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## G = C8.A4order 96 = 25·3

### The central extension by C8 of A4

Aliases: C8.A4, Q8.C12, C8SL2(𝔽3), SL2(𝔽3).C4, C8○D4⋊C3, C8(C4.A4), C4.5(C2×A4), C2.3(C4×A4), C4.A4.3C2, C4○D4.2C6, SmallGroup(96,74)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C8.A4
 Chief series C1 — C2 — Q8 — C4○D4 — C4.A4 — C8.A4
 Lower central Q8 — C8.A4
 Upper central C1 — C8

Generators and relations for C8.A4
G = < a,b,c,d | a8=d3=1, b2=c2=a4, ab=ba, ac=ca, ad=da, cbc-1=a4b, dbd-1=a4bc, dcd-1=b >

Character table of C8.A4

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 6 4 4 1 1 6 4 4 1 1 1 1 6 6 4 4 4 4 4 4 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 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 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 -1 -1 -1 -1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 linear of order 6 ρ5 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 -1 -1 -1 -1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 linear of order 6 ρ7 1 1 -1 1 1 -1 -1 1 1 1 -i -i i i i -i -1 -1 -1 -1 i -i i i -i -i i -i linear of order 4 ρ8 1 1 -1 1 1 -1 -1 1 1 1 i i -i -i -i i -1 -1 -1 -1 -i i -i -i i i -i i linear of order 4 ρ9 1 1 -1 ζ3 ζ32 -1 -1 1 ζ32 ζ3 -i -i i i i -i ζ65 ζ6 ζ65 ζ6 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 linear of order 12 ρ10 1 1 -1 ζ32 ζ3 -1 -1 1 ζ3 ζ32 i i -i -i -i i ζ6 ζ65 ζ6 ζ65 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 linear of order 12 ρ11 1 1 -1 ζ32 ζ3 -1 -1 1 ζ3 ζ32 -i -i i i i -i ζ6 ζ65 ζ6 ζ65 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 linear of order 12 ρ12 1 1 -1 ζ3 ζ32 -1 -1 1 ζ32 ζ3 i i -i -i -i i ζ65 ζ6 ζ65 ζ6 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 linear of order 12 ρ13 2 -2 0 -1 -1 2i -2i 0 1 1 2ζ8 2ζ85 2ζ83 2ζ87 0 0 -i -i i i ζ83 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ85 complex faithful ρ14 2 -2 0 -1 -1 -2i 2i 0 1 1 2ζ83 2ζ87 2ζ8 2ζ85 0 0 i i -i -i ζ8 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ87 complex faithful ρ15 2 -2 0 -1 -1 -2i 2i 0 1 1 2ζ87 2ζ83 2ζ85 2ζ8 0 0 i i -i -i ζ85 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ83 complex faithful ρ16 2 -2 0 -1 -1 2i -2i 0 1 1 2ζ85 2ζ8 2ζ87 2ζ83 0 0 -i -i i i ζ87 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ8 complex faithful ρ17 2 -2 0 ζ6 ζ65 -2i 2i 0 ζ3 ζ32 2ζ87 2ζ83 2ζ85 2ζ8 0 0 ζ82ζ32 ζ82ζ3 ζ86ζ32 ζ86ζ3 ζ85ζ32 ζ87ζ3 ζ8ζ3 ζ85ζ3 ζ83ζ32 ζ87ζ32 ζ8ζ32 ζ83ζ3 complex faithful ρ18 2 -2 0 ζ65 ζ6 -2i 2i 0 ζ32 ζ3 2ζ83 2ζ87 2ζ8 2ζ85 0 0 ζ82ζ3 ζ82ζ32 ζ86ζ3 ζ86ζ32 ζ8ζ3 ζ83ζ32 ζ85ζ32 ζ8ζ32 ζ87ζ3 ζ83ζ3 ζ85ζ3 ζ87ζ32 complex faithful ρ19 2 -2 0 ζ65 ζ6 -2i 2i 0 ζ32 ζ3 2ζ87 2ζ83 2ζ85 2ζ8 0 0 ζ82ζ3 ζ82ζ32 ζ86ζ3 ζ86ζ32 ζ85ζ3 ζ87ζ32 ζ8ζ32 ζ85ζ32 ζ83ζ3 ζ87ζ3 ζ8ζ3 ζ83ζ32 complex faithful ρ20 2 -2 0 ζ6 ζ65 -2i 2i 0 ζ3 ζ32 2ζ83 2ζ87 2ζ8 2ζ85 0 0 ζ82ζ32 ζ82ζ3 ζ86ζ32 ζ86ζ3 ζ8ζ32 ζ83ζ3 ζ85ζ3 ζ8ζ3 ζ87ζ32 ζ83ζ32 ζ85ζ32 ζ87ζ3 complex faithful ρ21 2 -2 0 ζ65 ζ6 2i -2i 0 ζ32 ζ3 2ζ85 2ζ8 2ζ87 2ζ83 0 0 ζ86ζ3 ζ86ζ32 ζ82ζ3 ζ82ζ32 ζ87ζ3 ζ85ζ32 ζ83ζ32 ζ87ζ32 ζ8ζ3 ζ85ζ3 ζ83ζ3 ζ8ζ32 complex faithful ρ22 2 -2 0 ζ6 ζ65 2i -2i 0 ζ3 ζ32 2ζ8 2ζ85 2ζ83 2ζ87 0 0 ζ86ζ32 ζ86ζ3 ζ82ζ32 ζ82ζ3 ζ83ζ32 ζ8ζ3 ζ87ζ3 ζ83ζ3 ζ85ζ32 ζ8ζ32 ζ87ζ32 ζ85ζ3 complex faithful ρ23 2 -2 0 ζ65 ζ6 2i -2i 0 ζ32 ζ3 2ζ8 2ζ85 2ζ83 2ζ87 0 0 ζ86ζ3 ζ86ζ32 ζ82ζ3 ζ82ζ32 ζ83ζ3 ζ8ζ32 ζ87ζ32 ζ83ζ32 ζ85ζ3 ζ8ζ3 ζ87ζ3 ζ85ζ32 complex faithful ρ24 2 -2 0 ζ6 ζ65 2i -2i 0 ζ3 ζ32 2ζ85 2ζ8 2ζ87 2ζ83 0 0 ζ86ζ32 ζ86ζ3 ζ82ζ32 ζ82ζ3 ζ87ζ32 ζ85ζ3 ζ83ζ3 ζ87ζ3 ζ8ζ32 ζ85ζ32 ζ83ζ32 ζ8ζ3 complex faithful ρ25 3 3 -1 0 0 3 3 -1 0 0 3 3 3 3 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ26 3 3 -1 0 0 3 3 -1 0 0 -3 -3 -3 -3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ27 3 3 1 0 0 -3 -3 -1 0 0 3i 3i -3i -3i i -i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4×A4 ρ28 3 3 1 0 0 -3 -3 -1 0 0 -3i -3i 3i 3i -i i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4×A4

