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## G = C72order 72 = 23·32

### Cyclic group

Aliases: C72, also denoted Z72, SmallGroup(72,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72
 Chief series C1 — C2 — C6 — C12 — C36 — C72
 Lower central C1 — C72
 Upper central C1 — C72

Generators and relations for C72
G = < a | a72=1 >

Smallest permutation representation of C72
Regular action on 72 points
Generators in S72
`(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)`

`G:=sub<Sym(72)| (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)>;`

`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) );`

`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)])`

C72 is a maximal subgroup of   C9⋊C16  Dic36  C8⋊D9  C72⋊C2  D72  Q8.C36

72 conjugacy classes

 class 1 2 3A 3B 4A 4B 6A 6B 8A 8B 8C 8D 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F 24A ··· 24H 36A ··· 36L 72A ··· 72X order 1 2 3 3 4 4 6 6 8 8 8 8 9 ··· 9 12 12 12 12 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 1 1 1 1 1 1 1 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C8 C9 C12 C18 C24 C36 C72 kernel C72 C36 C24 C18 C12 C9 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 6 4 6 8 12 24

Matrix representation of C72 in GL1(𝔽73) generated by

 28
`G:=sub<GL(1,GF(73))| [28] >;`

C72 in GAP, Magma, Sage, TeX

`C_{72}`
`% in TeX`

`G:=Group("C72");`
`// GroupNames label`

`G:=SmallGroup(72,2);`
`// by ID`

`G=gap.SmallGroup(72,2);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,-3,-2,30,66,102]);`
`// Polycyclic`

`G:=Group<a|a^72=1>;`
`// generators/relations`

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