direct product, abelian, monomial
Aliases: C2×C6×C30, SmallGroup(360,162)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C6×C30 |
| C1 — C2×C6×C30 |
| C1 — C2×C6×C30 |
Generators and relations for C2×C6×C30
G = < a,b,c | a2=b6=c30=1, ab=ba, ac=ca, bc=cb >
Subgroups: 192, all normal (8 characteristic)
C1, C2, C3, C22, C5, C6, C23, C32, C10, C2×C6, C15, C3×C6, C2×C10, C22×C6, C30, C62, C22×C10, C3×C15, C2×C30, C2×C62, C3×C30, C22×C30, C6×C30, C2×C6×C30
Quotients: C1, C2, C3, C22, C5, C6, C23, C32, C10, C2×C6, C15, C3×C6, C2×C10, C22×C6, C30, C62, C22×C10, C3×C15, C2×C30, C2×C62, C3×C30, C22×C30, C6×C30, C2×C6×C30
►Regular action on 360 points
Generators in S360
(1 225)(2 226)(3 227)(4 228)(5 229)(6 230)(7 231)(8 232)(9 233)(10 234)(11 235)(12 236)(13 237)(14 238)(15 239)(16 240)(17 211)(18 212)(19 213)(20 214)(21 215)(22 216)(23 217)(24 218)(25 219)(26 220)(27 221)(28 222)(29 223)(30 224)(31 186)(32 187)(33 188)(34 189)(35 190)(36 191)(37 192)(38 193)(39 194)(40 195)(41 196)(42 197)(43 198)(44 199)(45 200)(46 201)(47 202)(48 203)(49 204)(50 205)(51 206)(52 207)(53 208)(54 209)(55 210)(56 181)(57 182)(58 183)(59 184)(60 185)(61 334)(62 335)(63 336)(64 337)(65 338)(66 339)(67 340)(68 341)(69 342)(70 343)(71 344)(72 345)(73 346)(74 347)(75 348)(76 349)(77 350)(78 351)(79 352)(80 353)(81 354)(82 355)(83 356)(84 357)(85 358)(86 359)(87 360)(88 331)(89 332)(90 333)(91 162)(92 163)(93 164)(94 165)(95 166)(96 167)(97 168)(98 169)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)(105 176)(106 177)(107 178)(108 179)(109 180)(110 151)(111 152)(112 153)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 161)(121 321)(122 322)(123 323)(124 324)(125 325)(126 326)(127 327)(128 328)(129 329)(130 330)(131 301)(132 302)(133 303)(134 304)(135 305)(136 306)(137 307)(138 308)(139 309)(140 310)(141 311)(142 312)(143 313)(144 314)(145 315)(146 316)(147 317)(148 318)(149 319)(150 320)(241 278)(242 279)(243 280)(244 281)(245 282)(246 283)(247 284)(248 285)(249 286)(250 287)(251 288)(252 289)(253 290)(254 291)(255 292)(256 293)(257 294)(258 295)(259 296)(260 297)(261 298)(262 299)(263 300)(264 271)(265 272)(266 273)(267 274)(268 275)(269 276)(270 277)
(1 124 268 54 332 108)(2 125 269 55 333 109)(3 126 270 56 334 110)(4 127 241 57 335 111)(5 128 242 58 336 112)(6 129 243 59 337 113)(7 130 244 60 338 114)(8 131 245 31 339 115)(9 132 246 32 340 116)(10 133 247 33 341 117)(11 134 248 34 342 118)(12 135 249 35 343 119)(13 136 250 36 344 120)(14 137 251 37 345 91)(15 138 252 38 346 92)(16 139 253 39 347 93)(17 140 254 40 348 94)(18 141 255 41 349 95)(19 142 256 42 350 96)(20 143 257 43 351 97)(21 144 258 44 352 98)(22 145 259 45 353 99)(23 146 260 46 354 100)(24 147 261 47 355 101)(25 148 262 48 356 102)(26 149 263 49 357 103)(27 150 264 50 358 104)(28 121 265 51 359 105)(29 122 266 52 360 106)(30 123 267 53 331 107)(61 151 227 326 277 181)(62 152 228 327 278 182)(63 153 229 328 279 183)(64 154 230 329 280 184)(65 155 231 330 281 185)(66 156 232 301 282 186)(67 157 233 302 283 187)(68 158 234 303 284 188)(69 159 235 304 285 189)(70 160 236 305 286 190)(71 161 237 306 287 191)(72 162 238 307 288 192)(73 163 239 308 289 193)(74 164 240 309 290 194)(75 165 211 310 291 195)(76 166 212 311 292 196)(77 167 213 312 293 197)(78 168 214 313 294 198)(79 169 215 314 295 199)(80 170 216 315 296 200)(81 171 217 316 297 201)(82 172 218 317 298 202)(83 173 219 318 299 203)(84 174 220 319 300 204)(85 175 221 320 271 205)(86 176 222 321 272 206)(87 177 223 322 273 207)(88 178 224 323 274 208)(89 179 225 324 275 209)(90 180 226 325 276 210)
