metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5⋊C8, C2.F5, C10.C4, Dic5.2C2, SmallGroup(40,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5⋊C8 |
Generators and relations for C5⋊C8
G = < a,b | a5=b8=1, bab-1=a3 >
Character table of C5⋊C8
| class | 1 | 2 | 4A | 4B | 5 | 8A | 8B | 8C | 8D | 10 | |
| size | 1 | 1 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | linear of order 4 |
| ρ5 | 1 | -1 | i | -i | 1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | 1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | i | -i | 1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | -i | i | 1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | linear of order 8 |
| ρ9 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from F5 |
| ρ10 | 4 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | symplectic faithful, Schur index 2 |
►Regular action on 40 points
Generators in S40
(1 35 27 20 11)(2 21 36 12 28)(3 13 22 29 37)(4 30 14 38 23)(5 39 31 24 15)(6 17 40 16 32)(7 9 18 25 33)(8 26 10 34 19)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)
G:=sub<Sym(40)| (1,35,27,20,11)(2,21,36,12,28)(3,13,22,29,37)(4,30,14,38,23)(5,39,31,24,15)(6,17,40,16,32)(7,9,18,25,33)(8,26,10,34,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)>;
G:=Group( (1,35,27,20,11)(2,21,36,12,28)(3,13,22,29,37)(4,30,14,38,23)(5,39,31,24,15)(6,17,40,16,32)(7,9,18,25,33)(8,26,10,34,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40) );
G=PermutationGroup([[(1,35,27,20,11),(2,21,36,12,28),(3,13,22,29,37),(4,30,14,38,23),(5,39,31,24,15),(6,17,40,16,32),(7,9,18,25,33),(8,26,10,34,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40)]])
C5⋊C8 is a maximal subgroup of
D5⋊C8 C4.F5 C22.F5 C15⋊C8 C25⋊C8 C52⋊3C8 C52⋊4C8 C52⋊5C8 CSU2(𝔽5) C2.S5 C35⋊C8 (C3×C6).F5 C5⋊F9 C55⋊C8
C5⋊C8 is a maximal quotient of
C5⋊C16 C15⋊C8 C25⋊C8 C52⋊3C8 C52⋊4C8 C52⋊5C8 C35⋊C8 (C3×C6).F5 C5⋊F9 C55⋊C8
Matrix representation of C5⋊C8 ►in GL4(𝔽3) generated by
| 0 | 2 | 1 | 2 |
| 1 | 1 | 1 | 1 |
| 1 | 2 | 2 | 0 |
| 0 | 2 | 0 | 2 |
| 2 | 0 | 2 | 0 |
| 0 | 1 | 0 | 2 |
| 2 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 |
G:=sub<GL(4,GF(3))| [0,1,1,0,2,1,2,2,1,1,2,0,2,1,0,2],[2,0,2,2,0,1,0,2,2,0,0,0,0,2,0,0] >;
C5⋊C8 in GAP, Magma, Sage, TeX
C_5\rtimes C_8
% in TeX
G:=Group("C5:C8"); // GroupNames label
G:=SmallGroup(40,3);
// by ID
G=gap.SmallGroup(40,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-5,8,21,259,263]);
// Polycyclic
G:=Group<a,b|a^5=b^8=1,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C5⋊C8 in TeX
Character table of C5⋊C8 in TeX