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## G = C4.A4order 48 = 24·3

### The central extension by C4 of A4

Aliases: C4.A4, Q8.C6, C4SL2(𝔽3), SL2(𝔽3)⋊2C2, C4○D4⋊C3, C2.3(C2×A4), SmallGroup(48,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C4.A4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C4.A4
 Lower central Q8 — C4.A4
 Upper central C1 — C4

Generators and relations for C4.A4
G = < a,b,c,d | a4=d3=1, b2=c2=a2, ab=ba, ac=ca, ad=da, cbc-1=a2b, dbd-1=a2bc, dcd-1=b >

Character table of C4.A4

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 12A 12B 12C 12D size 1 1 6 4 4 1 1 6 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 -1 ζ32 ζ3 -1 -1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ5 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ6 1 1 -1 ζ3 ζ32 -1 -1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ7 2 -2 0 -1 -1 2i -2i 0 1 1 i i -i -i complex faithful ρ8 2 -2 0 -1 -1 -2i 2i 0 1 1 -i -i i i complex faithful ρ9 2 -2 0 ζ65 ζ6 -2i 2i 0 ζ32 ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 complex faithful ρ10 2 -2 0 ζ65 ζ6 2i -2i 0 ζ32 ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 complex faithful ρ11 2 -2 0 ζ6 ζ65 2i -2i 0 ζ3 ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 complex faithful ρ12 2 -2 0 ζ6 ζ65 -2i 2i 0 ζ3 ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 complex faithful ρ13 3 3 1 0 0 -3 -3 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 -1 0 0 3 3 -1 0 0 0 0 0 0 orthogonal lifted from A4

Permutation representations of C4.A4
On 16 points - transitive group 16T60
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8 3 6)(2 5 4 7)(9 13 11 15)(10 14 12 16)
(1 15 3 13)(2 16 4 14)(5 12 7 10)(6 9 8 11)
(5 16 10)(6 13 11)(7 14 12)(8 15 9)```

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

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

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

`G:=TransitiveGroup(16,60);`

On 24 points - transitive group 24T21
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)
(1 4 3 2)(5 23 7 21)(6 24 8 22)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 12 3 10)(2 9 4 11)(5 6 7 8)(13 18 15 20)(14 19 16 17)(21 24 23 22)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 18 10)(6 19 11)(7 20 12)(8 17 9)```

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

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

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

`G:=TransitiveGroup(24,21);`

C4.A4 is a maximal subgroup of
U2(𝔽3)  C8.A4  C4.S4  C4.6S4  C4.3S4  Q8.A4  D4.A4  Dic3.A4  C4○D4⋊A4  2+ 1+4.3C6  C4.A5  Dic5.A4  Dic7.2A4  Q8.F7  C28.A4
C4.A4 is a maximal quotient of
C4×SL2(𝔽3)  Q8.C18  Dic3.A4  C424C4⋊C3  C232D4⋊C3  (C22×C4).A4  C23.19(C2×A4)  C4○D4⋊A4  Dic5.A4  Dic7.2A4  Q8.F7  C28.A4

Matrix representation of C4.A4 in GL2(𝔽5) generated by

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

C4.A4 in GAP, Magma, Sage, TeX

`C_4.A_4`
`% in TeX`

`G:=Group("C4.A4");`
`// GroupNames label`

`G:=SmallGroup(48,33);`
`// by ID`

`G=gap.SmallGroup(48,33);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,2,-2,120,97,72,188,133,58]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=d^3=1,b^2=c^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b*c,d*c*d^-1=b>;`
`// generators/relations`

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