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## G = C8.S4order 192 = 26·3

### 2nd non-split extension by C8 of S4 acting via S4/A4=C2

Aliases: C8.2S4, Q8.3D12, SL2(𝔽3).3D4, C2.9(C4⋊S4), C4.19(C2×S4), C4.S4.C2, C8○D4.1S3, C8.A4.1C2, C4○D4.8D6, C4.A4.7C22, SmallGroup(192,962)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C4.A4 — C8.S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — C4.S4 — C8.S4
 Lower central SL2(𝔽3) — C4.A4 — C8.S4
 Upper central C1 — C2 — C4 — C8

Generators and relations for C8.S4
G = < a,b,c,d,e | a8=d3=1, b2=c2=e2=a4, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a4b, dbd-1=a4bc, ebe-1=bc, dcd-1=b, ece-1=a4c, ede-1=d-1 >

Subgroups: 229 in 59 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, D4, Q8, Q8, Dic3, C12, C2×C8, M4(2), SD16, Q16, C2×Q8, C4○D4, C24, SL2(𝔽3), Dic6, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, Dic12, CSU2(𝔽3), C4.A4, D4.5D4, C8.A4, C4.S4, C8.S4
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2×S4, C4⋊S4, C8.S4

Character table of C8.S4

 class 1 2A 2B 3 4A 4B 4C 4D 6 8A 8B 8C 8D 8E 12A 12B 24A 24B 24C 24D size 1 1 6 8 2 6 24 24 8 2 2 12 24 24 8 8 8 8 8 8 ρ1 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 linear of order 2 ρ3 1 1 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 -1 -1 linear of order 2 ρ5 2 2 -2 2 -2 2 0 0 2 0 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 2 -1 2 2 0 0 -1 -2 -2 -2 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ7 2 2 2 -1 2 2 0 0 -1 2 2 2 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 -2 -1 -2 2 0 0 -1 0 0 0 0 0 1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ9 2 2 -2 -1 -2 2 0 0 -1 0 0 0 0 0 1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ10 3 3 -1 0 3 -1 -1 1 0 -3 -3 1 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ11 3 3 -1 0 3 -1 1 -1 0 -3 -3 1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ12 3 3 -1 0 3 -1 1 1 0 3 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ13 3 3 -1 0 3 -1 -1 -1 0 3 3 -1 1 1 0 0 0 0 0 0 orthogonal lifted from S4 ρ14 4 -4 0 -2 0 0 0 0 2 -2√2 2√2 0 0 0 0 0 -√2 -√2 √2 √2 symplectic faithful, Schur index 2 ρ15 4 -4 0 -2 0 0 0 0 2 2√2 -2√2 0 0 0 0 0 √2 √2 -√2 -√2 symplectic faithful, Schur index 2 ρ16 4 -4 0 1 0 0 0 0 -1 -2√2 2√2 0 0 0 -√3 √3 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83+ζ8ζ32 symplectic faithful, Schur index 2 ρ17 4 -4 0 1 0 0 0 0 -1 2√2 -2√2 0 0 0 √3 -√3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87+ζ85ζ3 symplectic faithful, Schur index 2 ρ18 4 -4 0 1 0 0 0 0 -1 2√2 -2√2 0 0 0 -√3 √3 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 symplectic faithful, Schur index 2 ρ19 4 -4 0 1 0 0 0 0 -1 -2√2 2√2 0 0 0 √3 -√3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 symplectic faithful, Schur index 2 ρ20 6 6 2 0 -6 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4

Smallest permutation representation of C8.S4
On 64 points
Generators in S64
```(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(9 61 13 57)(10 62 14 58)(11 63 15 59)(12 64 16 60)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 31)(17 64 21 60)(18 57 22 61)(19 58 23 62)(20 59 24 63)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)
(17 64 27)(18 57 28)(19 58 29)(20 59 30)(21 60 31)(22 61 32)(23 62 25)(24 63 26)(33 43 51)(34 44 52)(35 45 53)(36 46 54)(37 47 55)(38 48 56)(39 41 49)(40 42 50)
(1 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 56 21 52)(18 55 22 51)(19 54 23 50)(20 53 24 49)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(41 59 45 63)(42 58 46 62)(43 57 47 61)(44 64 48 60)```

`G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,61,13,57)(10,62,14,58)(11,63,15,59)(12,64,16,60)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (17,64,27)(18,57,28)(19,58,29)(20,59,30)(21,60,31)(22,61,32)(23,62,25)(24,63,26)(33,43,51)(34,44,52)(35,45,53)(36,46,54)(37,47,55)(38,48,56)(39,41,49)(40,42,50), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,59,45,63)(42,58,46,62)(43,57,47,61)(44,64,48,60)>;`

`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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,61,13,57)(10,62,14,58)(11,63,15,59)(12,64,16,60)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (17,64,27)(18,57,28)(19,58,29)(20,59,30)(21,60,31)(22,61,32)(23,62,25)(24,63,26)(33,43,51)(34,44,52)(35,45,53)(36,46,54)(37,47,55)(38,48,56)(39,41,49)(40,42,50), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,59,45,63)(42,58,46,62)(43,57,47,61)(44,64,48,60) );`

`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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,53,5,49),(2,54,6,50),(3,55,7,51),(4,56,8,52),(9,61,13,57),(10,62,14,58),(11,63,15,59),(12,64,16,60),(17,31,21,27),(18,32,22,28),(19,25,23,29),(20,26,24,30),(33,43,37,47),(34,44,38,48),(35,45,39,41),(36,46,40,42)], [(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,32,13,28),(10,25,14,29),(11,26,15,30),(12,27,16,31),(17,64,21,60),(18,57,22,61),(19,58,23,62),(20,59,24,63),(41,49,45,53),(42,50,46,54),(43,51,47,55),(44,52,48,56)], [(17,64,27),(18,57,28),(19,58,29),(20,59,30),(21,60,31),(22,61,32),(23,62,25),(24,63,26),(33,43,51),(34,44,52),(35,45,53),(36,46,54),(37,47,55),(38,48,56),(39,41,49),(40,42,50)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,56,21,52),(18,55,22,51),(19,54,23,50),(20,53,24,49),(25,40,29,36),(26,39,30,35),(27,38,31,34),(28,37,32,33),(41,59,45,63),(42,58,46,62),(43,57,47,61),(44,64,48,60)]])`

Matrix representation of C8.S4 in GL4(𝔽7) generated by

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

C8.S4 in GAP, Magma, Sage, TeX

`C_8.S_4`
`% in TeX`

`G:=Group("C8.S4");`
`// GroupNames label`

`G:=SmallGroup(192,962);`
`// by ID`

`G=gap.SmallGroup(192,962);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,708,2102,520,451,1684,655,172,1013,404,285,124]);`
`// Polycyclic`

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

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