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G = C33⋊4C8order 216 = 23·33

2nd semidirect product of C33 and C8 acting via C8/C2=C4

Aliases: C334C8, C6.(C32⋊C4), C3⋊(C322C8), C324(C3⋊C8), C2.(C33⋊C4), C3⋊Dic3.2S3, (C32×C6).2C4, (C3×C6).5Dic3, (C3×C3⋊Dic3).4C2, SmallGroup(216,118)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊4C8
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊4C8
 Lower central C33 — C33⋊4C8
 Upper central C1 — C2

Generators and relations for C334C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b-1, dcd-1=c-1 >

Character table of C334C8

 class 1 2 3A 3B 3C 3D 3E 3F 3G 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B size 1 1 2 4 4 4 4 4 4 9 9 2 4 4 4 4 4 4 27 27 27 27 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 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 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -i i i -i -1 -1 linear of order 4 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 i -i -i i -1 -1 linear of order 4 ρ5 1 -1 1 1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 -1 -1 ζ83 ζ85 ζ8 ζ87 -i i linear of order 8 ρ6 1 -1 1 1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 -1 -1 ζ87 ζ8 ζ85 ζ83 -i i linear of order 8 ρ7 1 -1 1 1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 -1 -1 ζ8 ζ87 ζ83 ζ85 i -i linear of order 8 ρ8 1 -1 1 1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 -1 -1 ζ85 ζ83 ζ87 ζ8 i -i linear of order 8 ρ9 2 2 -1 2 -1 2 -1 -1 -1 2 2 -1 -1 2 -1 -1 -1 2 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ10 2 2 -1 2 -1 2 -1 -1 -1 -2 -2 -1 -1 2 -1 -1 -1 2 0 0 0 0 1 1 symplectic lifted from Dic3, Schur index 2 ρ11 2 -2 -1 2 -1 2 -1 -1 -1 2i -2i 1 1 -2 1 1 1 -2 0 0 0 0 -i i complex lifted from C3⋊C8 ρ12 2 -2 -1 2 -1 2 -1 -1 -1 -2i 2i 1 1 -2 1 1 1 -2 0 0 0 0 i -i complex lifted from C3⋊C8 ρ13 4 4 4 1 -2 -2 1 1 -2 0 0 4 -2 1 1 1 -2 -2 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ14 4 4 4 -2 1 1 -2 -2 1 0 0 4 1 -2 -2 -2 1 1 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ15 4 -4 4 -2 1 1 -2 -2 1 0 0 -4 -1 2 2 2 -1 -1 0 0 0 0 0 0 symplectic lifted from C32⋊2C8, Schur index 2 ρ16 4 -4 4 1 -2 -2 1 1 -2 0 0 -4 2 -1 -1 -1 2 2 0 0 0 0 0 0 symplectic lifted from C32⋊2C8, Schur index 2 ρ17 4 4 -2 1 1 -2 -1+3√-3/2 -1-3√-3/2 1 0 0 -2 1 1 -1+3√-3/2 -1-3√-3/2 1 -2 0 0 0 0 0 0 complex lifted from C33⋊C4 ρ18 4 4 -2 1 1 -2 -1-3√-3/2 -1+3√-3/2 1 0 0 -2 1 1 -1-3√-3/2 -1+3√-3/2 1 -2 0 0 0 0 0 0 complex lifted from C33⋊C4 ρ19 4 -4 -2 -2 -1+3√-3/2 1 1 1 -1-3√-3/2 0 0 2 1-3√-3/2 2 -1 -1 1+3√-3/2 -1 0 0 0 0 0 0 complex faithful ρ20 4 4 -2 -2 -1-3√-3/2 1 1 1 -1+3√-3/2 0 0 -2 -1-3√-3/2 -2 1 1 -1+3√-3/2 1 0 0 0 0 0 0 complex lifted from C33⋊C4 ρ21 4 -4 -2 -2 -1-3√-3/2 1 1 1 -1+3√-3/2 0 0 2 1+3√-3/2 2 -1 -1 1-3√-3/2 -1 0 0 0 0 0 0 complex faithful ρ22 4 -4 -2 1 1 -2 -1+3√-3/2 -1-3√-3/2 1 0 0 2 -1 -1 1-3√-3/2 1+3√-3/2 -1 2 0 0 0 0 0 0 complex faithful ρ23 4 -4 -2 1 1 -2 -1-3√-3/2 -1+3√-3/2 1 0 0 2 -1 -1 1+3√-3/2 1-3√-3/2 -1 2 0 0 0 0 0 0 complex faithful ρ24 4 4 -2 -2 -1+3√-3/2 1 1 1 -1-3√-3/2 0 0 -2 -1+3√-3/2 -2 1 1 -1-3√-3/2 1 0 0 0 0 0 0 complex lifted from C33⋊C4

Permutation representations of C334C8
On 24 points - transitive group 24T552
Generators in S24
```(2 22 13)(4 15 24)(6 18 9)(8 11 20)
(1 12 21)(2 22 13)(3 23 14)(4 15 24)(5 16 17)(6 18 9)(7 19 10)(8 11 20)
(1 21 12)(2 13 22)(3 23 14)(4 15 24)(5 17 16)(6 9 18)(7 19 10)(8 11 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)```

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

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

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

`G:=TransitiveGroup(24,552);`

C334C8 is a maximal subgroup of
S3×C322C8  C335(C2×C8)  C33⋊M4(2)  C332M4(2)  C33⋊D8  C336SD16  C337SD16  C33⋊Q16  C337(C2×C8)  C334M4(2)  C3312M4(2)
C334C8 is a maximal quotient of
C334C16

Matrix representation of C334C8 in GL4(𝔽7) generated by

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

C334C8 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_4C_8`
`% in TeX`

`G:=Group("C3^3:4C8");`
`// GroupNames label`

`G:=SmallGroup(216,118);`
`// by ID`

`G=gap.SmallGroup(216,118);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,3,-3,12,31,963,201,964,730,5189]);`
`// Polycyclic`

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

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