Smallest permutation representation of C8.A4
On 32 points
Generators in S32
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)
(1 23 5 19)(2 24 6 20)(3 17 7 21)(4 18 8 22)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)
(9 17 27)(10 18 28)(11 19 29)(12 20 30)(13 21 31)(14 22 32)(15 23 25)(16 24 26)```

`G:=sub<Sym(32)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(9,31,13,27)(10,32,14,28)(11,25,15,29)(12,26,16,30), (9,17,27)(10,18,28)(11,19,29)(12,20,30)(13,21,31)(14,22,32)(15,23,25)(16,24,26)>;`

`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)(25,26,27,28,29,30,31,32), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(9,31,13,27)(10,32,14,28)(11,25,15,29)(12,26,16,30), (9,17,27)(10,18,28)(11,19,29)(12,20,30)(13,21,31)(14,22,32)(15,23,25)(16,24,26) );`

`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),(25,26,27,28,29,30,31,32)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26)], [(1,23,5,19),(2,24,6,20),(3,17,7,21),(4,18,8,22),(9,31,13,27),(10,32,14,28),(11,25,15,29),(12,26,16,30)], [(9,17,27),(10,18,28),(11,19,29),(12,20,30),(13,21,31),(14,22,32),(15,23,25),(16,24,26)])`

C8.A4 is a maximal subgroup of
C8.7S4  C16.A4  C8.S4  CU2(𝔽3)  C8.5S4  C8.4S4  C8.3S4  M4(2).A4  Q16.A4  SD16.A4  D8.A4  SL2(𝔽3).Dic3  C8.A5  SL2(𝔽3).Dic5  SL2(𝔽3).F5
C8.A4 is a maximal quotient of
C8×SL2(𝔽3)  Q8.C36  SL2(𝔽3).Dic3  SL2(𝔽3).Dic5  SL2(𝔽3).F5

Matrix representation of C8.A4 in GL2(𝔽17) generated by

 2 0 0 2
,
 4 0 3 13
,
 13 5 0 4
,
 16 13 13 0
`G:=sub<GL(2,GF(17))| [2,0,0,2],[4,3,0,13],[13,0,5,4],[16,13,13,0] >;`

C8.A4 in GAP, Magma, Sage, TeX

`C_8.A_4`
`% in TeX`

`G:=Group("C8.A4");`
`// GroupNames label`

`G:=SmallGroup(96,74);`
`// by ID`

`G=gap.SmallGroup(96,74);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,-2,2,-2,36,158,297,117,550,202,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=d^3=1,b^2=c^2=a^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,d*c*d^-1=b>;`
`// generators/relations`

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