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G = C120order 120 = 23·3·5

Cyclic group

direct product, cyclic, abelian, monomial

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

Series: Derived Chief Lower central Upper central

C1 — C120
C1C2C4C20C60 — C120
C1 — C120
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 3A3B4A4B5A5B5C5D6A6B8A8B8C8D10A10B10C10D12A12B12C12D15A···15H20A···20H24A···24H30A···30H40A···40P60A···60P120A···120AF
order1233445555668888101010101212121215···1520···2024···2430···3040···4060···60120···120
size1111111111111111111111111···11···11···11···11···11···11···1

120 irreducible representations

dim1111111111111111
type++
imageC1C2C3C4C5C6C8C10C12C15C20C24C30C40C60C120
kernelC120C60C40C30C24C20C15C12C10C8C6C5C4C3C2C1
# reps1122424448888161632

Matrix representation of C120 in GL2(𝔽11) generated by

410
88
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

Export

Subgroup lattice of C120 in TeX

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