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## G = Q8⋊3S3order 48 = 24·3

### The semidirect product of Q8 and S3 acting through Inn(Q8)

Aliases: Q83S3, Q8Dic3, C4.7D6, D124C2, C6.8C23, C12.7C22, D6.3C22, Dic3.5C22, (C4×S3)⋊3C2, C33(C4○D4), (C3×Q8)⋊3C2, C2.9(C22×S3), SmallGroup(48,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Q8⋊3S3
 Chief series C1 — C3 — C6 — D6 — C4×S3 — Q8⋊3S3
 Lower central C3 — C6 — Q8⋊3S3
 Upper central C1 — C2 — Q8

Generators and relations for Q83S3
G = < a,b,c,d | a4=c3=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Character table of Q83S3

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6 12A 12B 12C size 1 1 6 6 6 2 2 2 2 3 3 2 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 -1 -1 linear of order 2 ρ6 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ7 1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 linear of order 2 ρ8 1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ9 2 2 0 0 0 -1 2 2 2 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 0 0 -1 -2 2 -2 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 0 0 0 -1 2 -2 -2 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ12 2 2 0 0 0 -1 -2 -2 2 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ13 2 -2 0 0 0 2 0 0 0 2i -2i -2 0 0 0 complex lifted from C4○D4 ρ14 2 -2 0 0 0 2 0 0 0 -2i 2i -2 0 0 0 complex lifted from C4○D4 ρ15 4 -4 0 0 0 -2 0 0 0 0 0 2 0 0 0 orthogonal faithful, Schur index 2

Permutation representations of Q83S3
On 24 points - transitive group 24T28
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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 19)(14 18)(15 17)(16 20)(21 24)(22 23)```

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

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

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

`G:=TransitiveGroup(24,28);`

Matrix representation of Q83S3 in GL4(𝔽5) generated by

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

Q83S3 in GAP, Magma, Sage, TeX

`Q_8\rtimes_3S_3`
`% in TeX`

`G:=Group("Q8:3S3");`
`// GroupNames label`

`G:=SmallGroup(48,41);`
`// by ID`

`G=gap.SmallGroup(48,41);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-3,46,182,97,42,804]);`
`// Polycyclic`

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

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