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G = C150order 150 = 2·3·52

Cyclic group

direct product, cyclic, abelian, monomial

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

Series: Derived Chief Lower central Upper central

C1 — C150
C1C5C25C75 — C150
C1 — C150
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 3A3B5A5B5C5D6A6B10A10B10C10D15A···15H25A···25T30A···30H50A···50T75A···75AN150A···150AN
order12335555661010101015···1525···2530···3050···5075···75150···150
size111111111111111···11···11···11···11···11···1

150 irreducible representations

dim111111111111
type++
imageC1C2C3C5C6C10C15C25C30C50C75C150
kernelC150C75C50C30C25C15C10C6C5C3C2C1
# reps1124248208204040

Matrix representation of C150 in GL3(𝔽151) generated by

15000
0910
0032
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

Export

Subgroup lattice of C150 in TeX

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