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## G = C36order 36 = 22·32

### Cyclic group

Aliases: C36, also denoted Z36, SmallGroup(36,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36
 Chief series C1 — C3 — C6 — C18 — C36
 Lower central C1 — C36
 Upper central C1 — C36

Generators and relations for C36
G = < a | a36=1 >

Smallest permutation representation of C36
Regular action on 36 points
Generators in S36
`(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)`

`G:=sub<Sym(36)| (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)>;`

`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) );`

`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)]])`

C36 is a maximal subgroup of
C9⋊C8  Dic18  D36  Q8.C18  C7⋊C36  He3.3C12  C132C36  C13⋊C36
C36 is a maximal quotient of
C7⋊C36  C132C36  C13⋊C36

Polynomial with Galois group C36 over ℚ
actionf(x)Disc(f)
36T1x36+x35+x34+x33+x32+x31+x30+x29+x28+x27+x26+x25+x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+13735

36 conjugacy classes

 class 1 2 3A 3B 4A 4B 6A 6B 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F 36A ··· 36L order 1 2 3 3 4 4 6 6 9 ··· 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C9 C12 C18 C36 kernel C36 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12

Matrix representation of C36 in GL1(𝔽37) generated by

 17
`G:=sub<GL(1,GF(37))| [17] >;`

C36 in GAP, Magma, Sage, TeX

`C_{36}`
`% in TeX`

`G:=Group("C36");`
`// GroupNames label`

`G:=SmallGroup(36,2);`
`// by ID`

`G=gap.SmallGroup(36,2);`
`# by ID`

`G:=PCGroup([4,-2,-3,-2,-3,24,53]);`
`// Polycyclic`

`G:=Group<a|a^36=1>;`
`// generators/relations`

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