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G = C2×C6×C36order 432 = 24·33

Abelian group of type [2,6,36]

direct product, abelian, monomial

Aliases: C2×C6×C36, SmallGroup(432,400)

Series: Derived Chief Lower central Upper central

C1 — C2×C6×C36
C1C3C6C3×C6C3×C18C3×C36C6×C36 — C2×C6×C36
C1 — C2×C6×C36
C1 — C2×C6×C36

Generators and relations for C2×C6×C36
 G = < a,b,c | a2=b6=c36=1, ab=ba, ac=ca, bc=cb >

Subgroups: 270, all normal (16 characteristic)
C1, C2, C2 [×6], C3, C3 [×3], C4 [×4], C22 [×7], C6, C6 [×27], C2×C4 [×6], C23, C9 [×3], C32, C12 [×16], C2×C6 [×28], C22×C4, C18 [×21], C3×C6, C3×C6 [×6], C2×C12 [×24], C22×C6, C22×C6 [×3], C3×C9, C36 [×12], C2×C18 [×21], C3×C12 [×4], C62 [×7], C22×C12, C22×C12 [×3], C3×C18, C3×C18 [×6], C2×C36 [×18], C22×C18 [×3], C6×C12 [×6], C2×C62, C3×C36 [×4], C6×C18 [×7], C22×C36 [×3], C2×C6×C12, C6×C36 [×6], C2×C6×C18, C2×C6×C36
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], C23, C9 [×3], C32, C12 [×16], C2×C6 [×28], C22×C4, C18 [×21], C3×C6 [×7], C2×C12 [×24], C22×C6 [×4], C3×C9, C36 [×12], C2×C18 [×21], C3×C12 [×4], C62 [×7], C22×C12 [×4], C3×C18 [×7], C2×C36 [×18], C22×C18 [×3], C6×C12 [×6], C2×C62, C3×C36 [×4], C6×C18 [×7], C22×C36 [×3], C2×C6×C12, C6×C36 [×6], C2×C6×C18, C2×C6×C36

