direct product, metabelian, soluble, monomial
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
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 |
►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 17 11 19)(10 20 12 18)
(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 9)(2 13 10)(3 14 11)(4 15 12)(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)(9,17,11,19)(10,20,12,18), (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), (1,16,9)(2,13,10)(3,14,11)(4,15,12)(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)(9,17,11,19)(10,20,12,18), (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), (1,16,9)(2,13,10)(3,14,11)(4,15,12)(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),(9,17,11,19),(10,20,12,18)], [(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)], [(1,16,9),(2,13,10),(3,14,11),(4,15,12),(5,19,21),(6,20,22),(7,17,23),(8,18,24)]])
G:=TransitiveGroup(24,86);
Q8×A4 is a maximal subgroup of
A4⋊2Q16 Q8⋊3S4 Q8⋊4S4
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
Export
Subgroup lattice of Q8×A4 in TeX
Character table of Q8×A4 in TeX