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## G = C2×C6×C30order 360 = 23·32·5

### Abelian group of type [2,6,30]

Aliases: C2×C6×C30, SmallGroup(360,162)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6×C30
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C6×C30 — C2×C6×C30
 Lower central C1 — C2×C6×C30
 Upper central 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 [×7], C3 [×4], C22 [×7], C5, C6 [×28], C23, C32, C10 [×7], C2×C6 [×28], C15 [×4], C3×C6 [×7], C2×C10 [×7], C22×C6 [×4], C30 [×28], C62 [×7], C22×C10, C3×C15, C2×C30 [×28], C2×C62, C3×C30 [×7], C22×C30 [×4], C6×C30 [×7], C2×C6×C30
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C5, C6 [×28], C23, C32, C10 [×7], C2×C6 [×28], C15 [×4], C3×C6 [×7], C2×C10 [×7], C22×C6 [×4], C30 [×28], C62 [×7], C22×C10, C3×C15, C2×C30 [×28], C2×C62, C3×C30 [×7], C22×C30 [×4], C6×C30 [×7], C2×C6×C30

Smallest permutation representation of C2×C6×C30
Regular action on 360 points
Generators in S360
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 150)(30 121)(31 344)(32 345)(33 346)(34 347)(35 348)(36 349)(37 350)(38 351)(39 352)(40 353)(41 354)(42 355)(43 356)(44 357)(45 358)(46 359)(47 360)(48 331)(49 332)(50 333)(51 334)(52 335)(53 336)(54 337)(55 338)(56 339)(57 340)(58 341)(59 342)(60 343)(61 118)(62 119)(63 120)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(151 274)(152 275)(153 276)(154 277)(155 278)(156 279)(157 280)(158 281)(159 282)(160 283)(161 284)(162 285)(163 286)(164 287)(165 288)(166 289)(167 290)(168 291)(169 292)(170 293)(171 294)(172 295)(173 296)(174 297)(175 298)(176 299)(177 300)(178 271)(179 272)(180 273)(181 315)(182 316)(183 317)(184 318)(185 319)(186 320)(187 321)(188 322)(189 323)(190 324)(191 325)(192 326)(193 327)(194 328)(195 329)(196 330)(197 301)(198 302)(199 303)(200 304)(201 305)(202 306)(203 307)(204 308)(205 309)(206 310)(207 311)(208 312)(209 313)(210 314)(211 265)(212 266)(213 267)(214 268)(215 269)(216 270)(217 241)(218 242)(219 243)(220 244)(221 245)(222 246)(223 247)(224 248)(225 249)(226 250)(227 251)(228 252)(229 253)(230 254)(231 255)(232 256)(233 257)(234 258)(235 259)(236 260)(237 261)(238 262)(239 263)(240 264)
(1 310 65 161 356 239)(2 311 66 162 357 240)(3 312 67 163 358 211)(4 313 68 164 359 212)(5 314 69 165 360 213)(6 315 70 166 331 214)(7 316 71 167 332 215)(8 317 72 168 333 216)(9 318 73 169 334 217)(10 319 74 170 335 218)(11 320 75 171 336 219)(12 321 76 172 337 220)(13 322 77 173 338 221)(14 323 78 174 339 222)(15 324 79 175 340 223)(16 325 80 176 341 224)(17 326 81 177 342 225)(18 327 82 178 343 226)(19 328 83 179 344 227)(20 329 84 180 345 228)(21 330 85 151 346 229)(22 301 86 152 347 230)(23 302 87 153 348 231)(24 303 88 154 349 232)(25 304 89 155 350 233)(26 305 90 156 351 234)(27 306 61 157 352 235)(28 307 62 158 353 236)(29 308 63 159 354 237)(30 309 64 160 355 238)(31 251 140 194 110 272)(32 252 141 195 111 273)(33 253 142 196 112 274)(34 254 143 197 113 275)(35 255 144 198 114 276)(36 256 145 199 115 277)(37 257 146 200 