non-abelian, soluble, monomial
Aliases: C8:1S4, A4:1D8, C22:D24, C23.10D12, C4:S4:1C2, (C8xA4):1C2, C2.8(C4:S4), C4.18(C2xS4), (C22xC8):2S3, (C2xA4).3D4, (C22xC4).15D6, (C4xA4).10C22, SmallGroup(192,961)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for A4:D8
G = < a,b,c,d,e | a2=b2=c3=d8=e2=1, cac-1=eae=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 442 in 79 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2xC4, D4, C23, C23, C12, A4, D6, C22:C4, C4:C4, C2xC8, D8, C22xC4, C2xD4, C24, D12, S4, C2xA4, D4:C4, C2.D8, C4:D4, C22xC8, C2xD8, D24, C4xA4, C2xS4, C8:7D4, C8xA4, C4:S4, A4:D8
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, S4, D24, C2xS4, C4:S4, A4:D8
Character table of A4:D8
class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 24A | 24B | 24C | 24D | |
size | 1 | 1 | 3 | 3 | 24 | 24 | 8 | 2 | 6 | 24 | 24 | 8 | 2 | 2 | 6 | 6 | 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 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 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 | 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 | -1 | -1 | linear of order 2 |
ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ7 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ8 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -√2 | √2 | √2 | -√2 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
ρ9 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | √2 | -√2 | -√2 | √2 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -√2 | √2 | √2 | -√2 | √3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | orthogonal lifted from D24 |
ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | √2 | -√2 | -√2 | √2 | -√3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | orthogonal lifted from D24 |
ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -√2 | √2 | √2 | -√2 | -√3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | orthogonal lifted from D24 |
ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | √2 | -√2 | -√2 | √2 | √3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | orthogonal lifted from D24 |
ρ16 | 3 | 3 | -1 | -1 | 1 | 1 | 0 | 3 | -1 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
ρ17 | 3 | 3 | -1 | -1 | 1 | -1 | 0 | 3 | -1 | 1 | -1 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
ρ18 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | -1 | 1 | 1 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
ρ19 | 3 | 3 | -1 | -1 | -1 | 1 | 0 | 3 | -1 | -1 | 1 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
ρ20 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | -6 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4:S4 |
ρ21 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3√2 | -3√2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
ρ22 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3√2 | 3√2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 22 11)(2 23 12)(3 24 13)(4 17 14)(5 18 15)(6 19 16)(7 20 9)(8 21 10)
(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 7)(2 6)(3 5)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 24)(16 23)
G:=sub<Sym(24)| (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,22,11)(2,23,12)(3,24,13)(4,17,14)(5,18,15)(6,19,16)(7,20,9)(8,21,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,7)(2,6)(3,5)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,24)(16,23)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,22,11)(2,23,12)(3,24,13)(4,17,14)(5,18,15)(6,19,16)(7,20,9)(8,21,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,7)(2,6)(3,5)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,24)(16,23) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24)], [(1,22,11),(2,23,12),(3,24,13),(4,17,14),(5,18,15),(6,19,16),(7,20,9),(8,21,10)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,7),(2,6),(3,5),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,24),(16,23)]])
G:=TransitiveGroup(24,324);
Matrix representation of A4:D8 ►in GL5(F73)
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 72 | 0 | 0 |
0 | 0 | 72 | 0 | 1 |
0 | 0 | 72 | 1 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 72 |
0 | 0 | 1 | 0 | 72 |
0 | 0 | 0 | 0 | 72 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 72 | 0 |
0 | 0 | 1 | 72 | 0 |
0 | 0 | 0 | 72 | 1 |
32 | 72 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
32 | 72 | 0 | 0 | 0 |
1 | 41 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,72,72,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,72,72,72,0,0,0,0,1],[32,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[32,1,0,0,0,72,41,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
A4:D8 in GAP, Magma, Sage, TeX
A_4\rtimes D_8
% in TeX
G:=Group("A4:D8");
// GroupNames label
G:=SmallGroup(192,961);
// by ID
G=gap.SmallGroup(192,961);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,85,92,254,58,1124,4037,285,2358,475]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=e^2=1,c*a*c^-1=e*a*e=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=c^-1,e*d*e=d^-1>;
// generators/relations
Export