(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)| (1,225)(2,226)(3,227)(4,228)(5,229)(6,230)(7,231)(8,232)(9,233)(10,234)(11,235)(12,236)(13,237)(14,238)(15,239)(16,240)(17,211)(18,212)(19,213)(20,214)(21,215)(22,216)(23,217)(24,218)(25,219)(26,220)(27,221)(28,222)(29,223)(30,224)(31,186)(32,187)(33,188)(34,189)(35,190)(36,191)(37,192)(38,193)(39,194)(40,195)(41,196)(42,197)(43,198)(44,199)(45,200)(46,201)(47,202)(48,203)(49,204)(50,205)(51,206)(52,207)(53,208)(54,209)(55,210)(56,181)(57,182)(58,183)(59,184)(60,185)(61,334)(62,335)(63,336)(64,337)(65,338)(66,339)(67,340)(68,341)(69,342)(70,343)(71,344)(72,345)(73,346)(74,347)(75,348)(76,349)(77,350)(78,351)(79,352)(80,353)(81,354)(82,355)(83,356)(84,357)(85,358)(86,359)(87,360)(88,331)(89,332)(90,333)(91,162)(92,163)(93,164)(94,165)(95,166)(96,167)(97,168)(98,169)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(109,180)(110,151)(111,152)(112,153)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,161)(121,321)(122,322)(123,323)(124,324)(125,325)(126,326)(127,327)(128,328)(129,329)(130,330)(131,301)(132,302)(133,303)(134,304)(135,305)(136,306)(137,307)(138,308)(139,309)(140,310)(141,311)(142,312)(143,313)(144,314)(145,315)(146,316)(147,317)(148,318)(149,319)(150,320)(241,278)(242,279)(243,280)(244,281)(245,282)(246,283)(247,284)(248,285)(249,286)(250,287)(251,288)(252,289)(253,290)(254,291)(255,292)(256,293)(257,294)(258,295)(259,296)(260,297)(261,298)(262,299)(263,300)(264,271)(265,272)(266,273)(267,274)(268,275)(269,276)(270,277), (1,124,268,54,332,108)(2,125,269,55,333,109)(3,126,270,56,334,110)(4,127,241,57,335,111)(5,128,242,58,336,112)(6,129,243,59,337,113)(7,130,244,60,338,114)(8,131,245,31,339,115)(9,132,246,32,340,116)(10,133,247,33,341,117)(11,134,248,34,342,118)(12,135,249,35,343,119)(13,136,250,36,344,120)(14,137,251,37,345,91)(15,138,252,38,346,92)(16,139,253,39,347,93)(17,140,254,40,348,94)(18,141,255,41,349,95)(19,142,256,42,350,96)(20,143,257,43,351,97)(21,144,258,44,352,98)(22,145,259,45,353,99)(23,146,260,46,354,100)(24,147,261,47,355,101)(25,148,262,48,356,102)(26,149,263,49,357,103)(27,150,264,50,358,104)(28,121,265,51,359,105)(29,122,266,52,360,106)(30,123,267,53,331,107)(61,151,227,326,277,181)(62,152,228,327,278,182)(63,153,229,328,279,183)(64,154,230,329,280,184)(65,155,231,330,281,185)(66,156,232,301,282,186)(67,157,233,302,283,187)(68,158,234,303,284,188)(69,159,235,304,285,189)(70,160,236,305,286,190)(71,161,237,306,287,191)(72,162,238,307,288,192)(73,163,239,308,289,193)(74,164,240,309,290,194)(75,165,211,310,291,195)(76,166,212,311,292,196)(77,167,213,312,293,197)(78,168,214,313,294,198)(79,169,215,314,295,199)(80,170,216,315,296,200)(81,171,217,316,297,201)(82,172,218,317,298,202)(83,173,219,318,299,203)(84,174,220,319,300,204)(85,175,221,320,271,205)(86,176,222,321,272,206)(87,177,223,322,273,207)(88,178,224,323,274,208)(89,179,225,324,275,209)(90,180,226,325,276,210), (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( (1,225)(2,226)(3,227)(4,228)(5,229)(6,230)(7,231)(8,232)(9,233)(10,234)(11,235)(12,236)(13,237)(14,238)(15,239)(16,240)(17,211)(18,212)(19,213)(20,214)(21,215)(22,216)(23,217)(24,218)(25,219)(26,220)(27,221)(28,222)(29,223)(30,224)(31,186)(32,187)(33,188)(34,189)(35,190)(36,191)(37,192)(38,193)(39,194)(40,195)(41,196)(42,197)(43