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G = Q8⋊A4order 96 = 25·3

1st semidirect product of Q8 and A4 acting via A4/C22=C3

non-abelian, soluble

Aliases: Q81A4, C23.5A4, C22⋊SL2(𝔽3), (C22×Q8)⋊3C3, C2.1(C22⋊A4), SmallGroup(96,203)

Series: Derived Chief Lower central Upper central

C1C2C22×Q8 — Q8⋊A4
C1C2Q8C22×Q8 — Q8⋊A4
C22×Q8 — Q8⋊A4
C1C2

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, eae-1=b, bc=cb, bd=db, ebe-1=ab, ece-1=cd=dc, ede-1=c >

3C2
3C2
16C3
3C22
3C22
3C4
3C4
3C4
3C4
16C6
3Q8
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3Q8
3Q8
3Q8
4A4
3C2×Q8
3C22×C4
3C2×Q8
3C2×Q8
3C2×Q8
4C2×A4
4SL2(𝔽3)
4SL2(𝔽3)
4SL2(𝔽3)
4SL2(𝔽3)

Character table of Q8⋊A4

 class 12A2B2C3A3B4A4B4C4D6A6B
 size 1133161666661616
ρ1111111111111    trivial
ρ21111ζ3ζ321111ζ32ζ3    linear of order 3
ρ31111ζ32ζ31111ζ3ζ32    linear of order 3
ρ42-22-2-1-1000011    symplectic lifted from SL2(𝔽3), Schur index 2
ρ52-22-2ζ65ζ60000ζ32ζ3    complex lifted from SL2(𝔽3)
ρ62-22-2ζ6ζ650000ζ3ζ32    complex lifted from SL2(𝔽3)
ρ7333300-1-1-1-100    orthogonal lifted from A4
ρ833-1-100-1-1-1300    orthogonal lifted from A4
ρ933-1-100-13-1-100    orthogonal lifted from A4
ρ1033-1-1003-1-1-100    orthogonal lifted from A4
ρ1133-1-100-1-13-100    orthogonal lifted from A4
ρ126-6-2200000000    symplectic faithful, Schur index 2

Permutation representations of Q8⋊A4
On 24 points - transitive group 24T88
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 22 7 24)(6 21 8 23)(13 19 15 17)(14 18 16 20)
(1 3)(2 4)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(1 23 14)(2 5 18)(3 21 16)(4 7 20)(6 15 10)(8 13 12)(9 22 19)(11 24 17)

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

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

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

G:=TransitiveGroup(24,88);

Q8⋊A4 is a maximal subgroup of   C24.A4  (C22×C4).A4  Q8.1S4  Q8⋊S4  C4○D4⋊A4  A4×SL2(𝔽3)
Q8⋊A4 is a maximal quotient of   C24.7A4  Q8⋊SL2(𝔽3)  C22⋊(Q8⋊C9)

Matrix representation of Q8⋊A4 in GL5(𝔽13)

412000
49000
000121
000120
001120
,
103000
13000
001200
001201
001210
,
10000
01000
000112
001012
000012
,
10000
01000
001200
001201
001210
,
104000
10000
00001
00100
00010

G:=sub<GL(5,GF(13))| [4,4,0,0,0,12,9,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[10,1,0,0,0,3,3,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[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],[10,1,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;

Q8⋊A4 in GAP, Magma, Sage, TeX

Q_8\rtimes A_4
% in TeX

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

G:=SmallGroup(96,203);
// by ID

G=gap.SmallGroup(96,203);
# by ID

G:=PCGroup([6,-3,-2,2,-2,2,-2,73,164,579,117,1084,202,88]);
// 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,e*a*e^-1=b,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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

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