direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D8, D4⋊C6, C8⋊1C6, C24⋊3C2, C6.14D4, C12.17C22, (C3×D4)⋊4C2, C4.1(C2×C6), C2.3(C3×D4), SmallGroup(48,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D8
G = < a,b,c | a3=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C3×D8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ17 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ18 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | -ζ83ζ3+ζ8ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ87ζ32+ζ85ζ32 | complex faithful |
| ρ19 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | -ζ87ζ32+ζ85ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ83ζ3+ζ8ζ3 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | -ζ87ζ3+ζ85ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ83ζ32+ζ8ζ32 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | -ζ83ζ32+ζ8ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ87ζ3+ζ85ζ3 | complex faithful |
►On 24 points - transitive group 24T40
Generators in S24
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 17)(8 9 18)
(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)
G:=sub<Sym(24)| (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,17)(8,9,18), (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)>;
G:=Group( (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,17)(8,9,18), (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) );
G=PermutationGroup([[(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,17),(8,9,18)], [(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)]])
G:=TransitiveGroup(24,40);
C3×D8 is a maximal subgroup of
C3⋊D16 D8.S3 D8⋊S3 D8⋊3S3 D8.A4 D56⋊C3 D4⋊F7
C3×D8 is a maximal quotient of D56⋊C3 D4⋊F7
Matrix representation of C3×D8 ►in GL2(𝔽7) generated by
| 4 | 0 |
| 0 | 4 |
| 0 | 6 |
| 1 | 3 |
| 3 | 1 |
| 6 | 4 |
G:=sub<GL(2,GF(7))| [4,0,0,4],[0,1,6,3],[3,6,1,4] >;
C3×D8 in GAP, Magma, Sage, TeX
C_3\times D_8
% in TeX
G:=Group("C3xD8"); // GroupNames label
G:=SmallGroup(48,25);
// by ID
G=gap.SmallGroup(48,25);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-2,141,723,368,58]);
// Polycyclic
G:=Group<a,b,c|a^3=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×D8 in TeX
Character table of C3×D8 in TeX