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## G = C33⋊2M4(2)  order 432 = 24·33

### 2nd semidirect product of C33 and M4(2) acting via M4(2)/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊2M4(2)
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊8(C2×C4) — C33⋊2M4(2)
 Lower central C33 — C32×C6 — C33⋊2M4(2)
 Upper central C1 — C2

Generators and relations for C332M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, dbd-1=ab=ba, ac=ca, dad-1=a-1b, eae=a-1, bc=cb, ebe=b-1, dcd-1=ece=c-1, ede=d5 >

Subgroups: 864 in 92 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, S3 [×9], C6, C6 [×4], C8 [×2], C2×C4, C32, C32 [×4], Dic3, Dic3 [×2], C12 [×3], D6 [×5], M4(2), C3⋊S3 [×9], C3×C6, C3×C6 [×4], C3⋊C8, C24, C4×S3 [×3], C33, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×5], C8⋊S3, C33⋊C2, C32×C6, C322C8, C322C8, C6.D6 [×2], C4×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, C32⋊M4(2), C3×C322C8, C334C8, C338(C2×C4), C332M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, M4(2), C4×S3, C32⋊C4, C8⋊S3, C2×C32⋊C4, C32⋊M4(2), S3×C32⋊C4, C332M4(2)

Character table of C332M4(2)

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D size 1 1 54 2 4 4 8 8 6 9 9 2 4 4 8 8 18 18 54 54 12 12 12 12 18 18 18 18 18 18 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 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 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 -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 -1 -1 -1 linear of order 2 ρ5 1 1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 i -i -i i 1 1 1 1 -1 -1 -i i i -i linear of order 4 ρ6 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 i -i i -i -1 -1 -1 -1 -1 -1 -i i i -i linear of order 4 ρ7 1 1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -i i i -i 1 1 1 1 -1 -1 i -i -i i linear of order 4 ρ8 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -i i -i i -1 -1 -1 -1 -1 -1 i -i -i i linear of order 4 ρ9 2 2 0 -1 2 2 -1 -1 0 2 2 -1 2 2 -1 -1 -2 -2 0 0 0 0 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 0 -1 2 2 -1 -1 0 2 2 -1 2 2 -1 -1 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 0 -1 2 2 -1 -1 0 -2 -2 -1 2 2 -1 -1 2i -2i 0 0 0 0 0 0 1 1 i -i -i i complex lifted from C4×S3 ρ12 2 -2 0 2 2 2 2 2 0 -2i 2i -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 complex lifted from M4(2) ρ13 2 -2 0 2 2 2 2 2 0 2i -2i -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from M4(2) ρ14 2 2 0 -1 2 2 -1 -1 0 -2 -2 -1 2 2 -1 -1 -2i 2i 0 0 0 0 0 0 1 1 -i i i -i complex lifted from C4×S3 ρ15 2 -2 0 -1 2 2 -1 -1 0 2i -2i 1 -2 -2 1 1 0 0 0 0 0 0 0 0 i -i 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 complex lifted from C8⋊S3 ρ16 2 -2 0 -1 2 2 -1 -1 0 -2i 2i 1 -2 -2 1 1 0 0 0 0 0 0 0 0 -i i 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 complex lifted from C8⋊S3 ρ17 2 -2 0 -1 2 2 -1 -1 0 -2i 2i 1 -2 -2 1 1 0 0 0 0 0 0 0 0 -i i 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 complex lifted from C8⋊S3 ρ18 2 -2 0 -1 2 2 -1 -1 0 2i -2i 1 -2 -2 1 1 0 0 0 0 0 0 0 0 i -i 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 complex lifted from C8⋊S3 ρ19 4 4 0 4 1 -2 -2 1 4 0 0 4 -2 1 -2 1 0 0 0 0 -2 -2 1 1 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ20 4 4 0 4 -2 1 1 -2 4 0 0 4 1 -2 1 -2 0 0 0 0 1 1 -2 -2 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ21 4 4 0 4 1 -2 -2 1 -4 0 0 4 -2 1 -2 1 0 0 0 0 2 2 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ22 4 4 0 4 -2 1 1 -2 -4 0 0 4 1 -2 1 -2 0 0 0 0 -1 -1 2 2 0 0 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ23 4 -4 0 4 1 -2 -2 1 0 0 0 -4 2 -1 2 -1 0 0 0 0 0 0 -3i 3i 0 0 0 0 0 0 complex lifted from C32⋊M4(2) ρ24 4 -4 0 4 -2 1 1 -2 0 0 0 -4 -1 2 -1 2 0 0 0 0 3i -3i 0 0 0 0 0 0 0 0 complex lifted from C32⋊M4(2) ρ25 4 -4 0 4 -2 1 1 -2 0 0 0 -4 -1 2 -1 2 0 0 0 0 -3i 3i 0 0 0 0 0 0 0 0 complex lifted from C32⋊M4(2) ρ26 4 -4 0 4 1 -2 -2 1 0 0 0 -4 2 -1 2 -1 0 0 0 0 0 0 3i -3i 0 0 0 0 0 0 complex lifted from C32⋊M4(2) ρ27 8 8 0 -4 2 -4 2 -1 0 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×C32⋊C4 ρ28 8 8 0 -4 -4 2 -1 2 0 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×C32⋊C4 ρ29 8 -8 0 -4 2 -4 2 -1 0 0 0 4 4 -2 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ30 8 -8 0 -4 -4 2 -1 2 0 0 0 4 -2 4 1 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C332M4(2)
On 24 points - transitive group 24T1309
Generators in S24
```(1 13 22)(2 14 23)(3 24 15)(4 17 16)(5 9 18)(6 10 19)(7 20 11)(8 21 12)
(1 22 13)(3 15 24)(5 18 9)(7 11 20)
(1 13 22)(2 23 14)(3 15 24)(4 17 16)(5 9 18)(6 19 10)(7 11 20)(8 21 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 6)(4 8)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)```

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

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

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

`G:=TransitiveGroup(24,1309);`

Matrix representation of C332M4(2) in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 1 0 0 0 0 0 0 72 0 0 72 0 0 0 0 0 0 1 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 1 2 0 0 71 0 1 2 0 0 72 1 0 1 0 0 72 71 2 2
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 70 0 0 0 0 3 67 0 0 0 0 0 0 0 0 0 72 0 0 72 0 0 0 0 0 71 0 1 1 0 0 1 72 0 72
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 1 0 0 0 0 0 1 0 0 72 72 1 2 0 0 0 1 0 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,72,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,71,72,72,0,0,72,0,1,71,0,0,1,1,0,2,0,0,2,2,1,2],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,3,0,0,0,0,70,67,0,0,0,0,0,0,0,72,71,1,0,0,0,0,0,72,0,0,0,0,1,0,0,0,72,0,1,72],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,72,0,72,0,0,0,1,0,72,1,0,0,0,0,1,0,0,0,1,1,2,0] >;`

C332M4(2) in GAP, Magma, Sage, TeX

`C_3^3\rtimes_2M_4(2)`
`% in TeX`

`G:=Group("C3^3:2M4(2)");`
`// GroupNames label`

`G:=SmallGroup(432,573);`
`// by ID`

`G=gap.SmallGroup(432,573);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,141,36,58,1411,298,1356,1027,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b,e*a*e=a^-1,b*c=c*b,e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;`
`// generators/relations`

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