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## G = Q8⋊3S4order 192 = 26·3

### The semidirect product of Q8 and S4 acting via S4/A4=C2

Aliases: Q83S4, A43SD16, A4⋊C83C2, C4⋊S4.2C2, C4.4(C2×S4), (Q8×A4)⋊1C2, (C2×A4).10D4, (C22×Q8)⋊1S3, (C22×C4).4D6, (C4×A4).4C22, C22⋊(Q82S3), C2.7(A4⋊D4), C23.20(C3⋊D4), SmallGroup(192,976)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C4×A4 — Q8⋊3S4
 Chief series C1 — C22 — A4 — C2×A4 — C4×A4 — C4⋊S4 — Q8⋊3S4
 Lower central A4 — C2×A4 — C4×A4 — Q8⋊3S4
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q83S4
G = < a,b,c,d,e,f | a4=c2=d2=e3=f2=1, b2=a2, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=a-1b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 354 in 77 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, D4, Q8, Q8, C23, C23, C12, A4, D6, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C3⋊C8, D12, C3×Q8, S4, C2×A4, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, Q82S3, C4×A4, C4×A4, C2×S4, Q8⋊D4, A4⋊C8, C4⋊S4, Q8×A4, Q83S4
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊D4, S4, Q82S3, C2×S4, A4⋊D4, Q83S4

Character table of Q83S4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6 8A 8B 8C 8D 12A 12B 12C size 1 1 3 3 24 8 2 4 6 12 24 8 12 12 12 12 16 16 16 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 2 2 0 2 -2 0 -2 0 0 2 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ6 2 2 2 2 0 -1 2 2 2 2 0 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 0 -1 2 -2 2 -2 0 -1 0 0 0 0 1 1 -1 orthogonal lifted from D6 ρ8 2 2 2 2 0 -1 -2 0 -2 0 0 -1 0 0 0 0 -√-3 √-3 1 complex lifted from C3⋊D4 ρ9 2 2 2 2 0 -1 -2 0 -2 0 0 -1 0 0 0 0 √-3 -√-3 1 complex lifted from C3⋊D4 ρ10 2 -2 -2 2 0 2 0 0 0 0 0 -2 √-2 -√-2 -√-2 √-2 0 0 0 complex lifted from SD16 ρ11 2 -2 -2 2 0 2 0 0 0 0 0 -2 -√-2 √-2 √-2 -√-2 0 0 0 complex lifted from SD16 ρ12 3 3 -1 -1 -1 0 3 -3 -1 1 1 0 -1 1 -1 1 0 0 0 orthogonal lifted from C2×S4 ρ13 3 3 -1 -1 -1 0 3 3 -1 -1 1 0 1 -1 1 -1 0 0 0 orthogonal lifted from S4 ρ14 3 3 -1 -1 1 0 3 3 -1 -1 -1 0 -1 1 -1 1 0 0 0 orthogonal lifted from S4 ρ15 3 3 -1 -1 1 0 3 -3 -1 1 -1 0 1 -1 1 -1 0 0 0 orthogonal lifted from C2×S4 ρ16 4 -4 -4 4 0 -2 0 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ17 6 6 -2 -2 0 0 -6 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ18 6 -6 2 -2 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 0 0 0 complex faithful ρ19 6 -6 2 -2 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 0 0 0 complex faithful

Permutation representations of Q83S4
On 24 points - transitive group 24T326
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 23 3 21)(2 22 4 24)(5 16 7 14)(6 15 8 13)(9 19 11 17)(10 18 12 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 11 16)(2 12 13)(3 9 14)(4 10 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(2 4)(5 20)(6 19)(7 18)(8 17)(9 14)(10 13)(11 16)(12 15)(21 22)(23 24)```

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

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

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

`G:=TransitiveGroup(24,326);`

Matrix representation of Q83S4 in GL5(𝔽73)

 72 12 0 0 0 12 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 72 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 72 0 0 0 1 72 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 72 0 0 1 0 72 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 1 0 0 0 72 0 0 0 0 72 0 1
,
 1 0 0 0 0 61 72 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(5,GF(73))| [72,12,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,72,72,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,1,0,0,0,0,0,0,1],[1,61,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;`

Q83S4 in GAP, Magma, Sage, TeX

`Q_8\rtimes_3S_4`
`% in TeX`

`G:=Group("Q8:3S4");`
`// GroupNames label`

`G:=SmallGroup(192,976);`
`// by ID`

`G=gap.SmallGroup(192,976);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,85,64,254,135,58,1124,4037,285,2358,475]);`
`// Polycyclic`

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

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