116 278)(38 258 147 201 117 279)(39 259 148 202 118 280)(40 260 149 203 119 281)(41 261 150 204 120 282)(42 262 121 205 91 283)(43 263 122 206 92 284)(44 264 123 207 93 285)(45 265 124 208 94 286)(46 266 125 209 95 287)(47 267 126 210 96 288)(48 268 127 181 97 289)(49 269 128 182 98 290)(50 270 129 183 99 291)(51 241 130 184 100 292)(52 242 131 185 101 293)(53 243 132 186 102 294)(54 244 133 187 103 295)(55 245 134 188 104 296)(56 246 135 189 105 297)(57 247 136 190 106 298)(58 248 137 191 107 299)(59 249 138 192 108 300)(60 250 139 193 109 271)
(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)| (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,121)(31,344)(32,345)(33,346)(34,347)(35,348)(36,349)(37,350)(38,351)(39,352)(40,353)(41,354)(42,355)(43,356)(44,357)(45,358)(46,359)(47,360)(48,331)(49,332)(50,333)(51,334)(52,335)(53,336)(54,337)(55,338)(56,339)(57,340)(58,341)(59,342)(60,343)(61,118)(62,119)(63,120)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111)(85,112)(86,113)(87,114)(88,115)(89,116)(90,117)(151,274)(152,275)(153,276)(154,277)(155,278)(156,279)(157,280)(158,281)(159,282)(160,283)(161,284)(162,285)(163,286)(164,287)(165,288)(166,289)(167,290)(168,291)(169,292)(170,293)(171,294)(172,295)(173,296)(174,297)(175,298)(176,299)(177,300)(178,271)(179,272)(180,273)(181,315)(182,316)(183,317)(184,318)(185,319)(186,320)(187,321)(188,322)(189,323)(190,324)(191,325)(192,326)(193,327)(194,328)(195,329)(196,330)(197,301)(198,302)(199,303)(200,304)(201,305)(202,306)(203,307)(204,308)(205,309)(206,310)(207,311)(208,312)(209,313)(210,314)(211,265)(212,266)(213,267)(214,268)(215,269)(216,270)(217,241)(218,242)(219,243)(220,244)(221,245)(222,246)(223,247)(224,248)(225,249)(226,250)(227,251)(228,252)(229,253)(230,254)(231,255)(232,256)(233,257)(234,258)(235,259)(236,260)(237,261)(238,262)(239,263)(240,264), (1,310,65,161,356,239)(2,311,66,162,357,240)(3,312,67,163,358,211)(4,313,68,164,359,212)(5,314,69,165,360,213)(6,315,70,166,331,214)(7,316,71,167,332,215)(8,317,72,168,333,216)(9,318,73,169,334,217)(10,319,74,170,335,218)(11,320,75,171,336,219)(12,321,76,172,337,220)(13,322,77,173,338,221)(14,323,78,174,339,222)(15,324,79,175,340,223)(16,325,80,176,341,224)(17,326,81,177,342,225)(18,327,82,178,343,226)(19,328,83,179,344,227)(20,329,84,180,345,228)(21,330,85,151,346,229)(22,301,86,152,347,230)(23,302,87,153,348,231)(24,303,88,154,349,232)(25,304,89,155,350,233)(26,305,90,156,351,234)(27,306,61,157,352,235)(28,307,62,158,353,236)(29,308,63,159,354,237)(30,309,64,160,355,238)(31,251,140,194,110,272)(32,252,141,195,111,273)(33,253,142,196,112,274)(34,254,143,197,113,275)(35,255,144,198,114,276)(36,256,145,199,115,277)(37,257,146,200,116,278)(38,258,147,201,117,279)(39,259,148,202,118,280)(40,260,149,203,119,281)(41,261,150,204,120,282)(42,262,121,205,91,283)(43,263,122,206,92,284)(44,264,123,207,93,285)(45,265,124,208,94,286)(46,266,125,209,95,287)(47,267,126,210,96,288)(48,268,127,181,97,289)(49,269,128,182,98,290)(50,270,129,183,99,291)(51,241,130,184,100