metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17⋊C8, D17.C4, C17⋊C4.C2, SmallGroup(136,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C17⋊C8 |
Generators and relations for C17⋊C8
G = < a,b | a17=b8=1, bab-1=a2 >
Character table of C17⋊C8
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 17A | 17B | |
| size | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 8 | 8 | |
| 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 | i | -i | -i | i | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | -i | i | i | -i | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | -i | i | ζ87 | ζ8 | ζ85 | ζ83 | 1 | 1 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | ζ83 | ζ85 | ζ8 | ζ87 | 1 | 1 | linear of order 8 |
| ρ7 | 1 | -1 | i | -i | ζ8 | ζ87 | ζ83 | ζ85 | 1 | 1 | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | ζ85 | ζ83 | ζ87 | ζ8 | 1 | 1 | linear of order 8 |
| ρ9 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√17/2 | -1-√17/2 | orthogonal faithful |
| ρ10 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√17/2 | -1+√17/2 | orthogonal faithful |
►On 17 points: primitive - transitive group 17T4
Generators in S17
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17)
(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7)
G:=sub<Sym(17)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)], [(2,10,14,16,17,9,5,3),(4,11,6,12,15,8,13,7)]])
G:=TransitiveGroup(17,4);
C17⋊C8 is a maximal subgroup of
F17 C51⋊C8
C17⋊C8 is a maximal quotient of C34.C8 C51⋊C8
Matrix representation of C17⋊C8 ►in GL8(𝔽2)
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(8,GF(2))| [0,0,0,1,0,0,0,0,0,1,1,0,0,1,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,1,1,0,1,0,1,1,1,0,0,1,1,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,1,1,1,1,0,1,0,0,0,1,1] >;
C17⋊C8 in GAP, Magma, Sage, TeX
C_{17}\rtimes C_8 % in TeX
G:=Group("C17:C8"); // GroupNames label
G:=SmallGroup(136,12);
// by ID
G=gap.SmallGroup(136,12);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,8,21,1155,839,523]);
// Polycyclic
G:=Group<a,b|a^17=b^8=1,b*a*b^-1=a^2>;
// generators/relations
Export
Subgroup lattice of C17⋊C8 in TeX
Character table of C17⋊C8 in TeX