metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17⋊C4, D17.C2, SmallGroup(68,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C17⋊C4 |
Generators and relations for C17⋊C4
G = < a,b | a17=b4=1, bab-1=a4 >
Character table of C17⋊C4
| class | 1 | 2 | 4A | 4B | 17A | 17B | 17C | 17D | |
| size | 1 | 17 | 17 | 17 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | orthogonal faithful |
| ρ6 | 4 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal faithful |
| ρ7 | 4 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal faithful |
| ρ8 | 4 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal faithful |
►On 17 points: primitive - transitive group 17T3
Generators in S17
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)
G:=sub<Sym(17)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)], [(2,14,17,5),(3,10,16,9),(4,6,15,13),(7,11,12,8)]])
G:=TransitiveGroup(17,3);
C17⋊C4 is a maximal subgroup of
C17⋊C8 C51⋊C4 C85⋊C4 C17⋊F5 C85⋊2C4 C17⋊Dic7
C17⋊C4 is a maximal quotient of C17⋊2C8 C51⋊C4 C85⋊C4 C17⋊F5 C85⋊2C4 C17⋊Dic7
Matrix representation of C17⋊C4 ►in GL4(𝔽137) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 136 | 63 | 27 | 63 |
| 1 | 0 | 0 | 0 |
| 136 | 63 | 27 | 63 |
| 136 | 1 | 74 | 109 |
| 109 | 74 | 1 | 136 |
G:=sub<GL(4,GF(137))| [0,0,0,136,1,0,0,63,0,1,0,27,0,0,1,63],[1,136,136,109,0,63,1,74,0,27,74,1,0,63,109,136] >;
C17⋊C4 in GAP, Magma, Sage, TeX
C_{17}\rtimes C_4 % in TeX
G:=Group("C17:C4"); // GroupNames label
G:=SmallGroup(68,3);
// by ID
G=gap.SmallGroup(68,3);
# by ID
G:=PCGroup([3,-2,-2,-17,6,470,293]);
// Polycyclic
G:=Group<a,b|a^17=b^4=1,b*a*b^-1=a^4>;
// generators/relations
Export
Subgroup lattice of C17⋊C4 in TeX
Character table of C17⋊C4 in TeX