direct product, abelian, monomial, 3-elementary
Aliases: C32×C42, SmallGroup(378,60)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C42 |
| C1 — C32×C42 |
| C1 — C32×C42 |
Generators and relations for C32×C42
G = < a,b,c | a3=b3=c42=1, ab=ba, ac=ca, bc=cb >
Subgroups: 112, all normal (8 characteristic)
C1, C2, C3, C6, C7, C32, C14, C3×C6, C21, C33, C42, C32×C6, C3×C21, C3×C42, C32×C21, C32×C42
Quotients: C1, C2, C3, C6, C7, C32, C14, C3×C6, C21, C33, C42, C32×C6, C3×C21, C3×C42, C32×C21, C32×C42
►Regular action on 378 points
Generators in S378
(1 320 151)(2 321 152)(3 322 153)(4 323 154)(5 324 155)(6 325 156)(7 326 157)(8 327 158)(9 328 159)(10 329 160)(11 330 161)(12 331 162)(13 332 163)(14 333 164)(15 334 165)(16 335 166)(17 336 167)(18 295 168)(19 296 127)(20 297 128)(21 298 129)(22 299 130)(23 300 131)(24 301 132)(25 302 133)(26 303 134)(27 304 135)(28 305 136)(29 306 137)(30 307 138)(31 308 139)(32 309 140)(33 310 141)(34 311 142)(35 312 143)(36 313 144)(37 314 145)(38 315 146)(39 316 147)(40 317 148)(41 318 149)(42 319 150)(43 110 206)(44 111 207)(45 112 208)(46 113 209)(47 114 210)(48 115 169)(49 116 170)(50 117 171)(51 118 172)(52 119 173)(53 120 174)(54 121 175)(55 122 176)(56 123 177)(57 124 178)(58 125 179)(59 126 180)(60 85 181)(61 86 182)(62 87 183)(63 88 184)(64 89 185)(65 90 186)(66 91 187)(67 92 188)(68 93 189)(69 94 190)(70 95 191)(71 96 192)(72 97 193)(73 98 194)(74 99 195)(75 100 196)(76 101 197)(77 102 198)(78 103 199)(79 104 200)(80 105 201)(81 106 202)(82 107 203)(83 108 204)(84 109 205)(211 293 363)(212 294 364)(213 253 365)(214 254 366)(215 255 367)(216 256 368)(217 257 369)(218 258 370)(219 259 371)(220 260 372)(221 261 373)(222 262 374)(223 263 375)(224 264 376)(225 265 377)(226 266 378)(227 267 337)(228 268 338)(229 269 339)(230 270 340)(231 271 341)(232 272 342)(233 273 343)(234 274 344)(235 275 345)(236 276 346)(237 277 347)(238 278 348)(239 279 349)(240 280 350)(241 281 351)(242 282 352)(243 283 353)(244 284 354)(245 285 355)(246 286 356)(247 287 357)(248 288 358)(249 289 359)(250 290 360)(251 291 361)(252 292 362)
(1 362 121)(2 363 122)(3 364 123)(4 365 124)(5 366 125)(6 367 126)(7 368 85)(8 369 86)(9 370 87)(10 371 88)(11 372 89)(12 373 90)(13 374 91)(14 375 92)(15 376 93)(16 377 94)(17 378 95)(18 337 96)(19 338 97)(20 339 98)(21 340 99)(22 341 100)(23 342 101)(24 343 102)(25 344 103)(26 345 104)(27 346 105)(28 347 106)(29 348 107)(30 349 108)(31 350 109)(32 351 110)(33 352 111)(34 353 112)(35 354 113)(36 355 114)(37 356 115)(38 357 116)(39 358 117)(40 359 118)(41 360 119)(42 361 120)(43 140 281)(44 141 282)(45 142 283)(46 143 284)(47 144 285)(48 145 286)(49 146 287)(50 147 288)(51 148 289)(52 149 290)(53 150 291)(54 151 292)(55 152 293)(56 153 294)(57 154 253)(58 155 254)(59 156 255)(60 157 256)(61 158 257)(62 159 258)(63 160 259)(64 161 260)(65 162 261)(66 163 262)(67 164 263)(68 165 264)(69 166 265)(70 167 266)(71 