Copied to
clipboard

G = A4⋊Q8order 96 = 25·3

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

non-abelian, soluble, monomial

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

C1C22C2×A4 — A4⋊Q8
C1C22A4C2×A4A4⋊C4 — A4⋊Q8
A4C2×A4 — A4⋊Q8
C1C2C4

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 >

3C2
3C2
4C3
3C22
3C22
3C4
6C4
6C4
6C4
6C4
4C6
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C2×C4
6Q8
6Q8
4Dic3
4Dic3
4C12
3C4⋊C4
3C4⋊C4
3C22⋊C4
3C4⋊C4
3C22⋊C4
3C2×Q8
4Dic6
3C22⋊Q8

Character table of A4⋊Q8

 class 12A2B2C34A4B4C4D4E4F612A12B
 size 113382612121212888
ρ111111111111111    trivial
ρ21111111-1-1-1-1111    linear of order 2
ρ311111-1-11-1-111-1-1    linear of order 2
ρ411111-1-1-111-11-1-1    linear of order 2
ρ52222-1-2-20000-111    orthogonal lifted from D6
ρ62222-1220000-1-1-1    orthogonal lifted from S3
ρ72-22-22000000-200    symplectic lifted from Q8, Schur index 2
ρ82-22-2-100000013-3    symplectic lifted from Dic6, Schur index 2
ρ92-22-2-10000001-33    symplectic lifted from Dic6, Schur index 2
ρ1033-1-10-3111-1-1000    orthogonal lifted from C2×S4
ρ1133-1-103-11-11-1000    orthogonal lifted from S4
ρ1233-1-103-1-11-11000    orthogonal lifted from S4
ρ1333-1-10-31-1-111000    orthogonal lifted from C2×S4
ρ146-6-220000000000    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)

10000
01000
001200
00010
0001212
,
10000
01000
00100
000120
0012012
,
10000
01000
00010
00121211
00001
,
80000
05000
001200
000120
000012
,
012000
10000
00100
00121211
00001

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

Export

Subgroup lattice of A4⋊Q8 in TeX
Character table of A4⋊Q8 in TeX

׿
×
𝔽