direct product, abelian, monomial
Aliases: C2×C6×C36, SmallGroup(432,400)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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, C3, C3, C4, C22, C6, C6, C2×C4, C23, C9, C32, C12, C2×C6, C22×C4, C18, C3×C6, C3×C6, C2×C12, C22×C6, C22×C6, C3×C9, C36, C2×C18, C3×C12, C62, C22×C12, C22×C12, C3×C18, C3×C18, C2×C36, C22×C18, C6×C12, C2×C62, C3×C36, C6×C18, C22×C36, C2×C6×C12, C6×C36, C2×C6×C18, C2×C6×C36
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C9, C32, C12, C2×C6, C22×C4, C18, C3×C6, C2×C12, C22×C6, C3×C9, C36, C2×C18, C3×C12, C62, C22×C12, C3×C18, C2×C36, C22×C18, C6×C12, C2×C62, C3×C36, C6×C18, C22×C36, C2×C6×C12, C6×C36, C2×C6×C18, C2×C6×C36
►Regular action on 432 points
Generators in S432
(1 207)(2 208)(3 209)(4 210)(5 211)(6 212)(7 213)(8 214)(9 215)(10 216)(11 181)(12 182)(13 183)(14 184)(15 185)(16 186)(17 187)(18 188)(19 189)(20 190)(21 191)(22 192)(23 193)(24 194)(25 195)(26 196)(27 197)(28 198)(29 199)(30 200)(31 201)(32 202)(33 203)(34 204)(35 205)(36 206)(37 336)(38 337)(39 338)(40 339)(41 340)(42 341)(43 342)(44 343)(45 344)(46 345)(47 346)(48 347)(49 348)(50 349)(51 350)(52 351)(53 352)(54 353)(55 354)(56 355)(57 356)(58 357)(59 358)(60 359)(61 360)(62 325)(63 326)(64 327)(65 328)(66 329)(67 330)(68 331)(69 332)(70 333)(71 334)(72 335)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 169)(85 170)(86 171)(87 172)(88 173)(89 174)(90 175)(91 176)(92 177)(93 178)(94 179)(95 180)(96 145)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 282)(110 283)(111 284)(112 285)(113 286)(114 287)(115 288)(116 253)(117 254)(118 255)(119 256)(120 257)(121 258)(122 259)(123 260)(124 261)(125 262)(126 263)(127 264)(128 265)(129 266)(130 267)(131 268)(132 269)(133 270)(134 271)(135 272)(136 273)(137 274)(138 275)(139 276)(140 277)(141 278)(142 279)(143 280)(144 281)(217 295)(218 296)(219 297)(220 298)(221 299)(222 300)(223 301)(224 302)(225 303)(226 304)(227 305)(228 306)(229 307)(230 308)(231 309)(232 310)(233 311)(234 312)(235 313)(236 314)(237 315)(238 316)(239 317)(240 318)(241 319)(242 320)(243 321)(244 322)(245 323)(246 324)(247 289)(248 290)(249 291)(250 292)(251 293)(252 294)(361 398)(362 399)(363 400)(364 401)(365 402)(366 403)(367 404)(368 405)(369 406)(370 407)(371 408)(372 409)(373 410)(374 411)(375 412)(376 413)(377 414)(378 415)(379 416)(380 417)(381 418)(382 419)(383 420)(384 421)(385 422)(386 423)(387 424)(388 425)(389 426)(390 427)(391 428)(392 429)(393 430)(394 431)(395 432)(396 397)
(1 83 136 44 291 386)(2 84 137 45 292 387)(3 85 138 46 293 388)(4 86 139 47 294 389)(5 87 140 48 295 390)(6 88 141 49 296 391)(7 89 142 50 297 392)(8 90 143 51 298 393)(9 91 144 52 299 394)(10 92 109 53 300 395)(11 93 110 54 301 396)(12 94 111 55 302 361)(13 95 112 56 303 362)(14 96 113 57 304 363)(15 97 114 58 305 364)(16 98 115 59 306 365)(17 99 116 60 307 366)(18 100 117 61 308 367)(19 101 118 62 309 368)(20 102 119 63 310 369)(21 103 120 64 311 370)(22 104 121 65 312 371)(23 105 122 66 313 372)(24 106 123 67 314 373)(25 107 124 68 315 374)(26 108 125 69 316 375)(27 73 126 70 317 376)(28 74 127 71 318 377)(29 75 128 72 319 378)(30 76 129 37 320 379)(31 