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## G = C2×C17⋊C8order 272 = 24·17

### Direct product of C2 and C17⋊C8

Aliases: C2×C17⋊C8, C34⋊C8, D17⋊C8, D34.C4, C17⋊(C2×C8), C17⋊C4.2C4, D17.(C2×C4), C17⋊C4.C22, (C2×C17⋊C4).2C2, SmallGroup(272,51)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C2×C17⋊C8
 Chief series C1 — C17 — D17 — C17⋊C4 — C17⋊C8 — C2×C17⋊C8
 Lower central C17 — C2×C17⋊C8
 Upper central C1 — C2

Generators and relations for C2×C17⋊C8
G = < a,b,c | a2=b17=c8=1, ab=ba, ac=ca, cbc-1=b2 >

Character table of C2×C17⋊C8

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H 17A 17B 34A 34B size 1 1 17 17 17 17 17 17 17 17 17 17 17 17 17 17 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 linear of order 2 ρ4 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 1 -1 -1 1 i i i -i -i -i -i i 1 1 -1 -1 linear of order 4 ρ6 1 1 1 1 -1 -1 -1 -1 i -i -i i i -i -i i 1 1 1 1 linear of order 4 ρ7 1 -1 1 -1 1 -1 -1 1 -i -i -i i i i i -i 1 1 -1 -1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 -1 -i i i -i -i i i -i 1 1 1 1 linear of order 4 ρ9 1 1 -1 -1 i -i i -i ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 1 1 1 1 linear of order 8 ρ10 1 -1 -1 1 -i -i i i ζ87 ζ85 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 1 1 -1 -1 linear of order 8 ρ11 1 -1 -1 1 i i -i -i ζ85 ζ87 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 1 1 -1 -1 linear of order 8 ρ12 1 1 -1 -1 -i i -i i ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 1 1 1 1 linear of order 8 ρ13 1 -1 -1 1 i i -i -i ζ8 ζ83 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 1 1 -1 -1 linear of order 8 ρ14 1 1 -1 -1 i -i i -i ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 1 1 1 1 linear of order 8 ρ15 1 1 -1 -1 -i i -i i ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 1 1 1 1 linear of order 8 ρ16 1 -1 -1 1 -i -i i i ζ83 ζ8 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 1 1 -1 -1 linear of order 8 ρ17 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√17/2 -1-√17/2 1+√17/2 1-√17/2 orthogonal faithful ρ18 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√17/2 -1+√17/2 1-√17/2 1+√17/2 orthogonal faithful ρ19 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√17/2 -1+√17/2 -1+√17/2 -1-√17/2 orthogonal lifted from C17⋊C8 ρ20 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√17/2 -1-√17/2 -1-√17/2 -1+√17/2 orthogonal lifted from C17⋊C8

Smallest permutation representation of C2×C17⋊C8
On 34 points
Generators in S34
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)
(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)
(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7)(19 27 31 33 34 26 22 20)(21 28 23 29 32 25 30 24)

G:=sub<Sym(34)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34), (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), (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7)(19,27,31,33,34,26,22,20)(21,28,23,29,32,25,30,24)>;

G:=Group( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34), (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), (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7)(19,27,31,33,34,26,22,20)(21,28,23,29,32,25,30,24) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34)], [(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)], [(2,10,14,16,17,9,5,3),(4,11,6,12,15,8,13,7),(19,27,31,33,34,26,22,20),(21,28,23,29,32,25,30,24)]])

Matrix representation of C2×C17⋊C8 in GL8(𝔽137)

 136 0 0 0 0 0 0 0 0 136 0 0 0 0 0 0 0 0 136 0 0 0 0 0 0 0 0 136 0 0 0 0 0 0 0 0 136 0 0 0 0 0 0 0 0 136 0 0 0 0 0 0 0 0 136 0 0 0 0 0 0 0 0 136
,
 2 1 22 115 136 135 114 136 3 1 22 115 136 135 114 136 2 2 22 115 136 135 114 136 2 1 23 115 136 135 114 136 2 1 22 116 136 135 114 136 2 1 22 115 0 135 114 136 2 1 22 115 136 136 114 136 2 1 22 115 136 135 115 136
,
 0 1 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 0 1 20 114 116 43 117 1 45 116 67 2 95 64 94 114 41 92 43 1 94 65 93 115 42 116 135 136 115 22 1 2 23 1

G:=sub<GL(8,GF(137))| [136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136],[2,3,2,2,2,2,2,2,1,1,2,1,1,1,1,1,22,22,22,23,22,22,22,22,115,115,115,115,116,115,115,115,136,136,136,136,136,0,136,136,135,135,135,135,135,135,136,135,114,114,114,114,114,114,114,115,136,136,136,136,136,136,136,136],[0,0,0,0,20,67,43,135,1,0,0,0,114,2,1,136,0,0,0,0,116,95,94,115,0,1,0,0,43,64,65,22,0,0,0,0,117,94,93,1,0,0,1,0,1,114,115,2,0,0,0,0,45,41,42,23,0,0,0,1,116,92,116,1] >;

C2×C17⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_{17}\rtimes C_8
% in TeX

G:=Group("C2xC17:C8");
// GroupNames label

G:=SmallGroup(272,51);
// by ID

G=gap.SmallGroup(272,51);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,20,42,3604,1314,819]);
// Polycyclic

G:=Group<a,b,c|a^2=b^17=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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