direct product, abelian, monomial, 3-elementary
Aliases: C33×C15, SmallGroup(405,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C33×C15 |
| C1 — C33×C15 |
| C1 — C33×C15 |
Generators and relations for C33×C15
G = < a,b,c,d | a3=b3=c3=d15=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 424, all normal (4 characteristic)
C1, C3, C5, C32, C15, C33, C3×C15, C34, C32×C15, C33×C15
Quotients: C1, C3, C5, C32, C15, C33, C3×C15, C34, C32×C15, C33×C15
►Regular action on 405 points
Generators in S405
(1 280 149)(2 281 150)(3 282 136)(4 283 137)(5 284 138)(6 285 139)(7 271 140)(8 272 141)(9 273 142)(10 274 143)(11 275 144)(12 276 145)(13 277 146)(14 278 147)(15 279 148)(16 290 151)(17 291 152)(18 292 153)(19 293 154)(20 294 155)(21 295 156)(22 296 157)(23 297 158)(24 298 159)(25 299 160)(26 300 161)(27 286 162)(28 287 163)(29 288 164)(30 289 165)(31 301 168)(32 302 169)(33 303 170)(34 304 171)(35 305 172)(36 306 173)(37 307 174)(38 308 175)(39 309 176)(40 310 177)(41 311 178)(42 312 179)(43 313 180)(44 314 166)(45 315 167)(46 329 190)(47 330 191)(48 316 192)(49 317 193)(50 318 194)(51 319 195)(52 320 181)(53 321 182)(54 322 183)(55 323 184)(56 324 185)(57 325 186)(58 326 187)(59 327 188)(60 328 189)(61 342 198)(62 343 199)(63 344 200)(64 345 201)(65 331 202)(66 332 203)(67 333 204)(68 334 205)(69 335 206)(70 336 207)(71 337 208)(72 338 209)(73 339 210)(74 340 196)(75 341 197)(76 348 215)(77 349 216)(78 350 217)(79 351 218)(80 352 219)(81 353 220)(82 354 221)(83 355 222)(84 356 223)(85 357 224)(86 358 225)(87 359 211)(88 360 212)(89 346 213)(90 347 214)(91 374 237)(92 375 238)(93 361 239)(94 362 240)(95 363 226)(96 364 227)(97 365 228)(98 366 229)(99 367 230)(100 368 231)(101 369 232)(102 370 233)(103 371 234)(104 372 235)(105 373 236)(106 389 250)(107 390 251)(108 376 252)(109 377 253)(110 378 254)(111 379 255)(112 380 241)(113 381 242)(114 382 243)(115 383 244)(116 384 245)(117 385 246)(118 386 247)(119 387 248)(120 388 249)(121 397 258)(122 398 259)(123 399 260)(124 400 261)(125 401 262)(126 402 263)(127 403 264)(128 404 265)(129 405 266)(130 391 267)(131 392 268)(132 393 269)(133 394 270)(134 395 256)(135 396 257)
(1 102 59)(2 103 60)(3 104 46)(4 105 47)(5 91 48)(6 92 49)(7 93 50)(8 94 51)(9 95 52)(10 96 53)(11 97 54)(12 98 55)(13 99 56)(14 100 57)(15 101 58)(16 112 65)(17 113 66)(18 114 67)(19 115 68)(20 116 69)(21 117 70)(22 118 71)(23 119 72)(24 120 73)(25 106 74)(26 107 75)(27 108 61)(28 109 62)(29 110 63)(30 111 64)(31 134 76)(32 135 77)(33 121 78)(34 122 79)(35 123 80)(36 124 81)(37 125 82)(38 126 83)(39 127 84)(40 128 85)(41 129 86)(42 130 87)(43 131 88)(44 132 89)(45 133 90)(136 235 190)(137 236 191)(138 237 192)(139 238 193)(140 239 194)(141 240 195)(142 226 181)(143 227 182)(144 228 183)(145 229 184)(146 230 185)(147 231 186)(148 232 187)(149 233 188)(150 234 189)(151 241 202)(152 242 203)(153 243 204)(154 244 205)(155 245 206)(156 246 207)(157 247 208)(158 248 209)(159 249 210)(160 250 196)(161 251 197)(162 252 198)(163 253 199)(164 254 200)(165 255 201)(166 269 213)(167 270 214)(168 256 215)(169 257 216)(170 258 217)(171 259 218)(172 260 219)(173 261 220)(174 262 221)(175 263 222)(176 264 223)(177 265 224)(178 266 225)(179 267 211)(180 268 212)(271 361 318)(272 362 319)(273 363 320)(274 364 321)(275 365 322)(276 366 323)(277 367 324)(278 368 325)(279 369 326)(280 370 327)(281 371 328)(282 372 329)(283 373 330)(284 374 316)(285 375 317)(286 376 342)(287 377 343)(288 378 344)(289 379 345)(290 380 331)(291 381 332)(292 382 333)(293 383 334)(294 384 335)(295 385 336)(296 386 337)(297 387 338)(298 388 339)(299 389 340)(300 390 341)(301 395 348)(302 396 349)(303 397 350)(304 398 351)(305 399 352)(306 400 353)(307 401 354)(308 402 355)(309 403 356)(310 404 357)(311 405 358)(312 391 359)(313 392 360)(314 393 346)(315 394 347)
