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## G = C3.A4order 36 = 22·32

### The central extension by C3 of A4

Aliases: C3.A4, C22⋊C9, (C2×C6).C3, SmallGroup(36,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3.A4
 Chief series C1 — C22 — C2×C6 — C3.A4
 Lower central C22 — C3.A4
 Upper central C1 — C3

Generators and relations for C3.A4
G = < a,b,c,d | a3=b2=c2=1, d3=a, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Character table of C3.A4

 class 1 2 3A 3B 6A 6B 9A 9B 9C 9D 9E 9F size 1 3 1 1 3 3 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ3 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 ζ32 ζ3 ζ3 ζ32 ζ9 ζ95 ζ92 ζ97 ζ94 ζ98 linear of order 9 ρ5 1 1 ζ32 ζ3 ζ3 ζ32 ζ97 ζ98 ζ95 ζ94 ζ9 ζ92 linear of order 9 ρ6 1 1 ζ3 ζ32 ζ32 ζ3 ζ92 ζ9 ζ94 ζ95 ζ98 ζ97 linear of order 9 ρ7 1 1 ζ3 ζ32 ζ32 ζ3 ζ98 ζ94 ζ97 ζ92 ζ95 ζ9 linear of order 9 ρ8 1 1 ζ32 ζ3 ζ3 ζ32 ζ94 ζ92 ζ98 ζ9 ζ97 ζ95 linear of order 9 ρ9 1 1 ζ3 ζ32 ζ32 ζ3 ζ95 ζ97 ζ9 ζ98 ζ92 ζ94 linear of order 9 ρ10 3 -1 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ11 3 -1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 0 0 0 0 0 0 complex faithful ρ12 3 -1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 0 0 0 0 0 0 complex faithful

Permutation representations of C3.A4
On 18 points - transitive group 18T7
Generators in S18
```(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(2 17)(3 18)(5 11)(6 12)(8 14)(9 15)
(1 16)(3 18)(4 10)(6 12)(7 13)(9 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,7);`

C3.A4 is a maximal subgroup of
C3.S4  C9×A4  C9⋊A4  C32.A4  C42⋊C9  C24⋊C9  C21.A4  C39.A4
C3.A4 is a maximal quotient of
Q8⋊C9  C9.A4  C42⋊C9  C24⋊C9  C21.A4  C39.A4

Matrix representation of C3.A4 in GL3(𝔽7) generated by

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

C3.A4 in GAP, Magma, Sage, TeX

`C_3.A_4`
`% in TeX`

`G:=Group("C3.A4");`
`// GroupNames label`

`G:=SmallGroup(36,3);`
`// by ID`

`G=gap.SmallGroup(36,3);`
`# by ID`

`G:=PCGroup([4,-3,-3,-2,2,12,218,435]);`
`// Polycyclic`

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

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