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## G = A4⋊Q8order 96 = 25·3

### The semidirect product of A4 and Q8 acting via Q8/C4=C2

Aliases: A4⋊Q8, C4.1S4, C22⋊Dic6, C23.1D6, A4⋊C4.C2, C2.3(C2×S4), (C4×A4).1C2, (C22×C4).2S3, (C2×A4).1C22, SmallGroup(96,185)

Series: Derived Chief Lower central Upper central

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

Generators and relations for A4⋊Q8
G = < a,b,c,d,e | a2=b2=c3=d4=1, e2=d2, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Character table of A4⋊Q8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 12A 12B size 1 1 3 3 8 2 6 12 12 12 12 8 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 2 2 -1 -2 -2 0 0 0 0 -1 1 1 orthogonal lifted from D6 ρ6 2 2 2 2 -1 2 2 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 -2 2 -2 2 0 0 0 0 0 0 -2 0 0 symplectic lifted from Q8, Schur index 2 ρ8 2 -2 2 -2 -1 0 0 0 0 0 0 1 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ9 2 -2 2 -2 -1 0 0 0 0 0 0 1 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ10 3 3 -1 -1 0 -3 1 1 1 -1 -1 0 0 0 orthogonal lifted from C2×S4 ρ11 3 3 -1 -1 0 3 -1 1 -1 1 -1 0 0 0 orthogonal lifted from S4 ρ12 3 3 -1 -1 0 3 -1 -1 1 -1 1 0 0 0 orthogonal lifted from S4 ρ13 3 3 -1 -1 0 -3 1 -1 -1 1 1 0 0 0 orthogonal lifted from C2×S4 ρ14 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of A4⋊Q8
On 24 points - transitive group 24T87
Generators in S24
```(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(1 16 11)(2 13 12)(3 14 9)(4 15 10)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(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)```

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

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

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

`G:=TransitiveGroup(24,87);`

On 24 points - transitive group 24T131
Generators in S24
```(1 7)(2 8)(3 5)(4 6)(9 15)(10 16)(11 13)(12 14)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 13)(10 14)(11 15)(12 16)(17 23)(18 24)(19 21)(20 22)
(1 15 23)(2 16 24)(3 13 21)(4 14 22)(5 9 17)(6 10 18)(7 11 19)(8 12 20)
(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 6 3 8)(2 5 4 7)(9 22 11 24)(10 21 12 23)(13 20 15 18)(14 19 16 17)```

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

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

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

`G:=TransitiveGroup(24,131);`

A4⋊Q8 is a maximal subgroup of
A4⋊Q16  C82S4  A4⋊SD16  A42Q16  C24.10D6  D42S4  Q8×S4  Dic3.S4  A4⋊Dic6  A5⋊Q8  A4⋊Dic10  C20.1S4
A4⋊Q8 is a maximal quotient of
Q8⋊Dic6  Q8.Dic6  SL2(𝔽3)⋊Q8  C24.3D6  C24.4D6  C12.1S4  Dic3.S4  A4⋊Dic6  A4⋊Dic10  C20.1S4

Matrix representation of A4⋊Q8 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 12 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 12 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 12 12 11 0 0 0 0 1
,
 8 0 0 0 0 0 5 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 0 12 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 12 12 11 0 0 0 0 1

`G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,12,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,11,1],[8,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,11,1] >;`

A4⋊Q8 in GAP, Magma, Sage, TeX

`A_4\rtimes Q_8`
`% in TeX`

`G:=Group("A4:Q8");`
`// GroupNames label`

`G:=SmallGroup(96,185);`
`// by ID`

`G=gap.SmallGroup(96,185);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-2,2,24,73,31,387,1444,202,869,347]);`
`// Polycyclic`

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

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