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## G = C81order 81 = 34

### Cyclic group

Aliases: C81, also denoted Z81, SmallGroup(81,1)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C81
 Chief series C1 — C3 — C9 — C27 — C81
 Lower central C1 — C81
 Upper central C1 — C81
 Jennings C1 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C9 — C9 — C9 — C9 — C9 — C27 — C27 — C81

Generators and relations for C81
G = < a | a81=1 >

Smallest permutation representation of C81
Regular action on 81 points
Generators in S81
`(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)`

`G:=sub<Sym(81)| (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)>;`

`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) );`

`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)]])`

C81 is a maximal subgroup of   D81  C243  C81⋊C3  C27.A4
C81 is a maximal quotient of   C243  C27.A4

81 conjugacy classes

 class 1 3A 3B 9A ··· 9F 27A ··· 27R 81A ··· 81BB order 1 3 3 9 ··· 9 27 ··· 27 81 ··· 81 size 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

81 irreducible representations

 dim 1 1 1 1 1 type + image C1 C3 C9 C27 C81 kernel C81 C27 C9 C3 C1 # reps 1 2 6 18 54

Matrix representation of C81 in GL1(𝔽163) generated by

 91
`G:=sub<GL(1,GF(163))| [91] >;`

C81 in GAP, Magma, Sage, TeX

`C_{81}`
`% in TeX`

`G:=Group("C81");`
`// GroupNames label`

`G:=SmallGroup(81,1);`
`// by ID`

`G=gap.SmallGroup(81,1);`
`# by ID`

`G:=PCGroup([4,-3,-3,-3,-3,12,29,46]);`
`// Polycyclic`

`G:=Group<a|a^81=1>;`
`// generators/relations`

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