Aliases: Q8⋊1A4, C23.5A4, C22⋊SL2(𝔽3), (C22×Q8)⋊3C3, C2.1(C22⋊A4), SmallGroup(96,203)
Series: Derived ►Chief ►Lower central ►Upper central
| C22×Q8 — Q8⋊A4 |
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 >
Character table of Q8⋊A4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | |
| size | 1 | 1 | 3 | 3 | 16 | 16 | 6 | 6 | 6 | 6 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 2 | -2 | 2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ5 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ6 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ7 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ8 | 3 | 3 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 3 | 0 | 0 | orthogonal lifted from A4 |
| ρ9 | 3 | 3 | -1 | -1 | 0 | 0 | -1 | 3 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ10 | 3 | 3 | -1 | -1 | 0 | 0 | 3 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | -1 | -1 | 0 | 0 | -1 | -1 | 3 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ12 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►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 9 3 11)(2 12 4 10)(5 22 7 24)(6 21 8 23)(13 17 15 19)(14 20 16 18)
(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 16)(2 5 18)(3 21 14)(4 7 20)(6 13 12)(8 15 10)(9 24 17)(11 22 19)
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), (1,9,3,11)(2,12,4,10)(5,22,7,24)(6,21,8,23)(13,17,15,19)(14,20,16,18), (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), (1,23,16)(2,5,18)(3,21,14)(4,7,20)(6,13,12)(8,15,10)(9,24,17)(11,22,19)>;
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), (1,9,3,11)(2,12,4,10)(5,22,7,24)(6,21,8,23)(13,17,15,19)(14,20,16,18), (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), (1,23,16)(2,5,18)(3,21,14)(4,7,20)(6,13,12)(8,15,10)(9,24,17)(11,22,19) );
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)], [(1,9,3,11),(2,12,4,10),(5,22,7,24),(6,21,8,23),(13,17,15,19),(14,20,16,18)], [(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)], [(1,23,16),(2,5,18),(3,21,14),(4,7,20),(6,13,12),(8,15,10),(9,24,17),(11,22,19)]])
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)
| 4 | 12 | 0 | 0 | 0 |
| 4 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 1 | 12 | 0 |
| 10 | 3 | 0 | 0 | 0 |
| 1 | 3 | 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 | 1 | 12 |
| 0 | 0 | 1 | 0 | 12 |
| 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 |
| 10 | 4 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
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
Export
Subgroup lattice of Q8⋊A4 in TeX
Character table of Q8⋊A4 in TeX