77 130 38 321 380)(32 78 131 39 322 381)(33 79 132 40 323 382)(34 80 133 41 324 383)(35 81 134 42 289 384)(36 82 135 43 290 385)(145 286 356 226 400 184)(146 287 357 227 401 185)(147 288 358 228 402 186)(148 253 359 229 403 187)(149 254 360 230 404 188)(150 255 325 231 405 189)(151 256 326 232 406 190)(152 257 327 233 407 191)(153 258 328 234 408 192)(154 259 329 235 409 193)(155 260 330 236 410 194)(156 261 331 237 411 195)(157 262 332 238 412 196)(158 263 333 239 413 197)(159 264 334 240 414 198)(160 265 335 241 415 199)(161 266 336 242 416 200)(162 267 337 243 417 201)(163 268 338 244 418 202)(164 269 339 245 419 203)(165 270 340 246 420 204)(166 271 341 247 421 205)(167 272 342 248 422 206)(168 273 343 249 423 207)(169 274 344 250 424 208)(170 275 345 251 425 209)(171 276 346 252 426 210)(172 277 347 217 427 211)(173 278 348 218 428 212)(174 279 349 219 429 213)(175 280 350 220 430 214)(176 281 351 221 431 215)(177 282 352 222 432 216)(178 283 353 223 397 181)(179 284 354 224 398 182)(180 285 355 225 399 183)
(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)| (1,207)(2,208)(3,209)(4,210)(5,211)(6,212)(7,213)(8,214)(9,215)(10,216)(11,181)(12,182)(13,183)(14,184)(15,185)(16,186)(17,187)(18,188)(19,189)(20,190)(21,191)(22,192)(23,193)(24,194)(25,195)(26,196)(27,197)(28,198)(29,199)(30,200)(31,201)(32,202)(33,203)(34,204)(35,205)(36,206)(37,336)(38,337)(39,338)(40,339)(41,340)(42,341)(43,342)(44,343)(45,344)(46,345)(47,346)(48,347)(49,348)(50,349)(51,350)(52,351)(53,352)(54,353)(55,354)(56,355)(57,356)(58,357)(59,358)(60,359)(61,360)(62,325)(63,326)(64,327)(65,328)(66,329)(67,330)(68,331)(69,332)(70,333)(71,334)(72,335)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,169)(85,170)(86,171)(87,172)(88,173)(89,174)(90,175)(91,176)(92,177)(93,178)(94,179)(95,180)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,282)(110,283)(111,284)(112,285)(113,286)(114,287)(115,288)(116,253)(117,254)(118,255)(119,256)(120,257)(121,258)(122,259)(123,260)(124,261)(125,262)(126,263)(127,264)(128,265)(129,266)(130,267)(131,268)(132,269)(133,270)(134,271)(135,272)(136,273)(137,274)(138,275)(139,276)(140,277)(141,278)(142,279)(143,280)(144,281)(217,295)(218,296)(219,297)(220,298)(221,299)(222,300)(223,301)(224,302)(225,303)(226,304)(227,305)(228,306)(229,307)(230,308)(231,309)(232,310)(233,311)(234,312)(235,313)(236,314)(237,315)(238,316)(239,317)(240,318)(241,319)(242,320)(243,321)(244,322)(245,323)(246,324)(247,289)(248,290)(249,291)(250,292)(251,293)(252,294)(361,398)(362,399)(363,400)(364,401)(365,402)(366,403)(367,404)(368,405)(369,406)(370,407)(371,408)(372,409)(373,410)(374,411)(375,412)(376,413)(377,414)(378,415)(379,416)(380,417)(381,418)(382,419)(383,420)(384,421)(385,422)(386,423)(387,424)(388,425)(389,426)(390,427)(391,428)(392,429)(393,430)(394,431)(395,432)(396,397), (1,83,136,44,291,386)(2,84,137,45,292,387)(3,85,138,46,293,388)(4,86,139,47,294,389)(5,87,140,48,295,390)(6,88,141,49,296,391)(7,89,142,50,297,392)(8,90,143,51,298,393)(9,91,144,52,299,394)(10,92,109,53,300,395)(11,93,110,54,301,396)(12,94,111,55,302,361)(13,95,112,56,303,362)(14,96,113,57,304,363)(15,97,114,58,305,364)(16,98,115,59,306,365)(