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## G = Q8×A4order 96 = 25·3

### Direct product of Q8 and A4

Aliases: Q8×A4, C22⋊(C3×Q8), C4.1(C2×A4), (C22×C4).C6, (C4×A4).3C2, (C22×Q8)⋊2C3, C23.7(C2×C6), C2.4(C22×A4), (C2×A4).8C22, SmallGroup(96,199)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — Q8×A4
 Chief series C1 — C22 — C23 — C2×A4 — C4×A4 — Q8×A4
 Lower central C22 — C23 — Q8×A4
 Upper central C1 — C2 — Q8

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

Character table of Q8×A4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 12A 12B 12C 12D 12E 12F size 1 1 3 3 4 4 2 2 2 6 6 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 1 trivial ρ2 1 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 -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 1 linear of order 2 ρ5 1 1 1 1 ζ3 ζ32 -1 1 -1 -1 -1 1 ζ3 ζ32 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 1 1 1 ζ32 ζ3 1 -1 -1 -1 1 -1 ζ32 ζ3 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 linear of order 6 ρ8 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 linear of order 3 ρ9 1 1 1 1 ζ3 ζ32 1 -1 -1 -1 1 -1 ζ3 ζ32 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 linear of order 6 ρ10 1 1 1 1 ζ32 ζ3 -1 -1 1 1 -1 -1 ζ32 ζ3 ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 linear of order 6 ρ11 1 1 1 1 ζ3 ζ32 -1 -1 1 1 -1 -1 ζ3 ζ32 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 linear of order 6 ρ12 1 1 1 1 ζ32 ζ3 -1 1 -1 -1 -1 1 ζ32 ζ3 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 linear of order 6 ρ13 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 2 -2 -1+√-3 -1-√-3 0 0 0 0 0 0 1-√-3 1+√-3 0 0 0 0 0 0 complex lifted from C3×Q8 ρ15 2 -2 2 -2 -1-√-3 -1+√-3 0 0 0 0 0 0 1+√-3 1-√-3 0 0 0 0 0 0 complex lifted from C3×Q8 ρ16 3 3 -1 -1 0 0 -3 3 -3 1 1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 3 3 -1 -1 0 0 3 3 3 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ18 3 3 -1 -1 0 0 -3 -3 3 -1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ19 3 3 -1 -1 0 0 3 -3 -3 1 -1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ20 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of Q8×A4
On 24 points - transitive group 24T86
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)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(1 16 11)(2 13 12)(3 14 9)(4 15 10)(5 19 21)(6 20 22)(7 17 23)(8 18 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), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (1,16,11)(2,13,12)(3,14,9)(4,15,10)(5,19,21)(6,20,22)(7,17,23)(8,18,24)>;

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

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

G:=TransitiveGroup(24,86);

Q8×A4 is a maximal subgroup of   A42Q16  Q83S4  Q84S4
Q8×A4 is a maximal quotient of   SL2(𝔽3)⋊3Q8

Matrix representation of Q8×A4 in GL5(𝔽13)

 12 12 0 0 0 2 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 0 0 0 0 10 5 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 12 0 1 0 0 12 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 12 0 0 0 1 12 0
,
 9 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 9 0 0 0 0 0 9 0

G:=sub<GL(5,GF(13))| [12,2,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,10,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[9,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,9,0,0,9,0,0] >;

Q8×A4 in GAP, Magma, Sage, TeX

Q_8\times A_4
% in TeX

G:=Group("Q8xA4");
// GroupNames label

G:=SmallGroup(96,199);
// by ID

G=gap.SmallGroup(96,199);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,2,72,169,79,376,665]);
// Polycyclic

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

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