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## G = C92order 92 = 22·23

### Cyclic group

Aliases: C92, also denoted Z92, SmallGroup(92,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C92
 Chief series C1 — C2 — C46 — C92
 Lower central C1 — C92
 Upper central C1 — C92

Generators and relations for C92
G = < a | a92=1 >

Smallest permutation representation of C92
Regular action on 92 points
Generators in S92
`(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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92)`

`G:=sub<Sym(92)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92)]])`

C92 is a maximal subgroup of   C23⋊C8  Dic46  D92

92 conjugacy classes

 class 1 2 4A 4B 23A ··· 23V 46A ··· 46V 92A ··· 92AR order 1 2 4 4 23 ··· 23 46 ··· 46 92 ··· 92 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

92 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C4 C23 C46 C92 kernel C92 C46 C23 C4 C2 C1 # reps 1 1 2 22 22 44

Matrix representation of C92 in GL1(𝔽277) generated by

 269
`G:=sub<GL(1,GF(277))| [269] >;`

C92 in GAP, Magma, Sage, TeX

`C_{92}`
`% in TeX`

`G:=Group("C92");`
`// GroupNames label`

`G:=SmallGroup(92,2);`
`// by ID`

`G=gap.SmallGroup(92,2);`
`# by ID`

`G:=PCGroup([3,-2,-23,-2,138]);`
`// Polycyclic`

`G:=Group<a|a^92=1>;`
`// generators/relations`

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