direct product, abelian, monomial, 3-elementary
Aliases: C3×C120, SmallGroup(360,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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
(1 197 288)(2 198 289)(3 199 290)(4 200 291)(5 201 292)(6 202 293)(7 203 294)(8 204 295)(9 205 296)(10 206 297)(11 207 298)(12 208 299)(13 209 300)(14 210 301)(15 211 302)(16 212 303)(17 213 304)(18 214 305)(19 215 306)(20 216 307)(21 217 308)(22 218 309)(23 219 310)(24 220 311)(25 221 312)(26 222 313)(27 223 314)(28 224 315)(29 225 316)(30 226 317)(31 227 318)(32 228 319)(33 229 320)(34 230 321)(35 231 322)(36 232 323)(37 233 324)(38 234 325)(39 235 326)(40 236 327)(41 237 328)(42 238 329)(43 239 330)(44 240 331)(45 121 332)(46 122 333)(47 123 334)(48 124 335)(49 125 336)(50 126 337)(51 127 338)(52 128 339)(53 129 340)(54 130 341)(55 131 342)(56 132 343)(57 133 344)(58 134 345)(59 135 346)(60 136 347)(61 137 348)(62 138 349)(63 139 350)(64 140 351)(65 141 352)(66 142 353)(67 143 354)(68 144 355)(69 145 356)(70 146 357)(71 147 358)(72 148 359)(73 149 360)(74 150 241)(75 151 242)(76 152 243)(77 153 244)(78 154 245)(79 155 246)(80 156 247)(81 157 248)(82 158 249)(83 159 250)(84 160 251)(85 161 252)(86 162 253)(87 163 254)(88 164 255)(89 165 256)(90 166 257)(91 167 258)(92 168 259)(93 169 260)(94 170 261)(95 171 262)(96 172 263)(97 173 264)(98 174 265)(99 175 266)(100 176 267)(101 177 268)(102 178 269)(103 179 270)(104 180 271)(105 181 272)(106 182 273)(107 183 274)(108 184 275)(109 185 276)(110 186 277)(111 187 278)(112 188 279)(113 189 280)(114 190 281)(115 191 282)(116 192 283)(117 193 284)(118 194 285)(119 195 286)(120 196 287)
(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)| (1,197,288)(2,198,289)(3,199,290)(4,200,291)(5,201,292)(6,202,293)(7,203,294)(8,204,295)(9,205,296)(10,206,297)(11,207,298)(12,208,299)(13,209,300)(14,210,301)(15,211,302)(16,212,303)(17,213,304)(18,214,305)(19,215,306)(20,216,307)(21,217,308)(22,218,309)(23,219,310)(24,220,311)(25,221,312)(26,222,313)(27,223,314)(28,224,315)(29,225,316)(30,226,317)(31,227,318)(32,228,319)(33,229,320)(34,230,321)(35,231,322)(36,232,323)(37,233,324)(38,234,325)(39,235,326)(40,236,327)(41,237,328)(42,238,329)(43,239,330)(44,240,331)(45,121,332)(46,122,333)(47,123,334)(48,124,335)(49,125,336)(50,126,337)(51,127,338)(52,128,339)(53,129,340)(54,130,341)(55,131,342)(56,132,343)(57,133,344)(58,134,345)(59,135,346)(60,136,347)(61,137,348)(62,138,349)(63,139,350)(64,140,351)(65,141,352)(66,142,353)(67,143,354)(68,144,355)(69,145,356)(70,146,357)(71,147,358)(72,148,359)(73,149,360)(74,150,241)(75,151,242)(76,152,243)(77,153,244)(78,154,245)(79,155,246)(80,156,247)(81,157,248)(82,158,249)(83,159,250)(84,160,251)(85,161,252)(86,162,253)(87,163,254)(88,164,255)(89,165,256)(90,166,257)(91,167,258)(92,168,259)(93,169,260)(94,170,261)(95,171,262)(96,172,263)(97,173,264)(98,174,265)(99,175,266)(100,176,267)(101,177,268)(102,178,269)(103,179,270)(104,180,271)(105,181,272)(106,182,273)(107,183,274)(108,184,275)(109,185,276)(110,186,277)(111,187,278)(112,188,279)(113,189,280)(114,190,281)(115,191,282)(116,192,283)(117,193,284)(118,194,285)(119,195,286)(120,196,287), (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( (1,197,288)(2,198,289)(3,199,290)(4,200,291)(5,201,292)(6,202,293)(7,203,294)(8,204,295)(9,205,296)(10,206,297)(11,207,298)(12,208,299)(13,209,300)(14,210,301)(15,211,302)(16,212,303)(17,213,304)(18,214,305)(19,215,306)(20,216,307)(21,217,308)(22,218,309)(23,219,310)(24,220,311)(25,221,312)(26,222,313)(27,223,314)(28,224,315)(29,225,316)(30,226,317)(31,227,318)(32,228,319)(33,229,320)(34,230,321)(35,231,322)(36,232,323)(37,233