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## G = Q8.S4order 192 = 26·3

### 2nd non-split extension by Q8 of S4 acting via S4/C22=S3

Aliases: Q8.2S4, C23.5S4, 2+ 1+4.3S3, C2.5(C22⋊S4), C23⋊A4.3C2, SmallGroup(192,1492)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2+ 1+4 — C23⋊A4 — Q8.S4
 Chief series C1 — C2 — Q8 — 2+ 1+4 — C23⋊A4 — Q8.S4
 Lower central C23⋊A4 — Q8.S4
 Upper central C1 — C2

Generators and relations for Q8.S4
G = < a,b,c,d,e,f | a4=c2=d2=e3=1, b2=f2=a2, bab-1=cac=dad=fbf-1=a-1, eae-1=a-1b, faf-1=dbd=a2b, bc=cb, ebe-1=a, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >

Subgroups: 333 in 66 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2 [×2], C3, C4 [×4], C22 [×5], C6, C8 [×2], C2×C4 [×4], D4 [×4], Q8 [×2], Q8 [×2], C23, C23 [×2], Dic3, A4 [×2], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×2], SL2(𝔽3) [×2], C2×A4 [×2], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, CSU2(𝔽3) [×2], A4⋊C4, D4.9D4, C23⋊A4, Q8.S4
Quotients: C1, C2, S3, S4 [×3], C22⋊S4, Q8.S4

Character table of Q8.S4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6 8A 8B size 1 1 6 12 32 6 6 12 12 24 32 24 24 ρ1 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 linear of order 2 ρ3 2 2 2 2 -1 2 2 0 0 0 -1 0 0 orthogonal lifted from S3 ρ4 3 3 3 -1 0 -1 -1 1 1 1 0 -1 -1 orthogonal lifted from S4 ρ5 3 3 -1 -1 0 3 -1 1 1 -1 0 -1 1 orthogonal lifted from S4 ρ6 3 3 3 -1 0 -1 -1 -1 -1 -1 0 1 1 orthogonal lifted from S4 ρ7 3 3 -1 -1 0 3 -1 -1 -1 1 0 1 -1 orthogonal lifted from S4 ρ8 3 3 -1 -1 0 -1 3 -1 -1 1 0 -1 1 orthogonal lifted from S4 ρ9 3 3 -1 -1 0 -1 3 1 1 -1 0 1 -1 orthogonal lifted from S4 ρ10 4 -4 0 0 1 0 0 2i -2i 0 -1 0 0 complex faithful ρ11 4 -4 0 0 1 0 0 -2i 2i 0 -1 0 0 complex faithful ρ12 6 6 -2 2 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from C22⋊S4 ρ13 8 -8 0 0 -1 0 0 0 0 0 1 0 0 symplectic faithful, Schur index 2

Permutation representations of Q8.S4
On 16 points - transitive group 16T443
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 3 11)(2 12 4 10)(5 14 7 16)(6 13 8 15)
(1 4)(2 3)(5 13)(6 16)(7 15)(8 14)(9 10)(11 12)
(1 11)(2 10)(3 9)(4 12)(5 6)(7 8)(13 16)(14 15)
(2 9 10)(4 11 12)(5 6 13)(7 8 15)
(1 16 3 14)(2 7 4 5)(6 10 8 12)(9 15 11 13)```

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

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

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

`G:=TransitiveGroup(16,443);`

On 16 points - transitive group 16T446
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 3 11)(2 12 4 10)(5 14 7 16)(6 13 8 15)
(2 4)(5 7)(10 12)(14 16)
(2 4)(6 8)(9 11)(14 16)
(2 9 10)(4 11 12)(5 14 8)(6 7 16)
(1 15 3 13)(2 8 4 6)(5 12 7 10)(9 14 11 16)```

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

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

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

`G:=TransitiveGroup(16,446);`

Matrix representation of Q8.S4 in GL4(𝔽5) generated by

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

Q8.S4 in GAP, Magma, Sage, TeX

`Q_8.S_4`
`% in TeX`

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

`G:=SmallGroup(192,1492);`
`// by ID`

`G=gap.SmallGroup(192,1492);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,57,254,135,171,262,1684,1271,718,172,1013,2532,530,285,124]);`
`// Polycyclic`

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

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