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## G = C120order 120 = 23·3·5

### Cyclic group

Aliases: C120, also denoted Z120, SmallGroup(120,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C120
 Chief series C1 — C2 — C4 — C20 — C60 — C120
 Lower central C1 — C120
 Upper central C1 — C120

Generators and relations for C120
G = < a | a120=1 >

Smallest permutation representation of C120
Regular action on 120 points
Generators in S120
`(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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)`

`G:=sub<Sym(120)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)])`

C120 is a maximal subgroup of   C153C16  C40⋊S3  C24⋊D5  D120  Dic60

120 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 12C 12D 15A ··· 15H 20A ··· 20H 24A ··· 24H 30A ··· 30H 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 3 3 4 4 5 5 5 5 6 6 8 8 8 8 10 10 10 10 12 12 12 12 15 ··· 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 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 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C5 C6 C8 C10 C12 C15 C20 C24 C30 C40 C60 C120 kernel C120 C60 C40 C30 C24 C20 C15 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 4 8 8 8 8 16 16 32

Matrix representation of C120 in GL2(𝔽11) generated by

 4 10 8 8
`G:=sub<GL(2,GF(11))| [4,8,10,8] >;`

C120 in GAP, Magma, Sage, TeX

`C_{120}`
`% in TeX`

`G:=Group("C120");`
`// GroupNames label`

`G:=SmallGroup(120,4);`
`// by ID`

`G=gap.SmallGroup(120,4);`
`# by ID`

`G:=PCGroup([5,-2,-3,-5,-2,-2,150,58]);`
`// Polycyclic`

`G:=Group<a|a^120=1>;`
`// generators/relations`

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