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G = C144order 144 = 24·32

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C144, also denoted Z144, SmallGroup(144,2)

Series: Derived Chief Lower central Upper central

C1 — C144
C1C2C4C12C24C72 — C144
C1 — C144
C1 — C144

Generators and relations for C144
 G = < a | a144=1 >


Smallest permutation representation of C144
Regular action on 144 points
Generators in S144
(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)

G:=sub<Sym(144)| (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)>;

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) );

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)])

C144 is a maximal subgroup of   C9⋊C32  C16⋊D9  D144  C144⋊C2  Dic72

144 conjugacy classes

class 1  2 3A3B4A4B6A6B8A8B8C8D9A···9F12A12B12C12D16A···16H18A···18F24A···24H36A···36L48A···48P72A···72X144A···144AV
order1233446688889···91212121216···1618···1824···2436···3648···4872···72144···144
size1111111111111···111111···11···11···11···11···11···11···1

144 irreducible representations

dim111111111111111
type++
imageC1C2C3C4C6C8C9C12C16C18C24C36C48C72C144
kernelC144C72C48C36C24C18C16C12C9C8C6C4C3C2C1
# reps1122246486812162448

Matrix representation of C144 in GL1(𝔽433) generated by

67
G:=sub<GL(1,GF(433))| [67] >;

C144 in GAP, Magma, Sage, TeX

C_{144}
% in TeX

G:=Group("C144");
// GroupNames label

G:=SmallGroup(144,2);
// by ID

G=gap.SmallGroup(144,2);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-2,-2,36,79,122,88]);
// Polycyclic

G:=Group<a|a^144=1>;
// generators/relations

Export

Subgroup lattice of C144 in TeX

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