17,99,116,60,307,366)(18,100,117,61,308,367)(19,101,118,62,309,368)(20,102,119,63,310,369)(21,103,120,64,311,370)(22,104,121,65,312,371)(23,105,122,66,313,372)(24,106,123,67,314,373)(25,107,124,68,315,374)(26,108,125,69,316,375)(27,73,126,70,317,376)(28,74,127,71,318,377)(29,75,128,72,319,378)(30,76,129,37,320,379)(31,77,130,38,321,380)(32,78,131,39,322,381)(33,79,132,40,323,382)(34,80,133,41,324,383)(35,81,134,42,289,384)(36,82,135,43,290,385)(145,286,356,226,400,184)(146,287,357,227,401,185)(147,288,358,228,402,186)(148,253,359,229,403,187)(149,254,360,230,404,188)(150,255,325,231,405,189)(151,256,326,232,406,190)(152,257,327,233,407,191)(153,258,328,234,408,192)(154,259,329,235,409,193)(155,260,330,236,410,194)(156,261,331,237,411,195)(157,262,332,238,412,196)(158,263,333,239,413,197)(159,264,334,240,414,198)(160,265,335,241,415,199)(161,266,336,242,416,200)(162,267,337,243,417,201)(163,268,338,244,418,202)(164,269,339,245,419,203)(165,270,340,246,420,204)(166,271,341,247,421,205)(167,272,342,248,422,206)(168,273,343,249,423,207)(169,274,344,250,424,208)(170,275,345,251,425,209)(171,276,346,252,426,210)(172,277,347,217,427,211)(173,278,348,218,428,212)(174,279,349,219,429,213)(175,280,350,220,430,214)(176,281,351,221,431,215)(177,282,352,222,432,216)(178,283,353,223,397,181)(179,284,354,224,398,182)(180,285,355,225,399,183), (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( (1,207)(2,208)(3,209)(4,210)(5,211)(6,212)(7,213)(8,214)(9,215)(10,216)(11,181)(12,182)(13,183)(14,184)(15,185)(16,186)(17,187)(18,188)(19,189)(20,190)(21,191)(22,192)(23,193)(24,194)(25,195)(26,196)(27,197)(28,198)(29,199)(30,200)(31,201)(32,202)(33,203)(34,204)(35,205)(36,206)(37,336)(38,337)(39,338)(40,339)(41,340)(42,341)(43,342)(44,343)(45,344)(46,345)(47,346)(48,347)(49,348)(50,349)(51,350)(52,351)(53,352)(54,353)(55,354)(56,355)(57,356)(58,357)(59,358)(60,359)(61,360)(62,325)(63,326)(64,327)(65,328)(66,329)(67,330)(68,331)(69,332)(70,333)(71,334)(72,335)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,169)(85,170)(86,171)(87,172)(88,173)(89,174)(90,175)(91,176)(92,177)(93,178)(94,179)(95,180)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,282)(110,283)(111,284)(112,285)(113,286)(114,287)(115,288)(116,253)(117,254)(118,255)(119,256)(120,257)(121,258)(122,259)(123,260)(124,261)(125,262)(126,263)(127,264)(128,265)(129,266)(130,267)(131,268)(132,269)(133,270)(134,271)(135,272)(136,273)(137,274)(138,275)(139,276)(140,277)(141,278)(142,279)(143,280)(144,281)(217,295)(218,296)(219,297)(220,298)(221,299)(222,300)(223,301)(224,302)(225,303)(226,304)(227,305)(228,306)(229,307)(230,308)(231,309)(232,310)(233,311)(234,312)(235,313)(236,314)(237,315)(238,316)(239,317)(240,318)(241,319)(242,320)(243,321)(244,322)(245,323)(246,324)(247,289)(248,290)(249,291)(250,292)(251,293)(252,294)(361,398)(362,399)(363,400)(364,401)(365,402)(366,403)(367,404)(368,405)(369,406)(370,407)(371,408)(372,409)(373,410)(374,411)(375,412)(376,413)(377,414)(378,415)(379,416)(380,417)(381,418)(382,419)(383,420)(384,421)(385,422)(386,423)(387,424)(388,425)(389,426)(390,427)(391,428)(392,429)(393,430)(394,431)(395,432)(396,397), (1,83,136,44,291,386)(2,84