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## G = C68order 68 = 22·17

### Cyclic group

Aliases: C68, also denoted Z68, SmallGroup(68,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C68
 Chief series C1 — C2 — C34 — C68
 Lower central C1 — C68
 Upper central C1 — C68

Generators and relations for C68
G = < a | a68=1 >

Smallest permutation representation of C68
Regular action on 68 points
Generators in S68
`(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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68)`

`G:=sub<Sym(68)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)]])`

C68 is a maximal subgroup of   C173C8  Dic34  D68

68 conjugacy classes

 class 1 2 4A 4B 17A ··· 17P 34A ··· 34P 68A ··· 68AF order 1 2 4 4 17 ··· 17 34 ··· 34 68 ··· 68 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

68 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C4 C17 C34 C68 kernel C68 C34 C17 C4 C2 C1 # reps 1 1 2 16 16 32

Matrix representation of C68 in GL1(𝔽137) generated by

 118
`G:=sub<GL(1,GF(137))| [118] >;`

C68 in GAP, Magma, Sage, TeX

`C_{68}`
`% in TeX`

`G:=Group("C68");`
`// GroupNames label`

`G:=SmallGroup(68,2);`
`// by ID`

`G=gap.SmallGroup(68,2);`
`# by ID`

`G:=PCGroup([3,-2,-17,-2,102]);`
`// Polycyclic`

`G:=Group<a|a^68=1>;`
`// generators/relations`

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