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G = D6≀C2order 288 = 25·32

Wreath product of D6 by C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — D6≀C2
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — C2×S3≀C2 — D6≀C2
 Lower central C32 — C2×C3⋊S3 — D6≀C2
 Upper central C1 — C2 — C22

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 [×9], C3 [×2], C4 [×3], C22, C22 [×23], S3 [×12], C6 [×10], C2×C4 [×3], D4 [×6], C23 [×10], C32, Dic3, C12, D6 [×35], C2×C6 [×9], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3 [×5], C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×16], C22×C6, C22≀C2, C3×Dic3, C32⋊C4 [×2], S32 [×4], S32 [×8], S3×C6 [×7], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×D4, S3×C23, C6.D6, C3⋊D12, C3×C3⋊D4, S3≀C2 [×4], C2×C32⋊C4 [×2], C2×S32, C2×S32 [×2], C2×S32 [×5], S3×C2×C6, C22×C3⋊S3, S32⋊C4 [×2], C62⋊C4, Dic3⋊D6, C2×S3≀C2 [×2], C22×S32, D6≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], 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

Permutation representations of D6≀C2
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 9 3 8)(2 11 4 10)(5 12)(6 7)
(8 9)(10 11)```

`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), (1,9,3,8)(2,11,4,10)(5,12)(6,7), (8,9)(10,11)>;`

`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), (1,9,3,8)(2,11,4,10)(5,12)(6,7), (8,9)(10,11) );`

`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)], [(1,9,3,8),(2,11,4,10),(5,12),(6,7)], [(8,9),(10,11)])`

`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 5 3 4 2 6)(7 10 8 11 9 12)(13 18 17 16 15 14)(19 20 21 22 23 24)
(1 16 4 13)(2 14 6 15)(3 18 5 17)(7 24 12 23)(8 20 11 21)(9 22 10 19)
(1 10)(2 12)(3 11)(4 9)(5 8)(6 7)(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), (1,5,3,4,2,6)(7,10,8,11,9,12)(13,18,17,16,15,14)(19,20,21,22,23,24), (1,16,4,13)(2,14,6,15)(3,18,5,17)(7,24,12,23)(8,20,11,21)(9,22,10,19), (1,10)(2,12)(3,11)(4,9)(5,8)(6,7)(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), (1,5,3,4,2,6)(7,10,8,11,9,12)(13,18,17,16,15,14)(19,20,21,22,23,24), (1,16,4,13)(2,14,6,15)(3,18,5,17)(7,24,12,23)(8,20,11,21)(9,22,10,19), (1,10)(2,12)(3,11)(4,9)(5,8)(6,7)(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)], [(1,5,3,4,2,6),(7,10,8,11,9,12),(13,18,17,16,15,14),(19,20,21,22,23,24)], [(1,16,4,13),(2,14,6,15),(3,18,5,17),(7,24,12,23),(8,20,11,21),(9,22,10,19)], [(1,10),(2,12),(3,11),(4,9),(5,8),(6,7),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20)])`

`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 20 11 24 9 22)(8 21 12 19 10 23)
(1 7 18 24)(2 19 17 12)(3 9 16 22)(4 21 15 10)(5 11 14 20)(6 23 13 8)
(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(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)(7,20,11,24,9,22)(8,21,12,19,10,23), (1,7,18,24)(2,19,17,12)(3,9,16,22)(4,21,15,10)(5,11,14,20)(6,23,13,8), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(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)(7,20,11,24,9,22)(8,21,12,19,10,23), (1,7,18,24)(2,19,17,12)(3,9,16,22)(4,21,15,10)(5,11,14,20)(6,23,13,8), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(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),(7,20,11,24,9,22),(8,21,12,19,10,23)], [(1,7,18,24),(2,19,17,12),(3,9,16,22),(4,21,15,10),(5,11,14,20),(6,23,13,8)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(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 7 9 3 12)(2 6 8 10 4 11)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 16 9 22)(2 19 10 13)(3 14 12 24)(4 23 11 15)(5 20 7 18)(6 17 8 21)
(1 10)(2 9)(3 6)(4 5)(7 11)(8 12)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)```

`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), (1,5,7,9,3,12)(2,6,8,10,4,11)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,16,9,22)(2,19,10,13)(3,14,12,24)(4,23,11,15)(5,20,7,18)(6,17,8,21), (1,10)(2,9)(3,6)(4,5)(7,11)(8,12)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23)>;`

`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), (1,5,7,9,3,12)(2,6,8,10,4,11)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,16,9,22)(2,19,10,13)(3,14,12,24)(4,23,11,15)(5,20,7,18)(6,17,8,21), (1,10)(2,9)(3,6)(4,5)(7,11)(8,12)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23) );`

`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)], [(1,5,7,9,3,12),(2,6,8,10,4,11),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,16,9,22),(2,19,10,13),(3,14,12,24),(4,23,11,15),(5,20,7,18),(6,17,8,21)], [(1,10),(2,9),(3,6),(4,5),(7,11),(8,12),(13,22),(14,21),(15,20),(16,19),(17,24),(18,23)])`

`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 23)(2 17 8 19)(3 15 9 21)(4 16 10 20)(5 14 11 22)(6 18 12 24)
(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)```

`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), (1,13,7,23)(2,17,8,19)(3,15,9,21)(4,16,10,20)(5,14,11,22)(6,18,12,24), (13,23)(14,22)(15,21)(16,20)(17,19)(18,24)>;`

`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), (1,13,7,23)(2,17,8,19)(3,15,9,21)(4,16,10,20)(5,14,11,22)(6,18,12,24), (13,23)(14,22)(15,21)(16,20)(17,19)(18,24) );`

`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)], [(1,13,7,23),(2,17,8,19),(3,15,9,21),(4,16,10,20),(5,14,11,22),(6,18,12,24)], [(13,23),(14,22),(15,21),(16,20),(17,19),(18,24)])`

`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 11 10 5 8 3)(2 12 9 6 7 4)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)
(1 19 4 24)(2 17 3 16)(5 14 9 13)(6 22 10 21)(7 15 11 18)(8 23 12 20)
(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), (1,11,10,5,8,3)(2,12,9,6,7,4)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,19,4,24)(2,17,3,16)(5,14,9,13)(6,22,10,21)(7,15,11,18)(8,23,12,20), (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), (1,11,10,5,8,3)(2,12,9,6,7,4)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,19,4,24)(2,17,3,16)(5,14,9,13)(6,22,10,21)(7,15,11,18)(8,23,12,20), (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)], [(1,11,10,5,8,3),(2,12,9,6,7,4),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23)], [(1,19,4,24),(2,17,3,16),(5,14,9,13),(6,22,10,21),(7,15,11,18),(8,23,12,20)], [(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 4 9 2 3 10)(5 11 8 6 12 7)(13 24)(14 22)(15 23)(16 21)(17 19)(18 20)
(1 16 7 24)(2 21 8 13)(3 17 11 23)(4 19 12 15)(5 14 10 20)(6 22 9 18)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(14 15)(17 18)(19 20)(22 23)```

`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), (1,4,9,2,3,10)(5,11,8,6,12,7)(13,24)(14,22)(15,23)(16,21)(17,19)(18,20), (1,16,7,24)(2,21,8,13)(3,17,11,23)(4,19,12,15)(5,14,10,20)(6,22,9,18), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(14,15)(17,18)(19,20)(22,23)>;`

`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), (1,4,9,2,3,10)(5,11,8,6,12,7)(13,24)(14,22)(15,23)(16,21)(17,19)(18,20), (1,16,7,24)(2,21,8,13)(3,17,11,23)(4,19,12,15)(5,14,10,20)(6,22,9,18), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(14,15)(17,18)(19,20)(22,23) );`

`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)], [(1,4,9,2,3,10),(5,11,8,6,12,7),(13,24),(14,22),(15,23),(16,21),(17,19),(18,20)], [(1,16,7,24),(2,21,8,13),(3,17,11,23),(4,19,12,15),(5,14,10,20),(6,22,9,18)], [(1,7),(2,8),(3,6),(4,5),(9,11),(10,12),(14,15),(17,18),(19,20),(22,23)])`

`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 5 3 4 2 6)(7 8 12 10 9 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 18)(2 16 3 14)(4 15)(5 13 6 17)(7 24 12 20)(8 19)(9 22)(10 21 11 23)
(1 9)(2 12)(3 7)(4 8)(5 11)(6 10)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)```

`G:=sub<Sym(24)| (13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,8,12,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,18)(2,16,3,14)(4,15)(5,13,6,17)(7,24,12,20)(8,19)(9,22)(10,21,11,23), (1,9)(2,12)(3,7)(4,8)(5,11)(6,10)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)>;`

`G:=Group( (13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,8,12,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,18)(2,16,3,14)(4,15)(5,13,6,17)(7,24,12,20)(8,19)(9,22)(10,21,11,23), (1,9)(2,12)(3,7)(4,8)(5,11)(6,10)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22) );`

`G=PermutationGroup([(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,5,3,4,2,6),(7,8,12,10,9,11),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,18),(2,16,3,14),(4,15),(5,13,6,17),(7,24,12,20),(8,19),(9,22),(10,21,11,23)], [(1,9),(2,12),(3,7),(4,8),(5,11),(6,10),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22)])`

`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 12 6 10 4 7)(2 11 5 9 3 8)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 20 12 15)(2 14 11 21)(3 18 9 23)(4 24 10 17)(5 16 8 19)(6 22 7 13)
(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)```

`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), (1,12,6,10,4,7)(2,11,5,9,3,8)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,20,12,15)(2,14,11,21)(3,18,9,23)(4,24,10,17)(5,16,8,19)(6,22,7,13), (13,22)(14,21)(15,20)(16,19)(17,24)(18,23)>;`

`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), (1,12,6,10,4,7)(2,11,5,9,3,8)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,20,12,15)(2,14,11,21)(3,18,9,23)(4,24,10,17)(5,16,8,19)(6,22,7,13), (13,22)(14,21)(15,20)(16,19)(17,24)(18,23) );`

`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)], [(1,12,6,10,4,7),(2,11,5,9,3,8),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,20,12,15),(2,14,11,21),(3,18,9,23),(4,24,10,17),(5,16,8,19),(6,22,7,13)], [(13,22),(14,21),(15,20),(16,19),(17,24),(18,23)])`

`G:=TransitiveGroup(24,691);`

Polynomial with Galois group D6≀C2 over ℚ
actionf(x)Disc(f)
12T125x12-12x10+53x8-107x6+98x4-34x2+2217·176·2932

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`

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