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## G = C62.2D4order 288 = 25·32

### 2nd non-split extension by C62 of D4 acting faithfully

Aliases: C62.2D4, C6.D6⋊C4, C32⋊(C23⋊C4), C22.2S3≀C2, C62⋊C41C2, Dic3⋊D6.1C2, C6.D121C2, (C2×S32)⋊C4, C2.8(S32⋊C4), (C2×C3⋊S3).9D4, (C3×C6).8(C22⋊C4), (C22×C3⋊S3).2C22, (C2×C3⋊S3).10(C2×C4), SmallGroup(288,386)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — C62.2D4
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — Dic3⋊D6 — C62.2D4
 Lower central C32 — C3×C6 — C2×C3⋊S3 — C62.2D4
 Upper central C1 — C2 — C22

Generators and relations for C62.2D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a3b, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=a3c-1 >

Subgroups: 640 in 96 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C4 [×3], C22, C22 [×5], S3 [×6], C6 [×6], C2×C4 [×3], D4 [×2], C23 [×2], C32, Dic3 [×2], C12 [×2], D6 [×11], C2×C6 [×3], C22⋊C4 [×2], C2×D4, C3×S3, C3⋊S3 [×2], C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4, C22×S3 [×3], C23⋊C4, C3×Dic3 [×2], C32⋊C4, S32, S3×C6, C2×C3⋊S3 [×2], C2×C3⋊S3, C62, D6⋊C4, S3×D4, C6.D6, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C32⋊C4, C2×S32, C22×C3⋊S3, C6.D12, C62⋊C4, Dic3⋊D6, C62.2D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C23⋊C4, S3≀C2, S32⋊C4, C62.2D4

Character table of C62.2D4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E size 1 1 2 12 18 18 4 4 12 12 12 36 36 4 4 4 4 8 24 12 12 12 12 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 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 i -i 1 i -i 1 1 -1 -1 -1 -1 -i i i -i 1 linear of order 4 ρ6 1 1 -1 1 1 -1 1 1 i -i -1 -i i 1 1 -1 -1 -1 1 -i i i -i -1 linear of order 4 ρ7 1 1 -1 1 1 -1 1 1 -i i -1 i -i 1 1 -1 -1 -1 1 i -i -i i -1 linear of order 4 ρ8 1 1 -1 -1 1 -1 1 1 -i i 1 -i i 1 1 -1 -1 -1 -1 i -i -i i 1 linear of order 4 ρ9 2 2 2 0 -2 -2 2 2 0 0 0 0 0 2 2 2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 -2 2 2 2 0 0 0 0 0 2 2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 -2 0 0 1 -2 0 0 2 0 0 1 -2 2 2 -1 1 0 0 0 0 -1 orthogonal lifted from S32⋊C4 ρ12 4 4 -4 2 0 0 1 -2 0 0 -2 0 0 1 -2 2 2 -1 -1 0 0 0 0 1 orthogonal lifted from S32⋊C4 ρ13 4 4 4 2 0 0 1 -2 0 0 2 0 0 1 -2 -2 -2 1 -1 0 0 0 0 -1 orthogonal lifted from S3≀C2 ρ14 4 4 4 -2 0 0 1 -2 0 0 -2 0 0 1 -2 -2 -2 1 1 0 0 0 0 1 orthogonal lifted from S3≀C2 ρ15 4 4 4 0 0 0 -2 1 2 2 0 0 0 -2 1 1 1 -2 0 -1 -1 -1 -1 0 orthogonal lifted from S3≀C2 ρ16 4 -4 0 0 0 0 4 4 0 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ17 4 4 4 0 0 0 -2 1 -2 -2 0 0 0 -2 1 1 1 -2 0 1 1 1 1 0 orthogonal lifted from S3≀C2 ρ18 4 -4 0 0 0 0 -2 1 0 0 0 0 0 2 -1 3 -3 0 0 √3 -√3 √3 -√3 0 orthogonal faithful ρ19 4 -4 0 0 0 0 -2 1 0 0 0 0 0 2 -1 3 -3 0 0 -√3 √3 -√3 √3 0 orthogonal faithful ρ20 4 4 -4 0 0 0 -2 1 -2i 2i 0 0 0 -2 1 -1 -1 2 0 -i i i -i 0 complex lifted from S32⋊C4 ρ21 4 4 -4 0 0 0 -2 1 2i -2i 0 0 0 -2 1 -1 -1 2 0 i -i -i i 0 complex lifted from S32⋊C4 ρ22 4 -4 0 0 0 0 -2 1 0 0 0 0 0 2 -1 -3 3 0 0 -√-3 -√-3 √-3 √-3 0 complex faithful ρ23 4 -4 0 0 0 0 -2 1 0 0 0 0 0 2 -1 -3 3 0 0 √-3 √-3 -√-3 -√-3 0 complex faithful ρ24 8 -8 0 0 0 0 2 -4 0 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C62.2D4
On 24 points - transitive group 24T596
Generators in S24
```(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 9 11 8 10 12)(13 17)(14 18)(15 16)(19 22)(20 23)(21 24)
(1 22 4 19)(2 20 6 21)(3 24 5 23)(7 17 10 18)(8 13 9 14)(11 16)(12 15)
(1 11)(2 7)(3 10)(4 12)(5 8)(6 9)(13 24)(14 20)(15 22)(16 19)(17 21)(18 23)```

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

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

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

`G:=TransitiveGroup(24,596);`

On 24 points - transitive group 24T597
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 11 9 10 8 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 21 4 24)(2 19 6 20)(3 23 5 22)(7 18)(8 16 9 14)(10 15)(11 13 12 17)
(1 18)(2 14)(3 16)(4 15)(5 17)(6 13)(7 24)(8 20)(9 22)(10 21)(11 23)(12 19)```

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

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

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

`G:=TransitiveGroup(24,597);`

On 24 points - transitive group 24T598
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 2 5 3 6)(7 12 8 10 9 11)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 20 5 23)(2 22 4 21)(3 24 6 19)(7 13)(8 15 9 17)(10 16)(11 18 12 14)
(1 13)(2 15)(3 17)(4 14)(5 16)(6 18)(7 23)(8 19)(9 21)(10 20)(11 22)(12 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), (1,4,2,5,3,6)(7,12,8,10,9,11)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,20,5,23)(2,22,4,21)(3,24,6,19)(7,13)(8,15,9,17)(10,16)(11,18,12,14), (1,13)(2,15)(3,17)(4,14)(5,16)(6,18)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24)>;`

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

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

`G:=TransitiveGroup(24,598);`

On 24 points - transitive group 24T602
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 11 9 10 8 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 19 11 17)(2 23 10 13)(3 21 12 15)(4 22 8 14)(5 20 7 16)(6 24 9 18)
(1 17)(2 13)(3 15)(4 14)(5 16)(6 18)(7 23)(8 19)(9 21)(10 20)(11 22)(12 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), (1,5,3,4,2,6)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,19,11,17)(2,23,10,13)(3,21,12,15)(4,22,8,14)(5,20,7,16)(6,24,9,18), (1,17)(2,13)(3,15)(4,14)(5,16)(6,18)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24)>;`

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

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

`G:=TransitiveGroup(24,602);`

On 24 points - transitive group 24T605
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 2 5 3 6)(7 12 8 10 9 11)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 23 10 13)(2 19 12 17)(3 21 11 15)(4 24 8 18)(5 20 7 16)(6 22 9 14)
(1 13)(2 15)(3 17)(4 14)(5 16)(6 18)(7 23)(8 19)(9 21)(10 20)(11 22)(12 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), (1,4,2,5,3,6)(7,12,8,10,9,11)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,23,10,13)(2,19,12,17)(3,21,11,15)(4,24,8,18)(5,20,7,16)(6,22,9,14), (1,13)(2,15)(3,17)(4,14)(5,16)(6,18)(7,23)(8,19)(9,21)(10,20)(11,22)(12,24)>;`

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

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

`G:=TransitiveGroup(24,605);`

On 24 points - transitive group 24T665
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 4 8 6 9)(2 11 3 7 5 10)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 19 7 22)(2 14 8 17)(3 16 12 15)(4 21 11 20)(5 18 9 13)(6 23 10 24)
(1 8)(4 9)(6 12)(13 15)(14 22)(16 20)(18 24)(21 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), (1,12,4,8,6,9)(2,11,3,7,5,10)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,19,7,22)(2,14,8,17)(3,16,12,15)(4,21,11,20)(5,18,9,13)(6,23,10,24), (1,8)(4,9)(6,12)(13,15)(14,22)(16,20)(18,24)(21,23)>;`

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

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

`G:=TransitiveGroup(24,665);`

Matrix representation of C62.2D4 in GL4(𝔽7) generated by

 6 4 6 1 1 3 0 5 4 5 6 3 3 5 0 6
,
 1 0 4 6 2 4 1 0 6 1 3 2 4 4 5 1
,
 5 1 4 1 6 1 2 1 5 0 3 0 1 6 3 5
,
 1 1 1 0 2 1 5 4 5 5 1 3 5 2 4 4
`G:=sub<GL(4,GF(7))| [6,1,4,3,4,3,5,5,6,0,6,0,1,5,3,6],[1,2,6,4,0,4,1,4,4,1,3,5,6,0,2,1],[5,6,5,1,1,1,0,6,4,2,3,3,1,1,0,5],[1,2,5,5,1,1,5,2,1,5,1,4,0,4,3,4] >;`

C62.2D4 in GAP, Magma, Sage, TeX

`C_6^2._2D_4`
`% in TeX`

`G:=Group("C6^2.2D4");`
`// GroupNames label`

`G:=SmallGroup(288,386);`
`// by ID`

`G=gap.SmallGroup(288,386);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,219,675,2693,2028,691,797,2372]);`
`// 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*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=a^3*c^-1>;`
`// generators/relations`

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