,292)(52,242,131,185,101,293)(53,243,132,186,102,294)(54,244,133,187,103,295)(55,245,134,188,104,296)(56,246,135,189,105,297)(57,247,136,190,106,298)(58,248,137,191,107,299)(59,249,138,192,108,300)(60,250,139,193,109,271), (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( (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,121)(31,344)(32,345)(33,346)(34,347)(35,348)(36,349)(37,350)(38,351)(39,352)(40,353)(41,354)(42,355)(43,356)(44,357)(45,358)(46,359)(47,360)(48,331)(49,332)(50,333)(51,334)(52,335)(53,336)(54,337)(55,338)(56,339)(57,340)(58,341)(59,342)(60,343)(61,118)(62,119)(63,120)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111)(85,112)(86,113)(87,114)(88,115)(89,116)(90,117)(151,274)(152,275)(153,276)(154,277)(155,278)(156,279)(157,280)(158,281)(159,282)(160,283)(161,284)(162,285)(163,286)(164,287)(165,288)(166,289)(167,290)(168,291)(169,292)(170,293)(171,294)(172,295)(173,296)(174,297)(175,298)(176,299)(177,300)(178,271)(179,272)(180,273)(181,315)(182,316)(183,317)(184,318)(185,319)(186,320)(187,321)(188,322)(189,323)(190,324)(191,325)(192,326)(193,327)(194,328)(195,329)(196,330)(197,301)(198,302)(199,303)(200,304)(201,305)(202,306)(203,307)(204,308)(205,309)(206,310)(207,311)(208,312)(209,313)(210,314)(211,265)(212,266)(213,267)(214,268)(215,269)(216,270)(217,241)(218,242)(219,243)(220,244)(221,245)(222,246)(223,247)(224,248)(225,249)(226,250)(227,251)(228,252)(229,253)(230,254)(231,255)(232,256)(233,257)(234,258)(235,259)(236,260)(237,261)(238,262)(239,263)(240,264), (1,310,65,161,356,239)(2,311,66,162,357,240)(3,312,67,163,358,211)(4,313,68,164,359,212)(5,314,69,165,360,213)(6,315,70,166,331,214)(7,316,71,167,332,215)(8,317,72,168,333,216)(9,318,73,169,334,217)(10,319,74,170,335,218)(11,320,75,171,336,219)(12,321,76,172,337,220)(13,322,77,173,338,221)(14,323,78,174,339,222)(15,324,79,175,340,223)(16,325,80,176,341,224)(17,326,81,177,342,225)(18,327,82,178,343,226)(19,328,83,179,344,227)(20,329,84,180,345,228)(21,330,85,151,346,229)(22,301,86,152,347,230)(23,302,87,153,348,231)(24,303,88,154,349,232)(25,304,89,155,350,233)(26,305,90,156,351,234)(27,306,61,157,352,235)(28,307,62,158,353,236)(29,308,63,159,354,237)(30,309,64,160,355,238)(31,251,140,194,110,272)(32,252,141,195,111,273)(33,253,142,196,112,274)(34,254,143,197,113,275)(35,255,144,198,114,276)(36,256,145,199,115,277)(37,257,146,200,116,278)(38,258,147,201,117,279)(39,259,148,202,118,280)(40,260,149,203,119,281)(41,261,150,204,120,282)(42,262,121,205,91,283)(43,263,122,206,92,284)(44,264,123,207,93,285)(45,265,124,208,94,286)(46,266,125,209,95,287)(47,267,126,210,96,288)(48,268,127,181,97,289)(49,269,128,182,98,290)(50,270,129,183,99,291)(51,241,130,184,100,292)(52,242,131,185,101,293)(53,243,132,186,102,294)(54,244,133,187,103,295)(55,245,134,188,104,296)(56,246,135,189,105,297)(57,247,136,190,106,298)(58,248,137,191,107,299)(59,249,138,192,108,300)(60,250,139,193,109,271), (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([(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