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## G = C32⋊M4(2)  order 144 = 24·32

### The semidirect product of C32 and M4(2) acting via M4(2)/C4=C4

Aliases: C321M4(2), C4.(C32⋊C4), (C3×C12).3C4, C322C83C2, C3⋊Dic3.8C22, (C2×C3⋊S3).6C4, (C4×C3⋊S3).7C2, (C3×C6).2(C2×C4), C2.4(C2×C32⋊C4), SmallGroup(144,131)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C32⋊M4(2)
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C32⋊M4(2)
 Lower central C32 — C3×C6 — C32⋊M4(2)
 Upper central C1 — C2 — C4

Generators and relations for C32⋊M4(2)
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=ab-1, dad=a-1, cbc-1=a-1b-1, dbd=b-1, dcd=c5 >

Character table of C32⋊M4(2)

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 18 4 4 2 9 9 4 4 18 18 18 18 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 1 1 -1 -1 -1 1 1 -i i i -i -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 1 1 1 -1 -1 1 1 -i -i i i 1 1 1 1 linear of order 4 ρ7 1 1 -1 1 1 1 -1 -1 1 1 i i -i -i 1 1 1 1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 1 1 i -i -i i -1 -1 -1 -1 linear of order 4 ρ9 2 -2 0 2 2 0 -2i 2i -2 -2 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ10 2 -2 0 2 2 0 2i -2i -2 -2 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ11 4 4 0 -2 1 -4 0 0 1 -2 0 0 0 0 -1 -1 2 2 orthogonal lifted from C2×C32⋊C4 ρ12 4 4 0 1 -2 4 0 0 -2 1 0 0 0 0 -2 -2 1 1 orthogonal lifted from C32⋊C4 ρ13 4 4 0 1 -2 -4 0 0 -2 1 0 0 0 0 2 2 -1 -1 orthogonal lifted from C2×C32⋊C4 ρ14 4 4 0 -2 1 4 0 0 1 -2 0 0 0 0 1 1 -2 -2 orthogonal lifted from C32⋊C4 ρ15 4 -4 0 -2 1 0 0 0 -1 2 0 0 0 0 -3i 3i 0 0 complex faithful ρ16 4 -4 0 1 -2 0 0 0 2 -1 0 0 0 0 0 0 3i -3i complex faithful ρ17 4 -4 0 -2 1 0 0 0 -1 2 0 0 0 0 3i -3i 0 0 complex faithful ρ18 4 -4 0 1 -2 0 0 0 2 -1 0 0 0 0 0 0 -3i 3i complex faithful

Permutation representations of C32⋊M4(2)
On 24 points - transitive group 24T243
Generators in S24
```(1 11 22)(2 12 23)(3 24 13)(4 17 14)(5 15 18)(6 16 19)(7 20 9)(8 21 10)
(2 23 12)(4 14 17)(6 19 16)(8 10 21)
(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 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)```

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

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

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

`G:=TransitiveGroup(24,243);`

C32⋊M4(2) is a maximal subgroup of
C4.S3≀C2  (C3×C12).D4  (C3×C24).C4  C8.(C32⋊C4)  C326C4≀C2  C327C4≀C2  C32⋊D8⋊C2  C32⋊Q16⋊C2  C3⋊S3⋊M4(2)  C62.(C2×C4)  C12⋊S3.C4  C332M4(2)  C334M4(2)
C32⋊M4(2) is a maximal quotient of
(C3×C12)⋊4C8  C322C8⋊C4  C62.6(C2×C4)  He31M4(2)  C332M4(2)  C334M4(2)

Matrix representation of C32⋊M4(2) in GL4(𝔽5) generated by

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

C32⋊M4(2) in GAP, Magma, Sage, TeX

`C_3^2\rtimes M_4(2)`
`% in TeX`

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

`G:=SmallGroup(144,131);`
`// by ID`

`G=gap.SmallGroup(144,131);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,121,55,50,3364,256,4613,881]);`
`// Polycyclic`

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

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