168 267)(72 127 268)(73 128 269)(74 129 270)(75 130 271)(76 131 272)(77 132 273)(78 133 274)(79 134 275)(80 135 276)(81 136 277)(82 137 278)(83 138 279)(84 139 280)(169 314 246)(170 315 247)(171 316 248)(172 317 249)(173 318 250)(174 319 251)(175 320 252)(176 321 211)(177 322 212)(178 323 213)(179 324 214)(180 325 215)(181 326 216)(182 327 217)(183 328 218)(184 329 219)(185 330 220)(186 331 221)(187 332 222)(188 333 223)(189 334 224)(190 335 225)(191 336 226)(192 295 227)(193 296 228)(194 297 229)(195 298 230)(196 299 231)(197 300 232)(198 301 233)(199 302 234)(200 303 235)(201 304 236)(202 305 237)(203 306 238)(204 307 239)(205 308 240)(206 309 241)(207 310 242)(208 311 243)(209 312 244)(210 313 245)
(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)(337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378)
G:=sub<Sym(378)| (1,320,151)(2,321,152)(3,322,153)(4,323,154)(5,324,155)(6,325,156)(7,326,157)(8,327,158)(9,328,159)(10,329,160)(11,330,161)(12,331,162)(13,332,163)(14,333,164)(15,334,165)(16,335,166)(17,336,167)(18,295,168)(19,296,127)(20,297,128)(21,298,129)(22,299,130)(23,300,131)(24,301,132)(25,302,133)(26,303,134)(27,304,135)(28,305,136)(29,306,137)(30,307,138)(31,308,139)(32,309,140)(33,310,141)(34,311,142)(35,312,143)(36,313,144)(37,314,145)(38,315,146)(39,316,147)(40,317,148)(41,318,149)(42,319,150)(43,110,206)(44,111,207)(45,112,208)(46,113,209)(47,114,210)(48,115,169)(49,116,170)(50,117,171)(51,118,172)(52,119,173)(53,120,174)(54,121,175)(55,122,176)(56,123,177)(57,124,178)(58,125,179)(59,126,180)(60,85,181)(61,86,182)(62,87,183)(63,88,184)(64,89,185)(65,90,186)(66,91,187)(67,92,188)(68,93,189)(69,94,190)(70,95,191)(71,96,192)(72,97,193)(73,98,194)(74,99,195)(75,100,196)(76,101,197)(77,102,198)(78,103,199)(79,104,200)(80,105,201)(81,106,202)(82,107,203)(83,108,204)(84,109,205)(211,293,363)(212,294,364)(213,253,365)(214,254,366)(215,255,367)(216,256,368)(217,257,369)(218,258,370)(219,259,371)(220,260,372)(221,261,373)(222,262,374)(223,263,375)(224,264,376)(225,265,377)(226,266,378)(227,267,337)(228,268,338)(229,269,339)(230,270,340)(231,271,341)(232,272,342)(233,273,343)(234,274,344)(235,275,345)(236,276,346)(237,277,347)(238,278,348)(239,279,349)(240,280,350)(241,281,351)(242,282,352)(243,283,353)(244,284,354)(245,285,355)(246,286,356)(247,287,357)(248,288,358)(249,289,359)(250,290,360)(251,291,361)(252,292,362), (1,362,121)(2,363,122)(3,364,123)(4,365,124)(5,366,125)(6,367,126)(7,368,85)(8,369,86)(9,370,87)(10,371,88)(11,372,89)(12,373,90)(13,374,91)(14,375,92)(15,376,93)(16,377,94)(17,378,95)(18,337,96)(19,338,97)(20,339,98)(21,340,99)(22,341,100)(23,342,101)(24,343,102)(25,344,103)(26,345,104)(27,346,105)(28,347,106)(29,348,107)(30,349,108)(31,350,109)(32,351,110)(33,352,111)(34,353,112)(35,354,113)(36,355,114)(37,356,115)(38,357,116)(39,358,117)(40,359,118)(41,360,119)(42,361,120)(43,140,281)(44,141,282)(45,142,283)(46,143,284