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## G = C150order 150 = 2·3·52

### Cyclic group

Aliases: C150, also denoted Z150, SmallGroup(150,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C150
 Chief series C1 — C5 — C25 — C75 — C150
 Lower central C1 — C150
 Upper central C1 — C150

Generators and relations for C150
G = < a | a150=1 >

Smallest permutation representation of C150
Regular action on 150 points
Generators in S150
`(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)`

`G:=sub<Sym(150)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)]])`

C150 is a maximal subgroup of   Dic75

150 conjugacy classes

 class 1 2 3A 3B 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 15A ··· 15H 25A ··· 25T 30A ··· 30H 50A ··· 50T 75A ··· 75AN 150A ··· 150AN order 1 2 3 3 5 5 5 5 6 6 10 10 10 10 15 ··· 15 25 ··· 25 30 ··· 30 50 ··· 50 75 ··· 75 150 ··· 150 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

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C5 C6 C10 C15 C25 C30 C50 C75 C150 kernel C150 C75 C50 C30 C25 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 2 4 2 4 8 20 8 20 40 40

Matrix representation of C150 in GL3(𝔽151) generated by

 150 0 0 0 91 0 0 0 32
`G:=sub<GL(3,GF(151))| [150,0,0,0,91,0,0,0,32] >;`

C150 in GAP, Magma, Sage, TeX

`C_{150}`
`% in TeX`

`G:=Group("C150");`
`// GroupNames label`

`G:=SmallGroup(150,4);`
`// by ID`

`G=gap.SmallGroup(150,4);`
`# by ID`

`G:=PCGroup([4,-2,-3,-5,-5,70]);`
`// Polycyclic`

`G:=Group<a|a^150=1>;`
`// generators/relations`

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