direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C8, C24⋊4C2, C8○Dic3, D6.2C4, C4.12D6, Dic3.2C4, C12.12C22, C8○(C3⋊C8), C3⋊C8⋊6C2, C3⋊1(C2×C8), C2.1(C4×S3), C6.1(C2×C4), (C4×S3).3C2, SmallGroup(48,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C8 |
Generators and relations for S3×C8
G = < a,b,c | a8=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of S3×C8
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 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 | -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 | -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 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | i | -i | -i | i | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | i | -i | -i | i | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -i | i | i | -i | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | i | i | -i | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | ζ8 | ζ83 | ζ85 | ζ87 | ζ85 | ζ83 | ζ87 | ζ8 | -i | i | ζ8 | ζ85 | ζ83 | ζ87 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | ζ87 | ζ85 | ζ83 | ζ8 | ζ87 | ζ8 | ζ85 | ζ83 | i | -i | ζ87 | ζ83 | ζ85 | ζ8 | linear of order 8 |
| ρ11 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | ζ83 | ζ8 | ζ87 | ζ85 | ζ87 | ζ8 | ζ85 | ζ83 | i | -i | ζ83 | ζ87 | ζ8 | ζ85 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | ζ85 | ζ87 | ζ8 | ζ83 | ζ85 | ζ83 | ζ87 | ζ8 | -i | i | ζ85 | ζ8 | ζ87 | ζ83 | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | ζ83 | ζ8 | ζ87 | ζ85 | ζ83 | ζ85 | ζ8 | ζ87 | i | -i | ζ83 | ζ87 | ζ8 | ζ85 | linear of order 8 |
| ρ14 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | ζ87 | ζ85 | ζ83 | ζ8 | ζ83 | ζ85 | ζ8 | ζ87 | i | -i | ζ87 | ζ83 | ζ85 | ζ8 | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | ζ8 | ζ83 | ζ85 | ζ87 | ζ8 | ζ87 | ζ83 | ζ85 | -i | i | ζ8 | ζ85 | ζ83 | ζ87 | linear of order 8 |
| ρ16 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | ζ85 | ζ87 | ζ8 | ζ83 | ζ8 | ζ87 | ζ83 | ζ85 | -i | i | ζ85 | ζ8 | ζ87 | ζ83 | linear of order 8 |
| ρ17 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ18 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 1 | 1 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ20 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 1 | 1 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ21 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | 2ζ87 | 2ζ85 | 2ζ83 | 2ζ8 | 0 | 0 | 0 | 0 | -i | i | ζ83 | ζ87 | ζ8 | ζ85 | complex faithful |
| ρ22 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | 2ζ8 | 2ζ83 | 2ζ85 | 2ζ87 | 0 | 0 | 0 | 0 | i | -i | ζ85 | ζ8 | ζ87 | ζ83 | complex faithful |
| ρ23 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | 2ζ83 | 2ζ8 | 2ζ87 | 2ζ85 | 0 | 0 | 0 | 0 | -i | i | ζ87 | ζ83 | ζ85 | ζ8 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | 2ζ85 | 2ζ87 | 2ζ8 | 2ζ83 | 0 | 0 | 0 | 0 | i | -i | ζ8 | ζ85 | ζ83 | ζ87 | complex faithful |
►On 24 points - transitive group 24T32
(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 13 21)(2 14 22)(3 15 23)(4 16 24)(5 9 17)(6 10 18)(7 11 19)(8 12 20)
(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 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), (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,9,17)(6,10,18)(7,11,19)(8,12,20), (9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)>;
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), (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,9,17)(6,10,18)(7,11,19)(8,12,20), (9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24) );
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)], [(1,13,21),(2,14,22),(3,15,23),(4,16,24),(5,9,17),(6,10,18),(7,11,19),(8,12,20)], [(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)]])
G:=TransitiveGroup(24,32);
S3×C8 is a maximal subgroup of
D6.C8 C8○D12 D12.C4 D8⋊3S3 Q8.7D6 D24⋊C2 C12.29D6 CU2(𝔽3) D15⋊2C8 D15⋊C8 D21⋊C8 C33⋊5(C2×C8) A5⋊C8 C8.A5
S3×C8 is a maximal quotient of
D6.C8 Dic3⋊C8 D6⋊C8 C12.29D6 D15⋊2C8 D15⋊C8 D21⋊C8 C33⋊5(C2×C8)
Matrix representation of S3×C8 ►in GL2(𝔽17) generated by
| 8 | 0 |
| 0 | 8 |
| 0 | 13 |
| 13 | 16 |
| 16 | 4 |
| 0 | 1 |
G:=sub<GL(2,GF(17))| [8,0,0,8],[0,13,13,16],[16,0,4,1] >;
S3×C8 in GAP, Magma, Sage, TeX
S_3\times C_8
% in TeX
G:=Group("S3xC8"); // GroupNames label
G:=SmallGroup(48,4);
// by ID
G=gap.SmallGroup(48,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,26,42,804]);
// Polycyclic
G:=Group<a,b,c|a^8=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of S3×C8 in TeX
Character table of S3×C8 in TeX