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## G = C32×C42order 378 = 2·33·7

### Abelian group of type [3,3,42]

Aliases: C32×C42, SmallGroup(378,60)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C42
 Chief series C1 — C7 — C21 — C3×C21 — C32×C21 — C32×C42
 Lower central C1 — C32×C42
 Upper central 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 [×13], C6 [×13], C7, C32 [×13], C14, C3×C6 [×13], C21 [×13], C33, C42 [×13], C32×C6, C3×C21 [×13], C3×C42 [×13], C32×C21, C32×C42
Quotients: C1, C2, C3 [×13], C6 [×13], C7, C32 [×13], C14, C3×C6 [×13], C21 [×13], C33, C42 [×13], C32×C6, C3×C21 [×13], C3×C42 [×13], C32×C21, C32×C42

Smallest permutation representation of C32×C42
Regular action on 378 points
Generators in S378
(1 193 276)(2 194 277)(3 195 278)(4 196 279)(5 197 280)(6 198 281)(7 199 282)(8 200 283)(9 201 284)(10 202 285)(11 203 286)(12 204 287)(13 205 288)(14 206 289)(15 207 290)(16 208 291)(17 209 292)(18 210 293)(19 169 294)(20 170 253)(21 171 254)(22 172 255)(23 173 256)(24 174 257)(25 175 258)(26 176 259)(27 177 260)(28 178 261)(29 179 262)(30 180 263)(31 181 264)(32 182 265)(33 183 266)(34 184 267)(35 185 268)(36 186 269)(37 187 270)(38 188 271)(39 189 272)(40 190 273)(41 191 274)(42 192 275)(43 251 326)(44 252 327)(45 211 328)(46 212 329)(47 213 330)(48 214 331)(49 215 332)(50 216 333)(51 217 334)(52 218 335)(53 219 336)(54 220 295)(55 221 296)(56 222 297)(57 223 298)(58 224 299)(59 225 300)(60 226 301)(61 227 302)(62 228 303)(63 229 304)(64 230 305)(65 231 306)(66 232 307)(67 233 308)(68 234 309)(69 235 310)(70 236 311)(71 237 312)(72 238 313)(73 239 314)(74 240 315)(75 241 316)(76 242 317)(77 243 318)(78 244 319)(79 245 320)(80 246 321)(81 247 322)(82 248 323)(83 249 324)(84 250 325)(85 168 374)(86 127 375)(87 128 376)(88 129 377)(89 130 378)(90 131 337)(91 132 338)(92 133 339)(93 134 340)(94 135 341)(95 136 342)(96 137 343)(97 138 344)(98 139 345)(99 140 346)(100 141 347)(101 142 348)(102 143 349)(103 144 350)(104 145 351)(105 146 352)(106 147 353)(107 148 354)(108 149 355)(109 150 356)(110 151 357)(111 152 358)(112 153 359)(113 154 360)(114 155 361)(115 156 362)(116 157 363)(117 158 364)(118 159 365)(119 160 366)(120 161 367)(121 162 368)(122 163 369)(123 164 370)(124 165 371)(125 166 372)(126 167 373)
(1 234 142)(2 235 143)(3 236 144)(4 237 145)(5 238 146)(6 239 147)(7 240 148)(8 241 149)(9 242 150)(10 243 151)(11 244 152)(12 245 153)(13 246 154)(14 247 155)(15 248 156)(16 249 157)(17 250 158)(18 251 159)(19 252 160)(20 211 161)(21 212 162)(22 213 163)(23 214 164)(24 215 165)(25 216 166)(26 217 167)(27 218 168)(28 219 127)(29 220 128)(30 221 129)(31 222 130)(32 223 131)(33 224 132)(34 225 133)(35 226 134)(36 227 135)(37 228 136)(38 229 137)(39 230 138)(40 231 139)(41 232 140)(42 233 141)(43 118 293)(44 119 294)(45 120 253)(46 121 254)(47 122 255)(48 123 256)(49 124 257)(50 125 258)(51 126 259)(52 85 260)(53 86 261)(54 87 262)(55 88 263)(56 89 264)(57 90 265)(58 91 266)(59 92 267)(60 93 268)(61 94 269)(62 95 270)(63 96 271)(64 97 272)(65 98 273)(66 99 274)(67 100 275)(68 101 276)(69 102 277)(70 103 278)(71 104 279)(72 105 280)(73 106 281)(74 107 282)(75 108 283)(76 109 284)(77 110 285)(78 111 286)(79 112 287)(80 113 288)(81 114 289)(82 115 290)(83 116 291)(84 117 292)(169 327 366)(170 328 367)(171 329 368)(172 330 369)(173 331 370)(174 332 371)(175 333 372)(176 334 373)(177 335 374)(178 336 375)(179 295 376)(180 296 377)(181 297 378)(182 298 337)(183 299 338)(184 300 339)(185 301 340)(186 302 341)(187 303 342)(188 304 343)(189 305 344)(190 306 345)(191 307 346)(192 308 347)(193 309 348)(194 310 349)(195 311 350)(196 312 351)(197 313 352)(198 314 353)(199 315 354)(200 316 355)(201 317 356)(202 318 357)(203 319 358)(204 320 359)(205 321 360)(206 322 361)(207 323 362)(208 324 363)(209 325 364)(210 326 365)
