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G = C62.32D4order 288 = 25·32

16th non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.32D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C62 — C3×C6.D4 — C62.32D4
 Lower central C32 — C3×C6 — C62 — C62.32D4
 Upper central C1 — C2 — C23

Generators and relations for C62.32D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1b3, cbc-1=dbd-1=b-1, dcd-1=a3b3c-1 >

Subgroups: 642 in 131 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×3], S3 [×4], C6 [×2], C6 [×11], C2×C4 [×3], D4 [×2], C23, C23, C32, Dic3 [×6], C12 [×2], D6 [×8], C2×C6 [×6], C2×C6 [×7], C22⋊C4 [×2], C2×D4, C3⋊S3, C3×C6, C3×C6 [×3], C2×Dic3 [×5], C3⋊D4 [×8], C2×C12 [×2], C22×S3 [×3], C22×C6 [×2], C22×C6, C23⋊C4, C3×Dic3 [×2], C3⋊Dic3, C2×C3⋊S3 [×2], C62, C62 [×2], C62, C6.D4 [×2], C3×C22⋊C4 [×2], C2×C3⋊D4 [×3], C6×Dic3 [×2], C2×C3⋊Dic3, C327D4 [×2], C22×C3⋊S3, C2×C62, C23.6D6 [×2], C3×C6.D4 [×2], C2×C327D4, C62.32D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C23⋊C4, S32, D6⋊C4 [×2], C6.D6, C3⋊D12 [×2], C23.6D6 [×2], C6.D12, C62.32D4

Permutation representations of C62.32D4
On 24 points - transitive group 24T583
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 2 6 3 4)(7 11 9 10 8 12)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 13)(2 17)(3 15)(4 14)(5 18)(6 16)(7 19 10 22)(8 23 11 20)(9 21 12 24)
(1 19 6 22)(2 21 4 24)(3 23 5 20)(7 16 10 13)(8 18 11 15)(9 14 12 17)```

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

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

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

`G:=TransitiveGroup(24,583);`

On 24 points - transitive group 24T615
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 23 4 7)(2 11 5 21)(3 19 6 9)(8 13 24 16)(10 15 20 18)(12 17 22 14)
(1 4 18 15)(2 14 13 3)(5 6 16 17)(7 23 20 10)(8 9 21 22)(11 19 24 12)```

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

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

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

`G:=TransitiveGroup(24,615);`

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 6A ··· 6F 6G ··· 6Q 12A ··· 12H order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 2 2 36 2 2 4 12 12 12 12 36 2 ··· 2 4 ··· 4 12 ··· 12

39 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C4 C4 S3 D4 D6 C4×S3 D12 C3⋊D4 C23⋊C4 S32 C6.D6 C3⋊D12 C23.6D6 C62.32D4 kernel C62.32D4 C3×C6.D4 C2×C32⋊7D4 C2×C3⋊Dic3 C22×C3⋊S3 C6.D4 C62 C22×C6 C2×C6 C2×C6 C2×C6 C32 C23 C22 C22 C3 C1 # reps 1 2 1 2 2 2 2 2 4 4 4 1 1 1 2 4 4

Matrix representation of C62.32D4 in GL4(𝔽7) generated by

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

C62.32D4 in GAP, Magma, Sage, TeX

`C_6^2._{32}D_4`
`% in TeX`

`G:=Group("C6^2.32D4");`
`// GroupNames label`

`G:=SmallGroup(288,229);`
`// by ID`

`G=gap.SmallGroup(288,229);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,219,675,346,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^3*b^3*c^-1>;`
`// generators/relations`

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