Smallest permutation representation of C2×C6×C36
Regular action on 432 points
Generators in S432
(1 126)(2 127)(3 128)(4 129)(5 130)(6 131)(7 132)(8 133)(9 134)(10 135)(11 136)(12 137)(13 138)(14 139)(15 140)(16 141)(17 142)(18 143)(19 144)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 121)(33 122)(34 123)(35 124)(36 125)(37 174)(38 175)(39 176)(40 177)(41 178)(42 179)(43 180)(44 145)(45 146)(46 147)(47 148)(48 149)(49 150)(50 151)(51 152)(52 153)(53 154)(54 155)(55 156)(56 157)(57 158)(58 159)(59 160)(60 161)(61 162)(62 163)(63 164)(64 165)(65 166)(66 167)(67 168)(68 169)(69 170)(70 171)(71 172)(72 173)(73 328)(74 329)(75 330)(76 331)(77 332)(78 333)(79 334)(80 335)(81 336)(82 337)(83 338)(84 339)(85 340)(86 341)(87 342)(88 343)(89 344)(90 345)(91 346)(92 347)(93 348)(94 349)(95 350)(96 351)(97 352)(98 353)(99 354)(100 355)(101 356)(102 357)(103 358)(104 359)(105 360)(106 325)(107 326)(108 327)(181 279)(182 280)(183 281)(184 282)(185 283)(186 284)(187 285)(188 286)(189 287)(190 288)(191 253)(192 254)(193 255)(194 256)(195 257)(196 258)(197 259)(198 260)(199 261)(200 262)(201 263)(202 264)(203 265)(204 266)(205 267)(206 268)(207 269)(208 270)(209 271)(210 272)(211 273)(212 274)(213 275)(214 276)(215 277)(216 278)(217 297)(218 298)(219 299)(220 300)(221 301)(222 302)(223 303)(224 304)(225 305)(226 306)(227 307)(228 308)(229 309)(230 310)(231 311)(232 312)(233 313)(234 314)(235 315)(236 316)(237 317)(238 318)(239 319)(240 320)(241 321)(242 322)(243 323)(244 324)(245 289)(246 290)(247 291)(248 292)(249 293)(250 294)(251 295)(252 296)(361 428)(362 429)(363 430)(364 431)(365 432)(366 397)(367 398)(368 399)(369 400)(370 401)(371 402)(372 403)(373 404)(374 405)(375 406)(376 407)(377 408)(378 409)(379 410)(380 411)(381 412)(382 413)(383 414)(384 415)(385 416)(386 417)(387 418)(388 419)(389 420)(390 421)(391 422)(392 423)(393 424)(394 425)(395 426)(396 427)
(1 415 295 164 205 88)(2 416 296 165 206 89)(3 417 297 166 207 90)(4 418 298 167 208 91)(5 419 299 168 209 92)(6 420 300 169 210 93)(7 421 301 170 211 94)(8 422 302 171 212 95)(9 423 303 172 213 96)(10 424 304 173 214 97)(11 425 305 174 215 98)(12 426 306 175 216 99)(13 427 307 176 181 100)(14 428 308 177 182 101)(15 429 309 178 183 102)(16 430 310 179 184 103)(17 431 311 180 185 104)(18 432 312 145 186 105)(19 397 313 146 187 106)(20 398 314 147 188 107)(21 399 315 148 189 108)(22 400 316 149 190 73)(23 401 317 150 191 74)(24 402 318 151 192 75)(25 403 319 152 193 76)(26 404 320 153 194 77)(27 405 321 154 195 78)(28 406 322 155 196 79)(29 407 323 156 197 80)(30 408 324 157 198 81)(31 409 289 158 199 82)(32 410 290 159 200 83)(33 411 291 160 201 84)(34 412 292 161 202 85)(35 413 293 162 203 86)(36 414 294 163 204 87)(37 277 353 136 394 225)(38 278 354 137 395 226)(39 279 355 138 396 227)(40 280 356 139 361 228)(41 281 357 140 362 229)(42 282 358 141 363 230)(43 283 359 142 364 231)(44 284 360 143 365 232)(45 285 325 144 366 233)(46 286 326 109 367 234)(47 287 327 110 368 235)(48 288 328 111 369 236)(49 253 329 112 370 237)(50 254 330 113 371 238)(51 255 331 114 372 239)(52 256 332 115 373 240)(53 257 333 116 374 241)(54 258 334 117 375 242)(55 259 335 118 376 243)(56 260 336 119 377 244)(57 261 337 120 378 245)(58 262 338 121 379 246)(59 263 339 122 380 247)(60 264 340 123 381 248)(61 265 341 124 382 249)(62 266 342 125 383 250)(63 267 343 126 384 251)(64 268 344 127 385 252)(65 269 345 128 386 217)(66 270 346 129 387 218)(67 271 347 130 388 219)(68 272 348 131 389 220)(69 273 349 132 390 221)(70 274 350 133 391 222)(71 275 351 134 392 223)(72 276 352 135 393 224)
(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)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396)(397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)

