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G = C4⋊(C32⋊C4)  order 144 = 24·32

The semidirect product of C4 and C32⋊C4 acting via C32⋊C4/C3⋊S3=C2

Aliases: C4⋊(C32⋊C4), (C3×C12)⋊2C4, C3⋊S3.4D4, C3⋊S3.2Q8, C323(C4⋊C4), C3⋊Dic35C4, (C4×C3⋊S3).9C2, (C3×C6).4(C2×C4), C2.5(C2×C32⋊C4), (C2×C32⋊C4).3C2, (C2×C3⋊S3).9C22, SmallGroup(144,133)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C4⋊(C32⋊C4)
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C4⋊(C32⋊C4)
 Lower central C32 — C3×C6 — C4⋊(C32⋊C4)
 Upper central C1 — C2 — C4

Generators and relations for C4⋊(C32⋊C4)
G = < a,b,c,d | a4=b3=c3=d4=1, ab=ba, ac=ca, dad-1=a-1, dcd-1=bc=cb, dbd-1=b-1c >

Character table of C4⋊(C32⋊C4)

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 12A 12B 12C 12D size 1 1 9 9 4 4 2 18 18 18 18 18 4 4 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 i -i i -i -1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 1 -i i -i i -1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 1 -1 -i -i i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 1 1 -1 i i -i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ9 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 4 4 0 0 -2 1 4 0 0 0 0 0 1 -2 1 1 -2 -2 orthogonal lifted from C32⋊C4 ρ12 4 4 0 0 1 -2 4 0 0 0 0 0 -2 1 -2 -2 1 1 orthogonal lifted from C32⋊C4 ρ13 4 4 0 0 1 -2 -4 0 0 0 0 0 -2 1 2 2 -1 -1 orthogonal lifted from C2×C32⋊C4 ρ14 4 4 0 0 -2 1 -4 0 0 0 0 0 1 -2 -1 -1 2 2 orthogonal lifted from C2×C32⋊C4 ρ15 4 -4 0 0 1 -2 0 0 0 0 0 0 2 -1 0 0 3i -3i complex faithful ρ16 4 -4 0 0 -2 1 0 0 0 0 0 0 -1 2 3i -3i 0 0 complex faithful ρ17 4 -4 0 0 1 -2 0 0 0 0 0 0 2 -1 0 0 -3i 3i complex faithful ρ18 4 -4 0 0 -2 1 0 0 0 0 0 0 -1 2 -3i 3i 0 0 complex faithful

Permutation representations of C4⋊(C32⋊C4)
On 24 points - transitive group 24T238
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)
(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 22)(2 21)(3 24)(4 23)(5 20 10 15)(6 19 11 14)(7 18 12 13)(8 17 9 16)```

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

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

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

`G:=TransitiveGroup(24,238);`

On 24 points - transitive group 24T239
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)
(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 24 3 22)(2 23 4 21)(5 20 12 13)(6 19 9 16)(7 18 10 15)(8 17 11 14)```

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

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

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

`G:=TransitiveGroup(24,239);`

Matrix representation of C4⋊(C32⋊C4) in GL4(𝔽5) generated by

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

C4⋊(C32⋊C4) in GAP, Magma, Sage, TeX

`C_4\rtimes (C_3^2\rtimes C_4)`
`% in TeX`

`G:=Group("C4:(C3^2:C4)");`
`// GroupNames label`

`G:=SmallGroup(144,133);`
`// by ID`

`G=gap.SmallGroup(144,133);`
`# by ID`

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

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

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