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G = C144order 144 = 24·32

Cyclic group

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C144
 Chief series C1 — C2 — C4 — C12 — C24 — C72 — C144
 Lower central C1 — C144
 Upper central 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 3A 3B 4A 4B 6A 6B 8A 8B 8C 8D 9A ··· 9F 12A 12B 12C 12D 16A ··· 16H 18A ··· 18F 24A ··· 24H 36A ··· 36L 48A ··· 48P 72A ··· 72X 144A ··· 144AV order 1 2 3 3 4 4 6 6 8 8 8 8 9 ··· 9 12 12 12 12 16 ··· 16 18 ··· 18 24 ··· 24 36 ··· 36 48 ··· 48 72 ··· 72 144 ··· 144 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

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C8 C9 C12 C16 C18 C24 C36 C48 C72 C144 kernel C144 C72 C48 C36 C24 C18 C16 C12 C9 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 6 4 8 6 8 12 16 24 48

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`

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