G:=sub<Sym(432)| (1,126)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,133)(9,134)(10,135)(11,136)(12,137)(13,138)(14,139)(15,140)(16,141)(17,142)(18,143)(19,144)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,174)(38,175)(39,176)(40,177)(41,178)(42,179)(43,180)(44,145)(45,146)(46,147)(47,148)(48,149)(49,150)(50,151)(51,152)(52,153)(53,154)(54,155)(55,156)(56,157)(57,158)(58,159)(59,160)(60,161)(61,162)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,169)(69,170)(70,171)(71,172)(72,173)(73,328)(74,329)(75,330)(76,331)(77,332)(78,333)(79,334)(80,335)(81,336)(82,337)(83,338)(84,339)(85,340)(86,341)(87,342)(88,343)(89,344)(90,345)(91,346)(92,347)(93,348)(94,349)(95,350)(96,351)(97,352)(98,353)(99,354)(100,355)(101,356)(102,357)(103,358)(104,359)(105,360)(106,325)(107,326)(108,327)(181,279)(182,280)(183,281)(184,282)(185,283)(186,284)(187,285)(188,286)(189,287)(190,288)(191,253)(192,254)(193,255)(194,256)(195,257)(196,258)(197,259)(198,260)(199,261)(200,262)(201,263)(202,264)(203,265)(204,266)(205,267)(206,268)(207,269)(208,270)(209,271)(210,272)(211,273)(212,274)(213,275)(214,276)(215,277)(216,278)(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)(240,320)(241,321)(242,322)(243,323)(244,324)(245,289)(246,290)(247,291)(248,292)(249,293)(250,294)(251,295)(252,296)(361,428)(362,429)(363,430)(364,431)(365,432)(366,397)(367,398)(368,399)(369,400)(370,401)(371,402)(372,403)(373,404)(374,405)(375,406)(376,407)(377,408)(378,409)(379,410)(380,411)(381,412)(382,413)(383,414)(384,415)(385,416)(386,417)(387,418)(388,419)(389,420)(390,421)(391,422)(392,423)(393,424)(394,425)(395,426)(396,427), (1,415,295,164,205,88)(2,416,296,165,206,89)(3,417,297,166,207,90)(4,418,298,167,208,91)(5,419,299,168,209,92)(6,420,300,169,210,93)(7,421,301,170,211,94)(8,422,302,171,212,95)(9,423,303,172,213,96)(10,424,304,173,214,97)(11,425,305,174,215,98)(12,426,306,175,216,99)(13,427,307,176,181,100)(14,428,308,177,182,101)(15,429,309,178,183,102)(16,430,310,179,184,103)(17,431,311,180,185,104)(18,432,312,145,186,105)(19,397,313,146,187,106)(20,398,314,147,188,107)(21,399,315,148,189,108)(22,400,316,149,190,73)(23,401,317,150,191,74)(24,402,318,151,192,75)(25,403,319,152,193,76)(26,404,320,153,194,77)(27,405,321,154,195,78)(28,406,322,155,196,79)(29,407,323,156,197,80)(30,408,324,157,198,81)(31,409,289,158,199,82)(32,410,290,159,200,83)(33,411,291,160,201,84)(34,412,292,161,202,85)(35,413,293,162,203,86)(36,414,294,163,204,87)(37,277,353,136,394,225)(38,278,354,137,395,226)(39,279,355,138,396,227)(40,280,356,139,361,228)(41,281,357,140,362,229)(42,282,358,141,363,230)(43,283,359,142,364,231)(44,284,360,143,365,232)(45,285,325,144,366,233)(46,286,326,109,367,234)(47,287,327,110,368,235)(48,288,328,111,369,236)(49,253,329,112,370,237)(50,254,330,113,371,238)(51,255,331,114,372,239)(52,256,332,115,373,240)(53,257,333,116,374,241)(54,258,334,117,375,242)(55,259,335,118,376,243)(56,260,336,119,377,244)(57,261,337,120,378,245)(58,262,338,121,379,246)(59,263,339,122,380,247)(60,264,340,123,381,248)(61,265,341,124,382,249)(62,266,342,125,383,250)(63,267,343,126,384,251)(64,268,344,127,385,252)(65,269,345,128,386,217)(66,270,346,129,387,218)(67,271,347,130,388,219)(68,272,348,131,389,220)(69,273,349,132,390,221)(70,274,350,133,391,222)(71,275,351,134,392,223)(72,276,352,135,393,224), (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)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)>;