(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)| (1,193,276)(2,194,277)(3,195,278)(4,196,279)(5,197,280)(6,198,281)(7,199,282)(8,200,283)(9,201,284)(10,202,285)(11,203,286)(12,204,287)(13,205,288)(14,206,289)(15,207,290)(16,208,291)(17,209,292)(18,210,293)(19,169,294)(20,170,253)(21,171,254)(22,172,255)(23,173,256)(24,174,257)(25,175,258)(26,176,259)(27,177,260)(28,178,261)(29,179,262)(30,180,263)(31,181,264)(32,182,265)(33,183,266)(34,184,267)(35,185,268)(36,186,269)(37,187,270)(38,188,271)(39,189,272)(40,190,273)(41,191,274)(42,192,275)(43,251,326)(44,252,327)(45,211,328)(46,212,329)(47,213,330)(48,214,331)(49,215,332)(50,216,333)(51,217,334)(52,218,335)(53,219,336)(54,220,295)(55,221,296)(56,222,297)(57,223,298)(58,224,299)(59,225,300)(60,226,301)(61,227,302)(62,228,303)(63,229,304)(64,230,305)(65,231,306)(66,232,307)(67,233,308)(68,234,309)(69,235,310)(70,236,311)(71,237,312)(72,238,313)(73,239,314)(74,240,315)(75,241,316)(76,242,317)(77,243,318)(78,244,319)(79,245,320)(80,246,321)(81,247,322)(82,248,323)(83,249,324)(84,250,325)(85,168,374)(86,127,375)(87,128,376)(88,129,377)(89,130,378)(90,131,337)(91,132,338)(92,133,339)(93,134,340)(94,135,341)(95,136,342)(96,137,343)(97,138,344)(98,139,345)(99,140,346)(100,141,347)(101,142,348)(102,143,349)(103,144,350)(104,145,351)(105,146,352)(106,147,353)(107,148,354)(108,149,355)(109,150,356)(110,151,357)(111,152,358)(112,153,359)(113,154,360)(114,155,361)(115,156,362)(116,157,363)(117,158,364)(118,159,365)(119,160,366)(120,161,367)(121,162,368)(122,163,369)(123,164,370)(124,165,371)(125,166,372)(126,167,373), (1,234,142)(2,235,143)(3,236,144)(4,237,145)(5,238,146)(6,239,147)(7,240,148)(8,241,149)(9,242,150)(10,243,151)(11,244,152)(12,245,153)(13,246,154)(14,247,155)(15,248,156)(16,249,157)(17,250,158)(18,251,159)(19,252,160)(20,211,161)(21,212,162)(22,213,163)(23,214,164)(24,215,165)(25,216,166)(26,217,167)(27,218,168)(28,219,127)(29,220,128)(30,221,129)(31,222,130)(32,223,131)(33,224,132)(34,225,133)(35,226,134)(36,227,135)(37,228,136)(38,229,137)(39,230,138)(40,231,139)(41,232,140)(42,233,141)(43,118,293)(44,119,294)(45,120,253)(46,121,254)(47,122,255)(48,123,256)(49,124,257)(50,125,258)(51,126,259)(52,85,260)(53,86,261)(54,87,262)(55,88,263)(56,89,264)(57,90,265)(58,91,266)(59,92,267)(60,93,268)(61,94,269)(62,95,270)(63,96,271)(64,97,272)(65,98,273)(66,99,274)(67,100,275)(68,101,276)(69,102,277)(70,103,278)(71,104,279)(72,105,280)(73,106,281)(74,107,282)(75,108,283)(76,109,284)(77,110,285)(78,111,286)(79,112,287)(80,113,288)(81,114,289)(82,115,290)(83,116,291)(84,117,292)(169,327,366)(170,328,367)(171,329,368)(172,330,369)(173,331,370)(174,332,371)(175,333,372)(176,334,373)(177,335,374)(178,336,375)(179,295,376)(180,296,377)(181,297,378)(182,298,337)(183,299,338)(184,300,339)(185,301,340)(186,302,341)(187,303,342)(188,304,343)(189,305,344)(190,306,345)(191,307,346)(192,308,347)(193,309,348)(194,310,349)(195,311,350)(196,312,