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## G = C54order 54 = 2·33

### Cyclic group

Aliases: C54, also denoted Z54, SmallGroup(54,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C54
 Chief series C1 — C3 — C9 — C27 — C54
 Lower central C1 — C54
 Upper central C1 — C54

Generators and relations for C54
G = < a | a54=1 >

Smallest permutation representation of C54
Regular action on 54 points
Generators in S54
`(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)`

`G:=sub<Sym(54)| (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)>;`

`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) );`

`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)])`

C54 is a maximal subgroup of   Dic27  Q8⋊C27  C7⋊C54
C54 is a maximal quotient of   C7⋊C54

54 conjugacy classes

 class 1 2 3A 3B 6A 6B 9A ··· 9F 18A ··· 18F 27A ··· 27R 54A ··· 54R order 1 2 3 3 6 6 9 ··· 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C9 C18 C27 C54 kernel C54 C27 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 6 6 18 18

Matrix representation of C54 in GL1(𝔽109) generated by

 36
`G:=sub<GL(1,GF(109))| [36] >;`

C54 in GAP, Magma, Sage, TeX

`C_{54}`
`% in TeX`

`G:=Group("C54");`
`// GroupNames label`

`G:=SmallGroup(54,2);`
`// by ID`

`G=gap.SmallGroup(54,2);`
`# by ID`

`G:=PCGroup([4,-2,-3,-3,-3,29,46]);`
`// Polycyclic`

`G:=Group<a|a^54=1>;`
`// generators/relations`

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