G:=Group( (1,126)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,133)(9,134)(10,135)(11,136)(12,137)(13,138)(14,139)(15,140)(16,141)(17,142)(18,143)(19,144)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,174)(38,175)(39,176)(40,177)(41,178)(42,179)(43,180)(44,145)(45,146)(46,147)(47,148)(48,149)(49,150)(50,151)(51,152)(52,153)(53,154)(54,155)(55,156)(56,157)(57,158)(58,159)(59,160)(60,161)(61,162)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,169)(69,170)(70,171)(71,172)(72,173)(73,328)(74,329)(75,330)(76,331)(77,332)(78,333)(79,334)(80,335)(81,336)(82,337)(83,338)(84,339)(85,340)(86,341)(87,342)(88,343)(89,344)(90,345)(91,346)(92,347)(93,348)(94,349)(95,350)(96,351)(97,352)(98,353)(99,354)(100,355)(101,356)(102,357)(103,358)(104,359)(105,360)(106,325)(107,326)(108,327)(181,279)(182,280)(183,281)(184,282)(185,283)(186,284)(187,285)(188,286)(189,287)(190,288)(191,253)(192,254)(193,255)(194,256)(195,257)(196,258)(197,259)(198,260)(199,261)(200,262)(201,263)(202,264)(203,265)(204,266)(205,267)(206,268)(207,269)(208,270)(209,271)(210,272)(211,273)(212,274)(213,275)(214,276)(215,277)(216,278)(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)(240,320)(241,321)(242,322)(243,323)(244,324)(245,289)(246,290)(247,291)(248,292)(249,293)(250,294)(251,295)(252,296)(361,428)(362,429)(363,430)(364,431)(365,432)(366,397)(367,398)(368,399)(369,400)(370,401)(371,402)(372,403)(373,404)(374,405)(375,406)(376,407)(377,408)(378,409)(379,410)(380,411)(381,412)(382,413)(383,414)(384,415)(385,416)(386,417)(387,418)(388,419)(389,420)(390,421)(391,422)(392,423)(393,424)(394,425)(395,426)(396,427), (1,415,295,164,205,88)(2,416,296,165,206,89)(3,417,297,166,207,90)(4,418,298,167,208,91)(5,419,299,168,209,92)(6,420,300,169,210,93)(7,421,301,170,211,94)(8,422,302,171,212,95)(9,423,303,172,213,96)(10,424,304,173,214,97)(11,425,305,174,215,98)(12,426,306,175,216,99)(13,427,307,176,181,100)(14,428,308,177,182,101)(15,429,309,178,183,102)(16,430,310,179,184,103)(17,431,311,180,185,104)(18,432,312,145,186,105)(19,397,313,146,187,106)(20,398,314,147,188,107)(21,399,315,148,189,108)(22,400,316,149,190,73)(23,401,317,150,191,74)(24,402,318,151,192,75)(25,403,319,152,193,76)(26,404,320,153,194,77)(27,405,321,154,195,78)(28,406,322,155,196,79)(29,407,323,156,197,80)(30,408,324,157,198,81)(31,409,289,158,199,82)(32,410,290,159,200,83)(33,411,291,160,201,84)(34,412,292,161,202,85)(35,413,293,162,203,86)(36,414,294,163,204,87)(37,277,353,136,394,225)(38,278,354,137,395,226)(39,279,355,138,396,227)(40,280,356,139,361,228)(41,281,357,140,362,229)(42,282,358,141,363,230)(43,283,359,142,364,231)(44,284,360,143,365,232)(45,285,325,144,366,233)(46,286,326,109,367,234)(47,287,327,110,368,235)(48,288,328,111,369,236)(49,253,329,112,370,237)(50,254,330,113,371,238)(51,255,331,114,372,239)(52,256,332,115,373,240)(53,257,333,116,374,241)(54,258,334,117,375,242)(55,259,335,118,376,243)(56,260,336,119,377,244)(57,261,337,120,378,245)(58,262,338,121,379,246)(59,263,339,122,380,247)(60,264,340,123,381,248)(61,265,341,124,382,249)(62,266,342,125,383,250)(63,267,343,126,384,251)(64,268,344,127,385,252)(65,269,345,128,386,217)(66,270,346,129,387,218)(67,271,347,130,388,219)(68,272,348,131,389,220)(69,273,349,132,390,221)(70,274,350,133,391,222)(71,275,351,134,392,223)(72,276,352,135,393,224), (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)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432) );

G=PermutationGroup([(1,126),(2,127),(3,128),(4,129),(5,130),(6,131),(7,132),(8,133),(9,134),(10,135),(11,136),(12,137),(13,138),(14,139),(15,140),(16,141),(17,142),(18,143),(19,144),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,121),(33,122),(34,123),(35,124),(36,125),(37,174),(38,175),(39,176),(40,177),(41,178),(42,179),(43,180),(44,145),(45,146),(46,147),(47,148),(48,149),(49,150),(50,151),(51,152),(52,153),(53,154),(54,155),(55,156),(56,157),(57,158),(58,159),(59,160),(60,161),(61,162),(62,163),(63,164),(64,165),(65,166),(66,167),(67,168),(68,169),(69,170),(70,171),(71,172),(72,173),(73,328),(74,329),(75,330),(76,331),(77,332),(78,333),(79,334),(80,335),(81,336),(82,337),(83,338),(84,339),(85,340),(86,341),(87,342),(88,343),(89,344),(90,345),(91,346),(92,347),(93,348),(94,349),(95,350),(96,351),(97,352),(98,353),(99,354),(100,355),(101,356),(102,357),(103,358),(104,359),(105,360),(106,325),(107,326),(108,327),(181,279),(182,280),(183,281),(184,282),(185,283),(186,284),(187,285),(188,286),(189,287),(190,288),(191,253),(192,254),(193,255),(194,256),(195,257),(196,258),(197,259),(198,260),(199,261),(200,262),(201,263),(202,264),(203,265),(204,266),(205,267),(206,268),(207,269),(208,270),(209,271),(210,272),(211,273),(212,274),(213,275),(214,276),(215,277),(216,278),(217,297),(218,298),(219,299),(220,300),(221,301),(222,302),(223,303),(224,304),(225,305),(226,306),(227,307),(228,308),(229,309),(230,310),(231,311),(232,312),(233,313),(234,314),(235,315),(236,316),(237,317),(238,318),(239,319),(240,320),(241,321),(242,322),(243,323),(244,324),(245,289),(246,290),(247,291),(248,292),(249,293),(250,294),(251,295),(252,296),(361,428),(362,429),(363,430),(364,431),(365,432),(366,397),(367,398),(368,399),(369,400),(370,401),(371,402),(372,403),(373,404),(374,405),(375,406),(376,407),(377,408),(378,409),(379,410),(380,411),(381,412),(382,413),(383,414),(384,415),(385,416),(386,417),(387,418),(388,419),(389,420),(390,421),(391,422),(392,423),(393,424),(394,425),(395,426),(396,427)], [(1,415,295,164,205,88),(2,416,296,165,206,89),(3,417,297,166,207,90),(4,418,298,167,208,91),(5,419,299,168,209,92),(6,420,300,169,210,93),(7,421,301,170,211,94),(8,422,302,171,212,95),(9,423,303,172,213,96),(10,424,304,173,214,97),(11,425,305,174,215,98),(12,426,306,175,216,99),(13,427,307,176,181,100),(14,428,308,177,182,101),(15,429,309,178,183,102),(16,430,310,179,184,103),(17,431,311,180,185,104),(18,432,312,145,186,105),(19,397,313,146,187,106),(20,398,314,147,188,107),(21,399,315,148,189,108),(22,400,316,149,190,73),(23,401,317,150,191,74),(24,402,318,151,192,75),(25,403,319,152,193,76),(26,404,320,153,194,77),(27,405,321,154,195,78),(28,406,322,155,196,79),(29,407,323,156,197,80),(30,408,324,157,198,81),(31,409,289,158,199,82),(32,410,290,159,200,83),(33,411,291,160,201,84),(34,412,292,161,202,85),(35,413,293,162,203,86),(36,414,294,163,204,87),(37,277,353,136,394,225),(38,278,354,137,395,226),(39,279,355,138,396,227),(40,280,356,139,361,228),(41,281,357,140,362,229),(42,282,358,141,363,230),(43,283,359,142,364,231),(44,284,360,143,365,232),(45,285,325,144,366,233),(46,286,326,109,367,234),(47,287,327,110,368,235),(48,288,328,111,369,236),(49,253,329,112,370,237),(50,254,330,113,371,238),(51,255,331,114,372,239),(52,256,332,115,373,240),(53,257,333,116,374,241),(54,258,334,117,375,242),(55,259,335,118,376,243),(56,260,336,119,377,244),(57,261,337,120,378,245),(58,262,338,121,379,246),(59,263,339,122,380,247),(60,264,340,123,381,248),(61,265,341,124,382,249),(62,266,342,125,383,250),(63,267,343,126,384,251),(64,268,344,127,385,252),(65,269,345,128,386,217),(66,270,346,129,387,218),(67,271,347,130,388,219),(68,272,348,131,389,220),(69,273,349,132,390,221),(70,274,350,133,391,222),(71,275,351,134,392,223),(72,276,352,135,393,224)], [(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),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396),(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)])

432 conjugacy classes

class 1 2A···2G3A···3H4A···4H6A···6BD9A···9R12A···12BL18A···18DV36A···36EN
order12···23···34···46···69···912···1218···1836···36
size11···11···11···11···11···11···11···11···1

432 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C3C4C6C6C6C6C9C12C12C18C18C36
kernelC2×C6×C36C6×C36C2×C6×C18C22×C36C2×C6×C12C6×C18C2×C36C22×C18C6×C12C2×C62C22×C12C2×C18C62C2×C12C22×C6C2×C6
# reps16162836612218481610818144

Matrix representation of C2×C6×C36 in GL3(𝔽37) generated by

100
010
0036
,
1000
0270
0027
,
3100
060
0028
G:=sub<GL(3,GF(37))| [1,0,0,0,1,0,0,0,36],[10,0,0,0,27,0,0,0,27],[31,0,0,0,6,0,0,0,28] >;

C2×C6×C36 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_{36}
% in TeX

G:=Group("C2xC6xC36");
// GroupNames label

G:=SmallGroup(432,400);
// by ID

G=gap.SmallGroup(432,400);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,528]);
// Polycyclic

G:=Group<a,b,c|a^2=b^6=c^36=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