)(47,144,285)(48,145,286)(49,146,287)(50,147,288)(51,148,289)(52,149,290)(53,150,291)(54,151,292)(55,152,293)(56,153,294)(57,154,253)(58,155,254)(59,156,255)(60,157,256)(61,158,257)(62,159,258)(63,160,259)(64,161,260)(65,162,261)(66,163,262)(67,164,263)(68,165,264)(69,166,265)(70,167,266)(71,168,267)(72,127,268)(73,128,269)(74,129,270)(75,130,271)(76,131,272)(77,132,273)(78,133,274)(79,134,275)(80,135,276)(81,136,277)(82,137,278)(83,138,279)(84,139,280)(169,314,246)(170,315,247)(171,316,248)(172,317,249)(173,318,250)(174,319,251)(175,320,252)(176,321,211)(177,322,212)(178,323,213)(179,324,214)(180,325,215)(181,326,216)(182,327,217)(183,328,218)(184,329,219)(185,330,220)(186,331,221)(187,332,222)(188,333,223)(189,334,224)(190,335,225)(191,336,226)(192,295,227)(193,296,228)(194,297,229)(195,298,230)(196,299,231)(197,300,232)(198,301,233)(199,302,234)(200,303,235)(201,304,236)(202,305,237)(203,306,238)(204,307,239)(205,308,240)(206,309,241)(207,310,242)(208,311,243)(209,312,244)(210,313,245), (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)(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378)>;
G:=Group( (1,320,151)(2,321,152)(3,322,153)(4,323,154)(5,324,155)(6,325,156)(7,326,157)(8,327,158)(9,328,159)(10,329,160)(11,330,161)(12,331,162)(13,332,163)(14,333,164)(15,334,165)(16,335,166)(17,336,167)(18,295,168)(19,296,127)(20,297,128)(21,298,129)(22,299,130)(23,300,131)(24,301,132)(25,302,133)(26,303,134)(27,304,135)(28,305,136)(29,306,137)(30,307,138)(31,308,139)(32,309,140)(33,310,141)(34,311,142)(35,312,143)(36,313,144)(37,314,145)(38,315,146)(39,316,147)(40,317,148)(41,318,149)(42,319,150)(43,110,206)(44,111,207)(45,112,208)(46,113,209)(47,114,210)(48,115,169)(49,116,170)(50,117,171)(51,118,172)(52,119,173)(53,120,174)(54,121,175)(55,122,176)(56,123,177)(57,124,178)(58,125,179)(59,126,180)(60,85,181)(61,86,182)(62,87,183)(63,88,184)(64,89,185)(65,90,186)(66,91,187)(67,92,188)(68,93,189)(69,94,190)(70,95,191)(71,96,192)(72,97,193)(73,98,194)(74,99,195)(75,100,196)(76,101,197)(77,102,198)(78,103,199)(79,104,200)(80,105,201)(81,106,202)(82,107,203)(83,108,204)(84,109,205)(211,293,363)(212,294,364)(213,253,365)(214,254,366)(215,255,367)(216,256,368)(217,257,369)(218,258,370)(219,259,371)(220,260,372)(221,261,373)(222,262,374)(223,263,375)(224,264,376)(225,265,377)(226,266,378)(227,267,337)(228,268,338)(229,269,339)(230,270,340)(231,271,341)(232,272,342)(233,273,343)(234,274,344)(235,275,345)(236,276,346)(237,277,347)(238,278,348)(239,279,349)(240,280,350)(241,281,351)(242,282,352)(243,283,353)(244,284,354)(245,285,355)(246,286,356)(247,287,357)(248,288,358)(249,289,359)(250,290,360)(251,291,361)(252,292,362), (1,362,121)(2,363,122)(3,364,123)(4,365,124)(5,366,125)(6,367,126)(7,368,85)(8,369,86)(9,370,87)(10,371,88)(11,372,89)(12,373,90)(13,374,91)(14,375,92)(15,376,93)(16,377,94)(17,378,95)(18,337,96)(19,338,97)(20,339,98)(21,340,99)(22,341,100