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## G = C17⋊C4order 68 = 22·17

### The semidirect product of C17 and C4 acting faithfully

Aliases: C17⋊C4, D17.C2, SmallGroup(68,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C17⋊C4
 Chief series C1 — C17 — D17 — C17⋊C4
 Lower central C17 — C17⋊C4
 Upper central C1

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

Permutation representations of C17⋊C4
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  C852C4  C17⋊Dic7
C17⋊C4 is a maximal quotient of   C172C8  C51⋊C4  C85⋊C4  C17⋊F5  C852C4  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

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