,137,45,292,387)(3,85,138,46,293,388)(4,86,139,47,294,389)(5,87,140,48,295,390)(6,88,141,49,296,391)(7,89,142,50,297,392)(8,90,143,51,298,393)(9,91,144,52,299,394)(10,92,109,53,300,395)(11,93,110,54,301,396)(12,94,111,55,302,361)(13,95,112,56,303,362)(14,96,113,57,304,363)(15,97,114,58,305,364)(16,98,115,59,306,365)(17,99,116,60,307,366)(18,100,117,61,308,367)(19,101,118,62,309,368)(20,102,119,63,310,369)(21,103,120,64,311,370)(22,104,121,65,312,371)(23,105,122,66,313,372)(24,106,123,67,314,373)(25,107,124,68,315,374)(26,108,125,69,316,375)(27,73,126,70,317,376)(28,74,127,71,318,377)(29,75,128,72,319,378)(30,76,129,37,320,379)(31,77,130,38,321,380)(32,78,131,39,322,381)(33,79,132,40,323,382)(34,80,133,41,324,383)(35,81,134,42,289,384)(36,82,135,43,290,385)(145,286,356,226,400,184)(146,287,357,227,401,185)(147,288,358,228,402,186)(148,253,359,229,403,187)(149,254,360,230,404,188)(150,255,325,231,405,189)(151,256,326,232,406,190)(152,257,327,233,407,191)(153,258,328,234,408,192)(154,259,329,235,409,193)(155,260,330,236,410,194)(156,261,331,237,411,195)(157,262,332,238,412,196)(158,263,333,239,413,197)(159,264,334,240,414,198)(160,265,335,241,415,199)(161,266,336,242,416,200)(162,267,337,243,417,201)(163,268,338,244,418,202)(164,269,339,245,419,203)(165,270,340,246,420,204)(166,271,341,247,421,205)(167,272,342,248,422,206)(168,273,343,249,423,207)(169,274,344,250,424,208)(170,275,345,251,425,209)(171,276,346,252,426,210)(172,277,347,217,427,211)(173,278,348,218,428,212)(174,279,349,219,429,213)(175,280,350,220,430,214)(176,281,351,221,431,215)(177,282,352,222,432,216)(178,283,353,223,397,181)(179,284,354,224,398,182)(180,285,355,225,399,183), (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([[(1,207),(2,208),(3,209),(4,210),(5,211),(6,212),(7,213),(8,214),(9,215),(10,216),(11,181),(12,182),(13,183),(14,184),(15,185),(16,186),(17,187),(18,188),(19,189),(20,190),(21,191),(22,192),(23,193),(24,194),(25,195),(26,196),(27,197),(28,198),(29,199),(30,200),(31,201),(32,202),(33,203),(34,204),(35,205),(36,206),(37,336),(38,337),(39,338),(40,339),(41,340),(42,341),(43,342),(44,343),(45,344),(46,345),(47,346),(48,347),(49,348),(50,349),(51,350),(52,351),(53,352),(54,353),(55,354),(56,355),(57,356),(58,357),(59,358),(60,359),(61,360),(62,325),(63,326),(64,327),(65,328),(66,329),(67,330),(68,331),(69,332),(70,333),(71,334),(72,335),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,164),(80,165),(81,166),(82,167),(83,168),(84,169),(85,170),(86,171),(87,172),(88,173),(89,174),(90,175),(91,176),(92,177),(93,178),(94,179),(95,180),(96,145),(97,146),(98,147),(99,148),(100,149),(101,150),(102,151),(103,152),(104,153),(105,154),(106,155),(107,156),(108,157),(109,282),(110,283),(111,284),(112,285),(113,286),(114,287),(115,288),(116,253),(117,254),(118,255),(119,256),(120,257),(121,258),(122,259),(123,260),(124,261),(125,262),(126,263),(127,264),(128,265),(129,266),(130,267),(131,268),(132,269),(133,270),(134,271),(135,272),(136,273),(137,274),(138,275),(139,276),(140,277),(141,278),(142,279),(143,280),(144,281),(217,295),(218,296),(219,297),(220,298),(221,299),(222,300),(223,301),(224,302),(225,303),(226,304),(227,305),(228