,198)(44,199)(45,200)(46,201)(47,202)(48,203)(49,204)(50,205)(51,206)(52,207)(53,208)(54,209)(55,210)(56,181)(57,182)(58,183)(59,184)(60,185)(61,334)(62,335)(63,336)(64,337)(65,338)(66,339)(67,340)(68,341)(69,342)(70,343)(71,344)(72,345)(73,346)(74,347)(75,348)(76,349)(77,350)(78,351)(79,352)(80,353)(81,354)(82,355)(83,356)(84,357)(85,358)(86,359)(87,360)(88,331)(89,332)(90,333)(91,162)(92,163)(93,164)(94,165)(95,166)(96,167)(97,168)(98,169)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(109,180)(110,151)(111,152)(112,153)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,161)(121,321)(122,322)(123,323)(124,324)(125,325)(126,326)(127,327)(128,328)(129,329)(130,330)(131,301)(132,302)(133,303)(134,304)(135,305)(136,306)(137,307)(138,308)(139,309)(140,310)(141,311)(142,312)(143,313)(144,314)(145,315)(146,316)(147,317)(148,318)(149,319)(150,320)(241,278)(242,279)(243,280)(244,281)(245,282)(246,283)(247,284)(248,285)(249,286)(250,287)(251,288)(252,289)(253,290)(254,291)(255,292)(256,293)(257,294)(258,295)(259,296)(260,297)(261,298)(262,299)(263,300)(264,271)(265,272)(266,273)(267,274)(268,275)(269,276)(270,277), (1,124,268,54,332,108)(2,125,269,55,333,109)(3,126,270,56,334,110)(4,127,241,57,335,111)(5,128,242,58,336,112)(6,129,243,59,337,113)(7,130,244,60,338,114)(8,131,245,31,339,115)(9,132,246,32,340,116)(10,133,247,33,341,117)(11,134,248,34,342,118)(12,135,249,35,343,119)(13,136,250,36,344,120)(14,137,251,37,345,91)(15,138,252,38,346,92)(16,139,253,39,347,93)(17,140,254,40,348,94)(18,141,255,41,349,95)(19,142,256,42,350,96)(20,143,257,43,351,97)(21,144,258,44,352,98)(22,145,259,45,353,99)(23,146,260,46,354,100)(24,147,261,47,355,101)(25,148,262,48,356,102)(26,149,263,49,357,103)(27,150,264,50,358,104)(28,121,265,51,359,105)(29,122,266,52,360,106)(30,123,267,53,331,107)(61,151,227,326,277,181)(62,152,228,327,278,182)(63,153,229,328,279,183)(64,154,230,329,280,184)(65,155,231,330,281,185)(66,156,232,301,282,186)(67,157,233,302,283,187)(68,158,234,303,284,188)(69,159,235,304,285,189)(70,160,236,305,286,190)(71,161,237,306,287,191)(72,162,238,307,288,192)(73,163,239,308,289,193)(74,164,240,309,290,194)(75,165,211,310,291,195)(76,166,212,311,292,196)(77,167,213,312,293,197)(78,168,214,313,294,198)(79,169,215,314,295,199)(80,170,216,315,296,200)(81,171,217,316,297,201)(82,172,218,317,298,202)(83,173,219,318,299,203)(84,174,220,319,300,204)(85,175,221,320,271,205)(86,176,222,321,272,206)(87,177,223,322,273,207)(88,178,224,323,274,208)(89,179,225,324,275,209)(90,180,226,325,276,210), (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([[(1,225),(2,226),(3,227),(4,228),(5,229),(6,230),(7,231),(8,232),(9,233),(10,234),(11,235),(12,236),(13,237),(14,238),(15,239),(16,240),(17,211),(18,212),(19,213),(20,214),(21,215),(22,216),(23,217),(24,218),(25,219),(26,220),(27,221),(28,222),(29,223),(30,224),(31,186),(32,187),(33,188),(34,189),(35,190),(36,191),(37,192),(38,193),(39,194),(40,195),(41,196),(42,197),(43,198),(44,199),(45,200),(46,201),(47,202),(48,203),(49,204),(50,205),(51,206),(52,207),(53,208),(54,209),(55,210),(56,181),(57,182),(58,183),(59,184),(60,185),(61,334),(62,335),(63,336),(64,337),(65,338),(66,339),(67,340),(68,341),(69,342),(70,343),(71,344),(72,345),(73,346),(74,347),(75,348),(76,349),(77,350),(78,351),(79,352),(80,353),(81,354),(82,355),(83,356),(84,357),(85,358),(86,359),(87,360),(88,331),(89,332),(90,333),(91,162),(92,163),(93,164),(94,165),(95,166),