,131),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,141),(21,142),(22,143),(23,144),(24,145),(25,146),(26,147),(27,148),(28,149),(29,150),(30,121),(31,344),(32,345),(33,346),(34,347),(35,348),(36,349),(37,350),(38,351),(39,352),(40,353),(41,354),(42,355),(43,356),(44,357),(45,358),(46,359),(47,360),(48,331),(49,332),(50,333),(51,334),(52,335),(53,336),(54,337),(55,338),(56,339),(57,340),(58,341),(59,342),(60,343),(61,118),(62,119),(63,120),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108),(82,109),(83,110),(84,111),(85,112),(86,113),(87,114),(88,115),(89,116),(90,117),(151,274),(152,275),(153,276),(154,277),(155,278),(156,279),(157,280),(158,281),(159,282),(160,283),(161,284),(162,285),(163,286),(164,287),(165,288),(166,289),(167,290),(168,291),(169,292),(170,293),(171,294),(172,295),(173,296),(174,297),(175,298),(176,299),(177,300),(178,271),(179,272),(180,273),(181,315),(182,316),(183,317),(184,318),(185,319),(186,320),(187,321),(188,322),(189,323),(190,324),(191,325),(192,326),(193,327),(194,328),(195,329),(196,330),(197,301),(198,302),(199,303),(200,304),(201,305),(202,306),(203,307),(204,308),(205,309),(206,310),(207,311),(208,312),(209,313),(210,314),(211,265),(212,266),(213,267),(214,268),(215,269),(216,270),(217,241),(218,242),(219,243),(220,244),(221,245),(222,246),(223,247),(224,248),(225,249),(226,250),(227,251),(228,252),(229,253),(230,254),(231,255),(232,256),(233,257),(234,258),(235,259),(236,260),(237,261),(238,262),(239,263),(240,264)], [(1,310,65,161,356,239),(2,311,66,162,357,240),(3,312,67,163,358,211),(4,313,68,164,359,212),(5,314,69,165,360,213),(6,315,70,166,331,214),(7,316,71,167,332,215),(8,317,72,168,333,216),(9,318,73,169,334,217),(10,319,74,170,335,218),(11,320,75,171,336,219),(12,321,76,172,337,220),(13,322,77,173,338,221),(14,323,78,174,339,222),(15,324,79,175,340,223),(16,325,80,176,341,224),(17,326,81,177,342,225),(18,327,82,178,343,226),(19,328,83,179,344,227),(20,329,84,180,345,228),(21,330,85,151,346,229),(22,301,86,152,347,230),(23,302,87,153,348,231),(24,303,88,154,349,232),(25,304,89,155,350,233),(26,305,90,156,351,234),(27,306,61,157,352,235),(28,307,62,158,353,236),(29,308,63,159,354,237),(30,309,64,160,355,238),(31,251,140,194,110,272),(32,252,141,195,111,273),(33,253,142,196,112,274),(34,254,143,197,113,275),(35,255,144,198,114,276),(36,256,145,199,115,277),(37,257,146,200,116,278),(38,258,147,201,117,279),(39,259,148,202,118,280),(40,260,149,203,119,281),(41,261,150,204,120,282),(42,262,121,205,91,283),(43,263,122,206,92,284),(44,264,123,207,93,285),(45,265,124,208,94,286),(46,266,125,209,95,287),(47,267,126,210,96,288),(48,268,127,181,97,289),(49,269,128,182,98,290),(50,270,129,183,99,291),(51,241,130,184,100,292),(52,242,131,185,101,293),(53,243,132,186,102,294),(54,244,133,187,103,295),(55,245,134,188,104,296),(56,246,135,189,105,297),(57,247,136,190,106,298),(58,248,137,191,107,299),(59,249,138,192,108,300),(60,250,139,193,109,271)], [(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

׿
×
𝔽