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## G = C62.96D6order 432 = 24·33

### 44th non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.96D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9(C2×C4) — C62.96D6
 Lower central C33 — C32×C6 — C62.96D6
 Upper central C1 — C2 — C22

Generators and relations for C62.96D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=a-1, dad-1=ab3, cbc-1=dbd-1=b-1, dcd-1=c5 >

Subgroups: 1032 in 218 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C62, C4○D12, D42S3, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C327D4, C3×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C3×C62, D6.3D6, D6.4D6, C339(C2×C4), C339D4, C335Q8, C6×C3⋊Dic3, C3×C327D4, C62.96D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D42S3, C2×S32, C324D6, D6.3D6, D6.4D6, C2×C324D6, C62.96D6

Permutation representations of C62.96D6
On 24 points - transitive group 24T1285
Generators in S24
```(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 15 17 19 21 23)(14 24 22 20 18 16)
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 24 22 20 18 16)
(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 15 7 21)(2 20 8 14)(3 13 9 19)(4 18 10 24)(5 23 11 17)(6 16 12 22)```

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

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

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

`G:=TransitiveGroup(24,1285);`

48 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D ··· 3H 4A 4B 4C 4D 4E 6A ··· 6E 6F ··· 6V 6W 6X 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 3 3 3 3 ··· 3 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 12 12 12 12 12 12 size 1 1 2 18 18 2 2 2 4 ··· 4 9 9 18 18 18 2 ··· 2 4 ··· 4 36 36 18 18 18 18 36 36

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 C4○D4 C4○D12 S32 D4⋊2S3 C2×S32 C32⋊4D6 D6.3D6 D6.4D6 C2×C32⋊4D6 C62.96D6 kernel C62.96D6 C33⋊9(C2×C4) C33⋊9D4 C33⋊5Q8 C6×C3⋊Dic3 C3×C32⋊7D4 C2×C3⋊Dic3 C32⋊7D4 C3⋊Dic3 C2×C3⋊S3 C62 C33 C32 C2×C6 C32 C6 C22 C3 C3 C2 C1 # reps 1 2 1 1 1 2 1 2 4 2 3 2 4 3 2 3 2 4 2 2 4

Matrix representation of C62.96D6 in GL4(𝔽7) generated by

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

C62.96D6 in GAP, Magma, Sage, TeX

`C_6^2._{96}D_6`
`% in TeX`

`G:=Group("C6^2.96D6");`
`// GroupNames label`

`G:=SmallGroup(432,693);`
`// by ID`

`G=gap.SmallGroup(432,693);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,1124,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^5>;`
`// generators/relations`

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