)(23,342,101)(24,343,102)(25,344,103)(26,345,104)(27,346,105)(28,347,106)(29,348,107)(30,349,108)(31,350,109)(32,351,110)(33,352,111)(34,353,112)(35,354,113)(36,355,114)(37,356,115)(38,357,116)(39,358,117)(40,359,118)(41,360,119)(42,361,120)(43,140,281)(44,141,282)(45,142,283)(46,143,284)(47,144,285)(48,145,286)(49,146,287)(50,147,288)(51,148,289)(52,149,290)(53,150,291)(54,151,292)(55,152,293)(56,153,294)(57,154,253)(58,155,254)(59,156,255)(60,157,256)(61,158,257)(62,159,258)(63,160,259)(64,161,260)(65,162,261)(66,163,262)(67,164,263)(68,165,264)(69,166,265)(70,167,266)(71,168,267)(72,127,268)(73,128,269)(74,129,270)(75,130,271)(76,131,272)(77,132,273)(78,133,274)(79,134,275)(80,135,276)(81,136,277)(82,137,278)(83,138,279)(84,139,280)(169,314,246)(170,315,247)(171,316,248)(172,317,249)(173,318,250)(174,319,251)(175,320,252)(176,321,211)(177,322,212)(178,323,213)(179,324,214)(180,325,215)(181,326,216)(182,327,217)(183,328,218)(184,329,219)(185,330,220)(186,331,221)(187,332,222)(188,333,223)(189,334,224)(190,335,225)(191,336,226)(192,295,227)(193,296,228)(194,297,229)(195,298,230)(196,299,231)(197,300,232)(198,301,233)(199,302,234)(200,303,235)(201,304,236)(202,305,237)(203,306,238)(204,307,239)(205,308,240)(206,309,241)(207,310,242)(208,311,243)(209,312,244)(210,313,245), (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)(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378) );
G=PermutationGroup([[(1,320,151),(2,321,152),(3,322,153),(4,323,154),(5,324,155),(6,325,156),(7,326,157),(8,327,158),(9,328,159),(10,329,160),(11,330,161),(12,331,162),(13,332,163),(14,333,164),(15,334,165),(16,335,166),(17,336,167),(18,295,168),(19,296,127),(20,297,128),(21,298,129),(22,299,130),(23,300,131),(24,301,132),(25,302,133),(26,303,134),(27,304,135),(28,305,136),(29,306,137),(30,307,138),(31,308,139),(32,309,140),(33,310,141),(34,311,142),(35,312,143),(36,313,144),(37,314,145),(38,315,146),(39,316,147),(40,317,148),(41,318,149),(42,319,150),(43,110,206),(44,111,207),(45,112,208),(46,113,209),(47,114,210),(48,115,169),(49,116,170),(50,117,171),(51,118,172),(52,119,173),(53,120,174),(54,121,175),(55,122,176),(56,123,177),(57,124,178),(58,125,179),(59,126,180),(60,85,181),(61,86,182),(62,87,183),(63,88,184),(64,89,185),(65,90,186),(66,91,187),(67,92,188),(68,93,189),(69,94,190),(70,95,191),(71,96,192),(72,97,193),(73,98,194),(74,99,195),(75,100,196),(76,101,197),(77,102,198),(78,103,199),(79,104,200),(80,105,201),(81,106,202),(82,107,203),(83,108,204),(84,109,205),(211,293,363),(212,294,364),(213,253,365),(214,254,366),(215,255,367),(216,256,368),(217,257,369),(218,258,370),(219,259,371),(220,260,372),(221,261,373),(222,262,374),(223,263,375),(224,264,376),(225,265,377),(226,266,378),(227,267,337),(228,268,338),(229,269,339),(230,270,340),(231,271,341),(232,272,342),(233,273,343),(234,274,344),(235,275,345),(236,276,346),(237,277,347),(238,278,348),(239,279,349),