351)(197,313,352)(198,314,353)(199,315,354)(200,316,355)(201,317,356)(202,318,357)(203,319,358)(204,320,359)(205,321,360)(206,322,361)(207,323,362)(208,324,363)(209,325,364)(210,326,365), (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( (1,193,276)(2,194,277)(3,195,278)(4,196,279)(5,197,280)(6,198,281)(7,199,282)(8,200,283)(9,201,284)(10,202,285)(11,203,286)(12,204,287)(13,205,288)(14,206,289)(15,207,290)(16,208,291)(17,209,292)(18,210,293)(19,169,294)(20,170,253)(21,171,254)(22,172,255)(23,173,256)(24,174,257)(25,175,258)(26,176,259)(27,177,260)(28,178,261)(29,179,262)(30,180,263)(31,181,264)(32,182,265)(33,183,266)(34,184,267)(35,185,268)(36,186,269)(37,187,270)(38,188,271)(39,189,272)(40,190,273)(41,191,274)(42,192,275)(43,251,326)(44,252,327)(45,211,328)(46,212,329)(47,213,330)(48,214,331)(49,215,332)(50,216,333)(51,217,334)(52,218,335)(53,219,336)(54,220,295)(55,221,296)(56,222,297)(57,223,298)(58,224,299)(59,225,300)(60,226,301)(61,227,302)(62,228,303)(63,229,304)(64,230,305)(65,231,306)(66,232,307)(67,233,308)(68,234,309)(69,235,310)(70,236,311)(71,237,312)(72,238,313)(73,239,314)(74,240,315)(75,241,316)(76,242,317)(77,243,318)(78,244,319)(79,245,320)(80,246,321)(81,247,322)(82,248,323)(83,249,324)(84,250,325)(85,168,374)(86,127,375)(87,128,376)(88,129,377)(89,130,378)(90,131,337)(91,132,338)(92,133,339)(93,134,340)(94,135,341)(95,136,342)(96,137,343)(97,138,344)(98,139,345)(99,140,346)(100,141,347)(101,142,348)(102,143,349)(103,144,350)(104,145,351)(105,146,352)(106,147,353)(107,148,354)(108,149,355)(109,150,356)(110,151,357)(111,152,358)(112,153,359)(113,154,360)(114,155,361)(115,156,362)(116,157,363)(117,158,364)(118,159,365)(119,160,366)(120,161,367)(121,162,368)(122,163,369)(123,164,370)(124,165,371)(125,166,372)(126,167,373), (1,234,142)(2,235,143)(3,236,144)(4,237,145)(5,238,146)(6,239,147)(7,240,148)(8,241,149)(9,242,150)(10,243,151)(11,244,152)(12,245,153)(13,246,154)(14,247,155)(15,248,156)(16,249,157)(17,250,158)(18,251,159)(19,252,160)(20,211,161)(21,212,162)(22,213,163)(23,214,164)(24,215,165)(25,216,166)(26,217,167)(27,218,168)(28,219,127)(29,220,128)(30,221,129)(31,222,130)(32,223,131)(33,224,132)(34,225,133)(35,226,134)(36,227,135)(37,228,136)(38,229,137)(39,230,138)(40,231,139)(41,232,140)(42,233,141)(43,118,293)(44,119,294)(45,120,253)(46,121,254)(47,122,255)(48,123,256)(49,124,257)(50,125,258)(51,126,259)(52,85,260)(53,86,261)(54,87,262)(55,88,263)(56,89,264)(57,90,265)(58,91,266)(59,92,267)(60,93,268)(61,94,269)(62,95,270)(63,96,271)(64,97,272)(65,98,273)(66,99,274)(67,100,275)(68,101,276)(69,102,277)(70,103,278)(71,104,279)(72,105,280)(73,106,281)(74,107,282)(75,108,283)(76,109,284)(77,110,285)(78,111,286)(79,112,287)(80,113,288)(81,114,289)(82,115,290)(83,116,291)(84,117,292)(169,327,366)(170,328,367)(171,329,368)(172,330,369)(173,331,370)(174,332,371)(175,333,372)(176,334,373)(177,335,374)(178,336,375)(179,295,376)(180,296,377)(181,297,378)(182,298,337)(183,299,338)(184,300,339)(185,301,340)(186,302,341)(187,303,342)(188,304,343)(189,305,344)(190,306,345)(191,307,346)(192,308,347)(193,309,348)(194,310,349)(195,311,350)(196,312,351)(197,313,352)(198,314,353)(199,315,354)(200,316,355)(201,317,356)(202,318,357)(203,319,358)(204,320,359)(205,321,360)(206,322,361)(207,323,362)(208,324,363)(209,325,364)(210,326,365), (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([(1,193,276),(2,194,277),(3,195,278),(4,196,279),(5,197,280),(6,198,281),(7,199,282),(8,200,283),(9,201,284),(10,202,285),(11,203,286),(12,204,287),(13,205,288),(14,206,289),(15,207,290),(16,208,291),(17,209,292),(18,210,293),(19,169,294),(20,170,253),(21,171,254),(22,172,255),(23,173,256),(24,174,257),(25,175,258),(26,176,259),(27,177,260),(28,178,261),(29,179,262),(30,180,263),(31,181,264),(32,182,265),(33,183,266),(34,184,267),(35,185,268),(36,186,269),(37,187,270),(38,188,271),(39,189,272),(40,190,273),(41,191,274),(42,192,275),(43,251,326),(44,252,327),(45,211,328),(46,212,329),(47,213,330),(48,214,331),(49,215,332),(50,216,333),(51,217,334),(52,218,335),(53,219,336),(54,220,295),(55,221,296),(56,222,297),(57,223,298),(58,224,299),(59,225,300),(60,226,301),(61,227,302),(62,228,303),(63,229,304),(64,230,305),(65,231,306),(66,232,307),(67,233,308),(68,234,309),(69,235,310),(70,236,311),(71,237,312),(72,238,313),(73,239,314),(74,240,315),(75,241,316),(76,242,317),(77,243,318),(78,244,319),(79,245,320),(80,246,321),(81,247,322),(82,248,323),(83,249,324),(84,250,325),(85,168,374),(86,127,375),(87,128,376),(88,129,377),(89,130,378),(90,131,337),(91,132,338),(92,133,339),(93,134,340),(94,135,341),(95,136,342),(96,137,343),(97,138,344),(98,139,345),(99,140,346),(100,141,347),(101,142,348),(102,143,349),(103,144,350),(104,145,351),(105,146,352),(106,147,353),(107,148,354),(108,149,355),(109,150,356),(110,151,357),(111,152,358),(112,153,359),(113,154,360),(114,155,361),(115,156,362),(116,157,363),(117,158,364),(118,159,365),(119,160,366),(120,161,367),(121,162,368),(122,163,369),(123,164,370),(124,165,371),(125,166,372),(126,167,373)], [(1,234,142),(2,235,143),(3,236,144),(4,237,145),(5,238,146),(6,239,147),(7,240,148),(8,241,149),(9,242,150),(10,243,151),(11,244,152),(12,245,153),(13,246,154),(14,247,155),(15,248,156),(16,249,157),(17,250,158),(18,251,159),(19,252,160),(20,211,161),(21,212,162),(22,213,163),(23,214,164),(24,215,165),(25,216,166),(26,217,167),(27,218,168),(28,219,127),(29,220,128),(30,221,129),(31,222,130),(32,223,131),(33,224,132),(34,225,133),(35,226,134),(36,227,135),(37,228,136),(38,229,137),(39,230,138),(40,231,139),(41,232,140),(42,233,141),(43,118,293),(44,119,294),(45,120,253),(46,121,254),(47,122,255),(48,123,256),(49,124,257),(50,125,258),(51,126,259),(52,85,260),(53,86,261),(54,87,262),(55,88,263),(56,89,264),(57,90,265),(58,91,266),(59,92,267),(60,93,268),(61,94,269),(62,95,270),(63,96,271),(64,97,272),(65,98,273),(66,99,274),(67,100,275),(68,101,276),(69,102,277),(70,103,278),(71,104,279),(72,105,280),(73,106,281),(74,107,282),(75,108,283),(76,109,284),(77,110,285),(78,111,286),(79,112,287),(80,113,288),(81,114,289),(82,115,290),(83,116,291),(84,117,292),(169,327,366),(170,328,367),(171,329,368),(172,330,369),(173,331,370),(174,332,371),(175,333,372),(176,334,373),(177,335,374),(178,336,375),(179,295,376),(180,296,377),(181,297,378),(182,298,337),(183,299,338),(184,300,339),(185,301,340),(186,302,341),(187,303,342),(188,304,343),(189,305,344),(190,306,345),(191,307,346),(192,308,347),(193,309,348),(194,310,349),(195,311,350),(196,312,351),(197,313,352),(198,314,353),(199,315,354),(200,316,355),(201,317,356),(202,318,357),(203,319,358),(204,320,359),(205,321,360),(206,322,361),(207,323,362),(208,324,363),(209,325,364),(210,326,365)], [(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

׿
×
𝔽