,306),(229,307),(230,308),(231,309),(232,310),(233,311),(234,312),(235,313),(236,314),(237,315),(238,316),(239,317),(240,318),(241,319),(242,320),(243,321),(244,322),(245,323),(246,324),(247,289),(248,290),(249,291),(250,292),(251,293),(252,294),(361,398),(362,399),(363,400),(364,401),(365,402),(366,403),(367,404),(368,405),(369,406),(370,407),(371,408),(372,409),(373,410),(374,411),(375,412),(376,413),(377,414),(378,415),(379,416),(380,417),(381,418),(382,419),(383,420),(384,421),(385,422),(386,423),(387,424),(388,425),(389,426),(390,427),(391,428),(392,429),(393,430),(394,431),(395,432),(396,397)], [(1,83,136,44,291,386),(2,84,137,45,292,387),(3,85,138,46,293,388),(4,86,139,47,294,389),(5,87,140,48,295,390),(6,88,141,49,296,391),(7,89,142,50,297,392),(8,90,143,51,298,393),(9,91,144,52,299,394),(10,92,109,53,300,395),(11,93,110,54,301,396),(12,94,111,55,302,361),(13,95,112,56,303,362),(14,96,113,57,304,363),(15,97,114,58,305,364),(16,98,115,59,306,365),(17,99,116,60,307,366),(18,100,117,61,308,367),(19,101,118,62,309,368),(20,102,119,63,310,369),(21,103,120,64,311,370),(22,104,121,65,312,371),(23,105,122,66,313,372),(24,106,123,67,314,373),(25,107,124,68,315,374),(26,108,125,69,316,375),(27,73,126,70,317,376),(28,74,127,71,318,377),(29,75,128,72,319,378),(30,76,129,37,320,379),(31,77,130,38,321,380),(32,78,131,39,322,381),(33,79,132,40,323,382),(34,80,133,41,324,383),(35,81,134,42,289,384),(36,82,135,43,290,385),(145,286,356,226,400,184),(146,287,357,227,401,185),(147,288,358,228,402,186),(148,253,359,229,403,187),(149,254,360,230,404,188),(150,255,325,231,405,189),(151,256,326,232,406,190),(152,257,327,233,407,191),(153,258,328,234,408,192),(154,259,329,235,409,193),(155,260,330,236,410,194),(156,261,331,237,411,195),(157,262,332,238,412,196),(158,263,333,239,413,197),(159,264,334,240,414,198),(160,265,335,241,415,199),(161,266,336,242,416,200),(162,267,337,243,417,201),(163,268,338,244,418,202),(164,269,339,245,419,203),(165,270,340,246,420,204),(166,271,341,247,421,205),(167,272,342,248,422,206),(168,273,343,249,423,207),(169,274,344,250,424,208),(170,275,345,251,425,209),(171,276,346,252,426,210),(172,277,347,217,427,211),(173,278,348,218,428,212),(174,279,349,219,429,213),(175,280,350,220,430,214),(176,281,351,221,431,215),(177,282,352,222,432,216),(178,283,353,223,397,181),(179,284,354,224,398,182),(180,285,355,225,399,183)], [(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 | ··· | 2G | 3A | ··· | 3H | 4A | ··· | 4H | 6A | ··· | 6BD | 9A | ··· | 9R | 12A | ··· | 12BL | 18A | ··· | 18DV | 36A | ··· | 36EN |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
432 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C4 | C6 | C6 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C36 |
| kernel | C2×C6×C36 | C6×C36 | C2×C6×C18 | C22×C36 | C2×C6×C12 | C6×C18 | C2×C36 | C22×C18 | C6×C12 | C2×C62 | C22×C12 | C2×C18 | C62 | C2×C12 | C22×C6 | C2×C6 |
| # reps | 1 | 6 | 1 | 6 | 2 | 8 | 36 | 6 | 12 | 2 | 18 | 48 | 16 | 108 | 18 | 144 |
Matrix representation of C2×C6×C36 ►in GL3(𝔽37) generated by
| 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 |
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