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G = C3×C120order 360 = 23·32·5

Abelian group of type [3,120]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C120, SmallGroup(360,38)

Series: Derived Chief Lower central Upper central

C1 — C3×C120
C1C2C4C20C60C3×C60 — C3×C120
C1 — C3×C120
C1 — C3×C120

Generators and relations for C3×C120
 G = < a,b | a3=b120=1, ab=ba >


Smallest permutation representation of C3×C120
Regular action on 360 points
Generators in S360
(1 140 271)(2 141 272)(3 142 273)(4 143 274)(5 144 275)(6 145 276)(7 146 277)(8 147 278)(9 148 279)(10 149 280)(11 150 281)(12 151 282)(13 152 283)(14 153 284)(15 154 285)(16 155 286)(17 156 287)(18 157 288)(19 158 289)(20 159 290)(21 160 291)(22 161 292)(23 162 293)(24 163 294)(25 164 295)(26 165 296)(27 166 297)(28 167 298)(29 168 299)(30 169 300)(31 170 301)(32 171 302)(33 172 303)(34 173 304)(35 174 305)(36 175 306)(37 176 307)(38 177 308)(39 178 309)(40 179 310)(41 180 311)(42 181 312)(43 182 313)(44 183 314)(45 184 315)(46 185 316)(47 186 317)(48 187 318)(49 188 319)(50 189 320)(51 190 321)(52 191 322)(53 192 323)(54 193 324)(55 194 325)(56 195 326)(57 196 327)(58 197 328)(59 198 329)(60 199 330)(61 200 331)(62 201 332)(63 202 333)(64 203 334)(65 204 335)(66 205 336)(67 206 337)(68 207 338)(69 208 339)(70 209 340)(71 210 341)(72 211 342)(73 212 343)(74 213 344)(75 214 345)(76 215 346)(77 216 347)(78 217 348)(79 218 349)(80 219 350)(81 220 351)(82 221 352)(83 222 353)(84 223 354)(85 224 355)(86 225 356)(87 226 357)(88 227 358)(89 228 359)(90 229 360)(91 230 241)(92 231 242)(93 232 243)(94 233 244)(95 234 245)(96 235 246)(97 236 247)(98 237 248)(99 238 249)(100 239 250)(101 240 251)(102 121 252)(103 122 253)(104 123 254)(105 124 255)(106 125 256)(107 126 257)(108 127 258)(109 128 259)(110 129 260)(111 130 261)(112 131 262)(113 132 263)(114 133 264)(115 134 265)(116 135 266)(117 136 267)(118 137 268)(119 138 269)(120 139 270)
(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)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)

G:=sub<Sym(360)| (1,140,271)(2,141,272)(3,142,273)(4,143,274)(5,144,275)(6,145,276)(7,146,277)(8,147,278)(9,148,279)(10,149,280)(11,150,281)(12,151,282)(13,152,283)(14,153,284)(15,154,285)(16,155,286)(17,156,287)(18,157,288)(19,158,289)(20,159,290)(21,160,291)(22,161,292)(23,162,293)(24,163,294)(25,164,295)(26,165,296)(27,166,297)(28,167,298)(29,168,299)(30,169,300)(31,170,301)(32,171,302)(33,172,303)(34,173,304)(35,174,305)(36,175,306)(37,176,307)(38,177,308)(39,178,309)(40,179,310)(41,180,311)(42,181,312)(43,182,313)(44,183,314)(45,184,315)(46,185,316)(47,186,317)(48,187,318)(49,188,319)(50,189,320)(51,190,321)(52,191,322)(53,192,323)(54,193,324)(55,194,325)(56,195,326)(57,196,327)(58,197,328)(59,198,329)(60,199,330)(61,200,331)(62,201,332)(63,202,333)(64,203,334)(65,204,335)(66,205,336)(67,206,337)(68,207,338)(69,208,339)(70,209,340)(71,210,341)(72,211,342)(73,212,343)(74,213,344)(75,214,345)(76,215,346)(77,216,347)(78,217,348)(79,218,349)(80,219,350)(81,220,351)(82,221,352)(83,222,353)(84,223,354)(85,224,355)(86,225,356)(87,226,357)(88,227,358)(89,228,359)(90,229,360)(91,230,241)(92,231,242)(93,232,243)(94,233,244)(95,234,245)(96,235,246)(97,236,247)(98,237,248)(99,238,249)(100,239,250)(101,240,251)(102,121,252)(103,122,253)(104,123,254)(105,124,255)(106,125,256)(107,126,257)(108,127,258)(109,128,259)(110,129,260)(111,130,261)(112,131,262)(113,132,263)(114,133,264)(115,134,265)(116,135,266)(117,136,267)(118,137,268)(119,138,269)(120,139,270), (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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)>;