,324)(38,234,325)(39,235,326)(40,236,327)(41,237,328)(42,238,329)(43,239,330)(44,240,331)(45,121,332)(46,122,333)(47,123,334)(48,124,335)(49,125,336)(50,126,337)(51,127,338)(52,128,339)(53,129,340)(54,130,341)(55,131,342)(56,132,343)(57,133,344)(58,134,345)(59,135,346)(60,136,347)(61,137,348)(62,138,349)(63,139,350)(64,140,351)(65,141,352)(66,142,353)(67,143,354)(68,144,355)(69,145,356)(70,146,357)(71,147,358)(72,148,359)(73,149,360)(74,150,241)(75,151,242)(76,152,243)(77,153,244)(78,154,245)(79,155,246)(80,156,247)(81,157,248)(82,158,249)(83,159,250)(84,160,251)(85,161,252)(86,162,253)(87,163,254)(88,164,255)(89,165,256)(90,166,257)(91,167,258)(92,168,259)(93,169,260)(94,170,261)(95,171,262)(96,172,263)(97,173,264)(98,174,265)(99,175,266)(100,176,267)(101,177,268)(102,178,269)(103,179,270)(104,180,271)(105,181,272)(106,182,273)(107,183,274)(108,184,275)(109,185,276)(110,186,277)(111,187,278)(112,188,279)(113,189,280)(114,190,281)(115,191,282)(116,192,283)(117,193,284)(118,194,285)(119,195,286)(120,196,287), (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([[(1,197,288),(2,198,289),(3,199,290),(4,200,291),(5,201,292),(6,202,293),(7,203,294),(8,204,295),(9,205,296),(10,206,297),(11,207,298),(12,208,299),(13,209,300),(14,210,301),(15,211,302),(16,212,303),(17,213,304),(18,214,305),(19,215,306),(20,216,307),(21,217,308),(22,218,309),(23,219,310),(24,220,311),(25,221,312),(26,222,313),(27,223,314),(28,224,315),(29,225,316),(30,226,317),(31,227,318),(32,228,319),(33,229,320),(34,230,321),(35,231,322),(36,232,323),(37,233,324),(38,234,325),(39,235,326),(40,236,327),(41,237,328),(42,238,329),(43,239,330),(44,240,331),(45,121,332),(46,122,333),(47,123,334),(48,124,335),(49,125,336),(50,126,337),(51,127,338),(52,128,339),(53,129,340),(54,130,341),(55,131,342),(56,132,343),(57,133,344),(58,134,345),(59,135,346),(60,136,347),(61,137,348),(62,138,349),(63,139,350),(64,140,351),(65,141,352),(66,142,353),(67,143,354),(68,144,355),(69,145,356),(70,146,357),(71,147,358),(72,148,359),(73,149,360),(74,150,241),(75,151,242),(76,152,243),(77,153,244),(78,154,245),(79,155,246),(80,156,247),(81,157,248),(82,158,249),(83,159,250),(84,160,251),(85,161,252),(86,162,253),(87,163,254),(88,164,255),(89,165,256),(90,166,257),(91,167,258),(92,168,259),(93,169,260),(94,170,261),(95,171,262),(96,172,263),(97,173,264),(98,174,265),(99,175,266),(100,176,267),(101,177,268),(102,178,269),(103,179,270),(104,180,271),(105,181,272),(106,182,273),(107,183,274),(108,184,275),(109,185,276),(110,186,277),(111,187,278),(112,188,279),(113,189,280),(114,190,281),(115,191,282),(116,192,283),(117,193,284),(118,194,285),(119,195,286),(120,196,287)], [(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 | ··· | 3H | 4A | 4B | 5A | 5B | 5C | 5D | 6A | ··· | 6H | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | ··· | 12P | 15A | ··· | 15AF | 20A | ··· | 20H | 24A | ··· | 24AF | 30A | ··· | 30AF | 40A | ··· | 40P | 60A | ··· | 60BL | 120A | ··· | 120DX |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| 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 |
360 irreducible representations
| dim | 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 | C20 | C24 | C30 | C40 | C60 | C120 |
| kernel | C3×C120 | C3×C60 | C120 | C3×C30 | C3×C24 | C60 | C3×C15 | C3×C12 | C30 | C24 | C3×C6 | C15 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 4 | 8 | 4 | 4 | 16 | 32 | 8 | 32 | 32 | 16 | 64 | 128 |
Matrix representation of C3×C120 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 225 | 0 |
| 0 | 0 | 1 |
| 48 | 0 | 0 |
| 0 | 225 | 0 |
| 0 | 0 | 119 |
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