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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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