G:=Group( (1,140,271)(2,141,272)(3,142,273)(4,143,274)(5,144,275)(6,145,276)(7,146,277)(8,147,278)(9,148,279)(10,149,280)(11,150,281)(12,151,282)(13,152,283)(14,153,284)(15,154,285)(16,155,286)(17,156,287)(18,157,288)(19,158,289)(20,159,290)(21,160,291)(22,161,292)(23,162,293)(24,163,294)(25,164,295)(26,165,296)(27,166,297)(28,167,298)(29,168,299)(30,169,300)(31,170,301)(32,171,302)(33,172,303)(34,173,304)(35,174,305)(36,175,306)(37,176,307)(38,177,308)(39,178,309)(40,179,310)(41,180,311)(42,181,312)(43,182,313)(44,183,314)(45,184,315)(46,185,316)(47,186,317)(48,187,318)(49,188,319)(50,189,320)(51,190,321)(52,191,322)(53,192,323)(54,193,324)(55,194,325)(56,195,326)(57,196,327)(58,197,328)(59,198,329)(60,199,330)(61,200,331)(62,201,332)(63,202,333)(64,203,334)(65,204,335)(66,205,336)(67,206,337)(68,207,338)(69,208,339)(70,209,340)(71,210,341)(72,211,342)(73,212,343)(74,213,344)(75,214,345)(76,215,346)(77,216,347)(78,217,348)(79,218,349)(80,219,350)(81,220,351)(82,221,352)(83,222,353)(84,223,354)(85,224,355)(86,225,356)(87,226,357)(88,227,358)(89,228,359)(90,229,360)(91,230,241)(92,231,242)(93,232,243)(94,233,244)(95,234,245)(96,235,246)(97,236,247)(98,237,248)(99,238,249)(100,239,250)(101,240,251)(102,121,252)(103,122,253)(104,123,254)(105,124,255)(106,125,256)(107,126,257)(108,127,258)(109,128,259)(110,129,260)(111,130,261)(112,131,262)(113,132,263)(114,133,264)(115,134,265)(116,135,266)(117,136,267)(118,137,268)(119,138,269)(120,139,270), (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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360) );

G=PermutationGroup([(1,140,271),(2,141,272),(3,142,273),(4,143,274),(5,144,275),(6,145,276),(7,146,277),(8,147,278),(9,148,279),(10,149,280),(11,150,281),(12,151,282),(13,152,283),(14,153,284),(15,154,285),(16,155,286),(17,156,287),(18,157,288),(19,158,289),(20,159,290),(21,160,291),(22,161,292),(23,162,293),(24,163,294),(25,164,295),(26,165,296),(27,166,297),(28,167,298),(29,168,299),(30,169,300),(31,170,301),(32,171,302),(33,172,303),(34,173,304),(35,174,305),(36,175,306),(37,176,307),(38,177,308),(39,178,309),(40,179,310),(41,180,311),(42,181,312),(43,182,313),(44,183,314),(45,184,315),(46,185,316),(47,186,317),(48,187,318),(49,188,319),(50,189,320),(51,190,321),(52,191,322),(53,192,323),(54,193,324),(55,194,325),(56,195,326),(57,196,327),(58,197,328),(59,198,329),(60,199,330),(61,200,331),(62,201,332),(63,202,333),(64,203,334),(65,204,335),(66,205,336),(67,206,337),(68,207,338),(69,208,339),(70,209,340),(71,210,341),(72,211,342),(73,212,343),(74,213,344),(75,214,345),(76,215,346),(77,216,347),(78,217,348),(79,218,349),(80,219,350),(81,220,351),(82,221,352),(83,222,353),(84,223,354),(85,224,355),(86,225,356),(87,226,357),(88,227,358),(89,228,359),(90,229,360),(91,230,241),(92,231,242),(93,232,243),(94,233,244),(95,234,245),(96,235,246),(97,236,247),(98,237,248),(99,238,249),(100,239,250),(101,240,251),(102,121,252),(103,122,253),(104,123,254),(105,124,255),(106,125,256),(107,126,257),(108,127,258),(109,128,259),(110,129,260),(111,130,261),(112,131,262),(113,132,263),(114,133,264),(115,134,265),(116,135,266),(117,136,267),(118,137,268),(119,138,269),(120,139,270)], [(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),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)])

360 conjugacy classes

class 1  2 3A···3H4A4B5A5B5C5D6A···6H8A8B8C8D10A10B10C10D12A···12P15A···15AF20A···20H24A···24AF30A···30AF40A···40P60A···60BL120A···120DX
order123···34455556···688881010101012···1215···1520···2024···2430···3040···4060···60120···120
size111···11111111···1111111111···11···11···11···11···11···11···11···1

360 irreducible representations

dim1111111111111111
type++
imageC1C2C3C4C5C6C8C10C12C15C20C24C30C40C60C120
kernelC3×C120C3×C60C120C3×C30C3×C24C60C3×C15C3×C12C30C24C3×C6C15C12C32C6C3
# reps118248441632832321664128

Matrix representation of C3×C120 in GL3(𝔽241) generated by

100
02250
001
,
4800
02250
00119
G:=sub<GL(3,GF(241))| [1,0,0,0,225,0,0,0,1],[48,0,0,0,225,0,0,0,119] >;

C3×C120 in GAP, Magma, Sage, TeX

C_3\times C_{120}
% in TeX

G:=Group("C3xC120");
// GroupNames label

G:=SmallGroup(360,38);
// by ID

G=gap.SmallGroup(360,38);
# by ID

G:=PCGroup([6,-2,-3,-3,-5,-2,-2,540,88]);
// Polycyclic

G:=Group<a,b|a^3=b^120=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C120 in TeX

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