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G = C17⋊C8order 136 = 23·17

The semidirect product of C17 and C8 acting faithfully

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

C1C17 — C17⋊C8
C1C17D17C17⋊C4 — C17⋊C8
C17 — C17⋊C8
C1

Generators and relations for C17⋊C8
 G = < a,b | a17=b8=1, bab-1=a2 >

17C2
17C4
17C8

Character table of C17⋊C8

 class 124A4B8A8B8C8D17A17B
 size 11717171717171788
ρ11111111111    trivial
ρ21111-1-1-1-111    linear of order 2
ρ311-1-1i-i-ii11    linear of order 4
ρ411-1-1-iii-i11    linear of order 4
ρ51-1-iiζ87ζ8ζ85ζ8311    linear of order 8
ρ61-1-iiζ83ζ85ζ8ζ8711    linear of order 8
ρ71-1i-iζ8ζ87ζ83ζ8511    linear of order 8
ρ81-1i-iζ85ζ83ζ87ζ811    linear of order 8
ρ980000000-1+17/2-1-17/2    orthogonal faithful
ρ1080000000-1-17/2-1+17/2    orthogonal faithful

Permutation representations of C17⋊C8
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)

00000111
01010011
01000010
10000010
00000001
01100011
01000001
00001010
,
10000101
00001100
00001011
01000000
00000110
00101110
00011111
00000011

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

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