(240,280,350),(241,281,351),(242,282,352),(243,283,353),(244,284,354),(245,285,355),(246,286,356),(247,287,357),(248,288,358),(249,289,359),(250,290,360),(251,291,361),(252,292,362)], [(1,362,121),(2,363,122),(3,364,123),(4,365,124),(5,366,125),(6,367,126),(7,368,85),(8,369,86),(9,370,87),(10,371,88),(11,372,89),(12,373,90),(13,374,91),(14,375,92),(15,376,93),(16,377,94),(17,378,95),(18,337,96),(19,338,97),(20,339,98),(21,340,99),(22,341,100),(23,342,101),(24,343,102),(25,344,103),(26,345,104),(27,346,105),(28,347,106),(29,348,107),(30,349,108),(31,350,109),(32,351,110),(33,352,111),(34,353,112),(35,354,113),(36,355,114),(37,356,115),(38,357,116),(39,358,117),(40,359,118),(41,360,119),(42,361,120),(43,140,281),(44,141,282),(45,142,283),(46,143,284),(47,144,285),(48,145,286),(49,146,287),(50,147,288),(51,148,289),(52,149,290),(53,150,291),(54,151,292),(55,152,293),(56,153,294),(57,154,253),(58,155,254),(59,156,255),(60,157,256),(61,158,257),(62,159,258),(63,160,259),(64,161,260),(65,162,261),(66,163,262),(67,164,263),(68,165,264),(69,166,265),(70,167,266),(71,168,267),(72,127,268),(73,128,269),(74,129,270),(75,130,271),(76,131,272),(77,132,273),(78,133,274),(79,134,275),(80,135,276),(81,136,277),(82,137,278),(83,138,279),(84,139,280),(169,314,246),(170,315,247),(171,316,248),(172,317,249),(173,318,250),(174,319,251),(175,320,252),(176,321,211),(177,322,212),(178,323,213),(179,324,214),(180,325,215),(181,326,216),(182,327,217),(183,328,218),(184,329,219),(185,330,220),(186,331,221),(187,332,222),(188,333,223),(189,334,224),(190,335,225),(191,336,226),(192,295,227),(193,296,228),(194,297,229),(195,298,230),(196,299,231),(197,300,232),(198,301,233),(199,302,234),(200,303,235),(201,304,236),(202,305,237),(203,306,238),(204,307,239),(205,308,240),(206,309,241),(207,310,242),(208,311,243),(209,312,244),(210,313,245)], [(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),(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378)]])
378 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 6A | ··· | 6Z | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21EZ | 42A | ··· | 42EZ |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
378 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 |
| kernel | C32×C42 | C32×C21 | C3×C42 | C3×C21 | C32×C6 | C33 | C3×C6 | C32 |
| # reps | 1 | 1 | 26 | 26 | 6 | 6 | 156 | 156 |
Matrix representation of C32×C42 ►in GL3(𝔽43) generated by
| 36 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 6 |
| 7 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 21 |
G:=sub<GL(3,GF(43))| [36,0,0,0,6,0,0,0,6],[36,0,0,0,36,0,0,0,6],[7,0,0,0,7,0,0,0,21] >;
C32×C42 in GAP, Magma, Sage, TeX
C_3^2\times C_{42} % in TeX
G:=Group("C3^2xC42"); // GroupNames label
G:=SmallGroup(378,60);
// by ID
G=gap.SmallGroup(378,60);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^42=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations