metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8:2D6, D8:2S3, D4:2D6, D6.6D4, C24:4C22, C12.2C23, Dic3.8D4, Dic6:1C22, D12.1C22, D4:S3:2C2, (C3xD8):4C2, (S3xD4):2C2, C3:C8:1C22, C8:S3:3C2, C24:C2:3C2, C3:2(C8:C22), D4.S3:1C2, C6.28(C2xD4), C2.16(S3xD4), D4:2S3:1C2, (C3xD4):2C22, C4.2(C22xS3), (C4xS3).1C22, SmallGroup(96,118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8:S3
G = < a,b,c,d | a8=b2=c3=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >
Subgroups: 194 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2xC6, M4(2), D8, D8, SD16, C2xD4, C4oD4, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C3xD4, C22xS3, C8:C22, C8:S3, C24:C2, D4:S3, D4.S3, C3xD8, S3xD4, D4:2S3, D8:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, C8:C22, S3xD4, D8:S3
Character table of D8:S3
class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 12 | 24A | 24B | |
size | 1 | 1 | 4 | 4 | 6 | 12 | 2 | 2 | 6 | 12 | 2 | 8 | 8 | 4 | 12 | 4 | 4 | 4 | |
ρ1 | 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 | linear of order 2 |
ρ3 | 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 | linear of order 2 |
ρ5 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | linear of order 2 |
ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | linear of order 2 |
ρ9 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | -1 | 1 | 1 | 2 | 0 | -1 | -1 | -1 | orthogonal lifted from D6 |
ρ12 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | -1 | -1 | 1 | -2 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
ρ13 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | 2 | 0 | 0 | -1 | 1 | -1 | -2 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
ρ14 | 2 | 2 | 0 | 0 | -2 | 0 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
ρ15 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from S3xD4 |
ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | √-6 | -√-6 | complex faithful |
ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -√-6 | √-6 | complex faithful |
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 17 16)(8 18 9)
(1 5)(3 7)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,17,16)(8,18,9), (1,5)(3,7)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)>;
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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,17,16)(8,18,9), (1,5)(3,7)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21) );
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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24)], [(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,17,16),(8,18,9)], [(1,5),(3,7),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)]])
G:=TransitiveGroup(24,141);
D8:S3 is a maximal subgroup of
D8:13D6 SD16:D6 D8:11D6 S3xC8:C22 D8:4D6 D8:5D6 D8:6D6 D8:D9 D24:S3 C24:6D6 Dic6:3D6 D12.D6 D12:9D6 D12:5D6 C24:8D6 D40:S3 C40:8D6 D60.C22 D30.8D4 D20:10D6 Dic6:D10 D8:D15
D8:S3 is a maximal quotient of
D4.S3:C4 D4:Dic6 Dic6:2D4 D4.Dic6 C4:C4.D6 C12:Q8:C2 C4:C4:19D6 D4:(C4xS3) D6:5SD16 D6.SD16 D6:C8:11C2 C3:C8:1D4 D4:3D12 C3:C8:D4 D4:S3:C4 D12.D4 Dic3.Q16 C24:4Q8 C8:S3:C4 D6.2Q16 C2.D8:S3 C8:3D12 C24:C2:C4 D12.2Q8 Dic3:D8 D8:Dic3 (C6xD8).C2 C24:11D4 D12:D4 Dic6:D4 C24:12D4 D8:D9 D24:S3 C24:6D6 Dic6:3D6 D12.D6 D12:9D6 D12:5D6 C24:8D6 D40:S3 C40:8D6 D60.C22 D30.8D4 D20:10D6 Dic6:D10 D8:D15
Matrix representation of D8:S3 ►in GL4(F5) generated by
0 | 4 | 3 | 3 |
1 | 4 | 3 | 0 |
0 | 2 | 0 | 4 |
0 | 0 | 1 | 1 |
4 | 2 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 3 | 0 | 1 |
0 | 2 | 1 | 0 |
3 | 0 | 0 | 2 |
0 | 0 | 1 | 1 |
4 | 4 | 4 | 3 |
1 | 0 | 0 | 1 |
4 | 0 | 0 | 2 |
0 | 1 | 1 | 1 |
0 | 0 | 4 | 3 |
0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [0,1,0,0,4,4,2,0,3,3,0,1,3,0,4,1],[4,0,0,0,2,1,3,2,0,0,0,1,0,0,1,0],[3,0,4,1,0,0,4,0,0,1,4,0,2,1,3,1],[4,0,0,0,0,1,0,0,0,1,4,0,2,1,3,1] >;
D8:S3 in GAP, Magma, Sage, TeX
D_8\rtimes S_3
% in TeX
G:=Group("D8:S3");
// GroupNames label
G:=SmallGroup(96,118);
// by ID
G=gap.SmallGroup(96,118);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,116,297,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export