(96,167),(97,168),(98,169),(99,170),(100,171),(101,172),(102,173),(103,174),(104,175),(105,176),(106,177),(107,178),(108,179),(109,180),(110,151),(111,152),(112,153),(113,154),(114,155),(115,156),(116,157),(117,158),(118,159),(119,160),(120,161),(121,321),(122,322),(123,323),(124,324),(125,325),(126,326),(127,327),(128,328),(129,329),(130,330),(131,301),(132,302),(133,303),(134,304),(135,305),(136,306),(137,307),(138,308),(139,309),(140,310),(141,311),(142,312),(143,313),(144,314),(145,315),(146,316),(147,317),(148,318),(149,319),(150,320),(241,278),(242,279),(243,280),(244,281),(245,282),(246,283),(247,284),(248,285),(249,286),(250,287),(251,288),(252,289),(253,290),(254,291),(255,292),(256,293),(257,294),(258,295),(259,296),(260,297),(261,298),(262,299),(263,300),(264,271),(265,272),(266,273),(267,274),(268,275),(269,276),(270,277)], [(1,124,268,54,332,108),(2,125,269,55,333,109),(3,126,270,56,334,110),(4,127,241,57,335,111),(5,128,242,58,336,112),(6,129,243,59,337,113),(7,130,244,60,338,114),(8,131,245,31,339,115),(9,132,246,32,340,116),(10,133,247,33,341,117),(11,134,248,34,342,118),(12,135,249,35,343,119),(13,136,250,36,344,120),(14,137,251,37,345,91),(15,138,252,38,346,92),(16,139,253,39,347,93),(17,140,254,40,348,94),(18,141,255,41,349,95),(19,142,256,42,350,96),(20,143,257,43,351,97),(21,144,258,44,352,98),(22,145,259,45,353,99),(23,146,260,46,354,100),(24,147,261,47,355,101),(25,148,262,48,356,102),(26,149,263,49,357,103),(27,150,264,50,358,104),(28,121,265,51,359,105),(29,122,266,52,360,106),(30,123,267,53,331,107),(61,151,227,326,277,181),(62,152,228,327,278,182),(63,153,229,328,279,183),(64,154,230,329,280,184),(65,155,231,330,281,185),(66,156,232,301,282,186),(67,157,233,302,283,187),(68,158,234,303,284,188),(69,159,235,304,285,189),(70,160,236,305,286,190),(71,161,237,306,287,191),(72,162,238,307,288,192),(73,163,239,308,289,193),(74,164,240,309,290,194),(75,165,211,310,291,195),(76,166,212,311,292,196),(77,167,213,312,293,197),(78,168,214,313,294,198),(79,169,215,314,295,199),(80,170,216,315,296,200),(81,171,217,316,297,201),(82,172,218,317,298,202),(83,173,219,318,299,203),(84,174,220,319,300,204),(85,175,221,320,271,205),(86,176,222,321,272,206),(87,177,223,322,273,207),(88,178,224,323,274,208),(89,179,225,324,275,209),(90,180,226,325,276,210)], [(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 | 2A | ··· | 2G | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 6A | ··· | 6BD | 10A | ··· | 10AB | 15A | ··· | 15AF | 30A | ··· | 30HP |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 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 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C2×C6×C30 | C6×C30 | C22×C30 | C2×C62 | C2×C30 | C62 | C22×C6 | C2×C6 |
| # reps | 1 | 7 | 8 | 4 | 56 | 28 | 32 | 224 |
Matrix representation of C2×C6×C30 ►in GL3(𝔽31) generated by
| 30 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 5 | 0 | 0 |
| 0 | 26 | 0 |
| 0 | 0 | 6 |
| 7 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 9 |
G:=sub<GL(3,GF(31))| [30,0,0,0,1,0,0,0,1],[5,0,0,0,26,0,0,0,6],[7,0,0,0,22,0,0,0,9] >;
C2×C6×C30 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_{30} % in TeX
G:=Group("C2xC6xC30"); // GroupNames label
G:=SmallGroup(360,162);
// by ID
G=gap.SmallGroup(360,162);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5]);
// Polycyclic
G:=Group<a,b,c|a^2=b^6=c^30=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations