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## G = C240order 240 = 24·3·5

### Cyclic group

Aliases: C240, also denoted Z240, SmallGroup(240,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C240
 Chief series C1 — C2 — C4 — C8 — C40 — C120 — C240
 Lower central C1 — C240
 Upper central C1 — C240

Generators and relations for C240
G = < a | a240=1 >

Smallest permutation representation of C240
Regular action on 240 points
Generators in S240
`(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)`

`G:=sub<Sym(240)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)])`

C240 is a maximal subgroup of   C153C32  C80⋊S3  D240  C48⋊D5  Dic120

240 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 12C 12D 15A ··· 15H 16A ··· 16H 20A ··· 20H 24A ··· 24H 30A ··· 30H 40A ··· 40P 48A ··· 48P 60A ··· 60P 80A ··· 80AF 120A ··· 120AF 240A ··· 240BL order 1 2 3 3 4 4 5 5 5 5 6 6 8 8 8 8 10 10 10 10 12 12 12 12 15 ··· 15 16 ··· 16 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 48 ··· 48 60 ··· 60 80 ··· 80 120 ··· 120 240 ··· 240 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 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

240 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C5 C6 C8 C10 C12 C15 C16 C20 C24 C30 C40 C48 C60 C80 C120 C240 kernel C240 C120 C80 C60 C48 C40 C30 C24 C20 C16 C15 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 4 8 8 8 8 8 16 16 16 32 32 64

Matrix representation of C240 in GL2(𝔽31) generated by

 0 17 1 4
`G:=sub<GL(2,GF(31))| [0,1,17,4] >;`

C240 in GAP, Magma, Sage, TeX

`C_{240}`
`% in TeX`

`G:=Group("C240");`
`// GroupNames label`

`G:=SmallGroup(240,4);`
`// by ID`

`G=gap.SmallGroup(240,4);`
`# by ID`

`G:=PCGroup([6,-2,-3,-5,-2,-2,-2,180,69,88]);`
`// Polycyclic`

`G:=Group<a|a^240=1>;`
`// generators/relations`

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