metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3⋊C8, C6.C4, C4.2S3, C2.Dic3, C12.2C2, SmallGroup(24,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C3⋊C8 |
Generators and relations for C3⋊C8
G = < a,b | a3=b8=1, bab-1=a-1 >
Character table of C3⋊C8
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | -1 | -1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | i | -i | -1 | ζ85 | ζ83 | ζ87 | ζ8 | i | -i | linear of order 8 |
| ρ6 | 1 | -1 | 1 | i | -i | -1 | ζ8 | ζ87 | ζ83 | ζ85 | i | -i | linear of order 8 |
| ρ7 | 1 | -1 | 1 | -i | i | -1 | ζ83 | ζ85 | ζ8 | ζ87 | -i | i | linear of order 8 |
| ρ8 | 1 | -1 | 1 | -i | i | -1 | ζ87 | ζ8 | ζ85 | ζ83 | -i | i | linear of order 8 |
| ρ9 | 2 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -1 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | -1 | -2i | 2i | 1 | 0 | 0 | 0 | 0 | i | -i | complex faithful |
| ρ12 | 2 | -2 | -1 | 2i | -2i | 1 | 0 | 0 | 0 | 0 | -i | i | complex faithful |
►Regular action on 24 points - transitive group 24T8
(1 21 9)(2 10 22)(3 23 11)(4 12 24)(5 17 13)(6 14 18)(7 19 15)(8 16 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,21,9)(2,10,22)(3,23,11)(4,12,24)(5,17,13)(6,14,18)(7,19,15)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (1,21,9)(2,10,22)(3,23,11)(4,12,24)(5,17,13)(6,14,18)(7,19,15)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,21,9),(2,10,22),(3,23,11),(4,12,24),(5,17,13),(6,14,18),(7,19,15),(8,16,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,8);
C3⋊C8 is a maximal subgroup of
S3×C8 C8⋊S3 D4⋊S3 D4.S3 Q8⋊2S3 C3⋊Q16 C32⋊4C8 A4⋊C8 U2(𝔽3) He3⋊C8 C33⋊4C8 C3⋊F9 C51⋊C8
C6p.C4: C4.Dic3 C9⋊C8 C15⋊3C8 C15⋊C8 C21⋊C8 C33⋊C8 C39⋊3C8 C39⋊C8 ...
C3⋊C8 is a maximal quotient of
A4⋊C8 C15⋊C8 C33⋊4C8 C3⋊F9 C39⋊C8 C51⋊3C8 C51⋊C8
C4p.S3: C3⋊C16 C9⋊C8 C32⋊4C8 C15⋊3C8 C21⋊C8 C33⋊C8 C39⋊3C8 C51⋊5C8 ...
Matrix representation of C3⋊C8 ►in GL2(𝔽5) generated by
| 4 | 1 |
| 4 | 0 |
| 4 | 3 |
| 2 | 1 |
G:=sub<GL(2,GF(5))| [4,4,1,0],[4,2,3,1] >;
C3⋊C8 in GAP, Magma, Sage, TeX
C_3\rtimes C_8
% in TeX
G:=Group("C3:C8"); // GroupNames label
G:=SmallGroup(24,1);
// by ID
G=gap.SmallGroup(24,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-3,8,21,259]);
// Polycyclic
G:=Group<a,b|a^3=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C3⋊C8 in TeX
Character table of C3⋊C8 in TeX