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## G = Q8.A4order 96 = 25·3

### The non-split extension by Q8 of A4 acting through Inn(Q8)

Aliases: Q8.A4, Q8SL2(𝔽3), 2+ 1+41C3, SL2(𝔽3)⋊4C22, C4○D4.C6, C4.A43C2, C4.2(C2×A4), Q8.2(C2×C6), C2.6(C22×A4), SmallGroup(96,201)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8.A4
G = < a,b,c,d,e | a4=e3=1, b2=c2=d2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece-1=a2cd, ede-1=c >

Character table of Q8.A4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 6A 6B 12A 12B 12C 12D 12E 12F size 1 1 6 6 6 4 4 2 2 2 6 4 4 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 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 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 linear of order 3 ρ6 1 1 -1 -1 1 ζ3 ζ32 -1 1 -1 1 ζ32 ζ3 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 linear of order 6 ρ7 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 linear of order 3 ρ8 1 1 1 -1 -1 ζ32 ζ3 -1 -1 1 1 ζ3 ζ32 ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 linear of order 6 ρ9 1 1 -1 1 -1 ζ32 ζ3 1 -1 -1 1 ζ3 ζ32 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 linear of order 6 ρ10 1 1 1 -1 -1 ζ3 ζ32 -1 -1 1 1 ζ32 ζ3 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 linear of order 6 ρ11 1 1 -1 1 -1 ζ3 ζ32 1 -1 -1 1 ζ32 ζ3 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 linear of order 6 ρ12 1 1 -1 -1 1 ζ32 ζ3 -1 1 -1 1 ζ3 ζ32 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 linear of order 6 ρ13 3 3 1 -1 1 0 0 3 -3 -3 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 -1 1 1 0 0 -3 -3 3 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 3 -1 -1 -1 0 0 3 3 3 -1 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ16 3 3 1 1 -1 0 0 -3 3 -3 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 4 -4 0 0 0 -2 -2 0 0 0 0 2 2 0 0 0 0 0 0 orthogonal faithful ρ18 4 -4 0 0 0 1-√-3 1+√-3 0 0 0 0 -1-√-3 -1+√-3 0 0 0 0 0 0 complex faithful ρ19 4 -4 0 0 0 1+√-3 1-√-3 0 0 0 0 -1+√-3 -1-√-3 0 0 0 0 0 0 complex faithful

Permutation representations of Q8.A4
On 24 points - transitive group 24T137
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 11 3 9)(2 10 4 12)(5 23 7 21)(6 22 8 24)(13 18 15 20)(14 17 16 19)
(1 4 3 2)(5 23 7 21)(6 24 8 22)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 12 3 10)(2 9 4 11)(5 6 7 8)(13 18 15 20)(14 19 16 17)(21 24 23 22)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 18 10)(6 19 11)(7 20 12)(8 17 9)```

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

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

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

`G:=TransitiveGroup(24,137);`

Q8.A4 is a maximal subgroup of
Q8.4S4  Q8.5S4  Q16.A4  SD16.A4  Q8.6S4  Q8.7S4  2- 1+43C6  Ω4+ (𝔽3)  Dic6.A4  Q8.A5  Dic10.A4
Q8.A4 is a maximal quotient of
C4○D4⋊C12  SL2(𝔽3)⋊6D4  Q8×SL2(𝔽3)  2+ 1+4⋊C9  Dic6.A4  Dic10.A4

Matrix representation of Q8.A4 in GL4(ℚ) generated by

 0 0 1 0 0 0 0 -1 -1 0 0 0 0 1 0 0
,
 0 1 0 0 -1 0 0 0 0 0 0 1 0 0 -1 0
,
 0 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 0
,
 0 0 -1 0 0 0 0 -1 1 0 0 0 0 1 0 0
,
 -1/2 1/2 1/2 1/2 -1/2 -1/2 -1/2 1/2 -1/2 1/2 -1/2 -1/2 -1/2 -1/2 1/2 -1/2
`G:=sub<GL(4,Rationals())| [0,0,-1,0,0,0,0,1,1,0,0,0,0,-1,0,0],[0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,1,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[0,0,1,0,0,0,0,1,-1,0,0,0,0,-1,0,0],[-1/2,-1/2,-1/2,-1/2,1/2,-1/2,1/2,-1/2,1/2,-1/2,-1/2,1/2,1/2,1/2,-1/2,-1/2] >;`

Q8.A4 in GAP, Magma, Sage, TeX

`Q_8.A_4`
`% in TeX`

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

`G:=SmallGroup(96,201);`
`// by ID`

`G=gap.SmallGroup(96,201);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,2,-2,288,601,295,159,117,286,202,88]);`
`// Polycyclic`

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

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