),(18,22)]])
G:=TransitiveGroup(24,691);
Polynomial with Galois group D6≀C2 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 12T125 | x12-12x10+53x8-107x6+98x4-34x2+2 | 217·176·2932 |
Matrix representation of D6≀C2 ►in GL4(ℤ) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | -1 | 0 |
| 1 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | -1 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | -1 | -1 |
G:=sub<GL(4,Integers())| [1,0,0,0,0,1,0,0,0,0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0,0,0,0,-1,0,0,0,0,-1],[0,0,1,0,0,0,0,1,1,-1,0,0,0,-1,0,0],[1,0,0,0,0,1,0,0,0,0,1,-1,0,0,0,-1] >;
D6≀C2 in GAP, Magma, Sage, TeX
D_6\wr C_2
% in TeX
G:=Group("D6wrC2"); // GroupNames label
G:=SmallGroup(288,889);
// by ID
G=gap.SmallGroup(288,889);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,219,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^3*b,d*a*d=a^-1,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export