(1 42 23)(2 43 24)(3 44 25)(4 45 26)(5 31 27)(6 32 28)(7 33 29)(8 34 30)(9 35 16)(10 36 17)(11 37 18)(12 38 19)(13 39 20)(14 40 21)(15 41 22)(46 89 74)(47 90 75)(48 76 61)(49 77 62)(50 78 63)(51 79 64)(52 80 65)(53 81 66)(54 82 67)(55 83 68)(56 84 69)(57 85 70)(58 86 71)(59 87 72)(60 88 73)(91 134 108)(92 135 109)(93 121 110)(94 122 111)(95 123 112)(96 124 113)(97 125 114)(98 126 115)(99 127 116)(100 128 117)(101 129 118)(102 130 119)(103 131 120)(104 132 106)(105 133 107)(136 166 160)(137 167 161)(138 168 162)(139 169 163)(140 170 164)(141 171 165)(142 172 151)(143 173 152)(144 174 153)(145 175 154)(146 176 155)(147 177 156)(148 178 157)(149 179 158)(150 180 159)(181 219 202)(182 220 203)(183 221 204)(184 222 205)(185 223 206)(186 224 207)(187 225 208)(188 211 209)(189 212 210)(190 213 196)(191 214 197)(192 215 198)(193 216 199)(194 217 200)(195 218 201)(226 260 241)(227 261 242)(228 262 243)(229 263 244)(230 264 245)(231 265 246)(232 266 247)(233 267 248)(234 268 249)(235 269 250)(236 270 251)(237 256 252)(238 257 253)(239 258 254)(240 259 255)(271 303 288)(272 304 289)(273 305 290)(274 306 291)(275 307 292)(276 308 293)(277 309 294)(278 310 295)(279 311 296)(280 312 297)(281 313 298)(282 314 299)(283 315 300)(284 301 286)(285 302 287)(316 348 342)(317 349 343)(318 350 344)(319 351 345)(320 352 331)(321 353 332)(322 354 333)(323 355 334)(324 356 335)(325 357 336)(326 358 337)(327 359 338)(328 360 339)(329 346 340)(330 347 341)(361 397 378)(362 398 379)(363 399 380)(364 400 381)(365 401 382)(366 402 383)(367 403 384)(368 404 385)(369 405 386)(370 391 387)(371 392 388)(372 393 389)(373 394 390)(374 395 376)(375 396 377)
(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)
G:=sub<Sym(405)| (1,280,149)(2,281,150)(3,282,136)(4,283,137)(5,284,138)(6,285,139)(7,271,140)(8,272,141)(9,273,142)(10,274,143)(11,275,144)(12,276,145)(13,277,146)(14,278,147)(15,279,148)(16,290,151)(17,291,152)(18,292,153)(19,293,154)(20,294,155)(21,295,156)(22,296,157)(23,297,158)(24,298,159)(25,299,160)(26,300,161)(27,286,162)(28,287,163)(29,288,164)(30,289,165)(31,301,168)(32,302,169)(33,303,170)(34,304,171)(35,305,172)(36,306,173)(37,307,174)(38,308,175)(39,309,176)(40,310,177)(41,311,178)(42,312,179)(43,313,180)(44,314,166)(45,315,167)(46,329,190)(47,330,191)(48,316,192)(49,317,193)(50,318,194)(51,319,195)(52,320,181)(53,321,182)(54,322,183)(55,323,184)(56,324,185)(57,325,186)(58,326,187)(59,327,188)(60,328,189)(61,342,198)(62,343,199)(63,344,200)(64,345,201)(65,331,202)(66,332,203)(67,333,204)(68,334,205)(69,335,206)(70,336,207)(71,337,208)(72,338,209)(73,339,210)(74,340,196)(75,341,197)(76,348,215)(77,349,216)(78,350,217)(79,351,218)(80,352,219)(81,353,220)(82,354,221)(83,355,222)(84,356,223)(85,357,224)(86,358,225)(87,359,211)(88,360,212)(89,346,213)(90,347,214)(91,374,237)(92,375,238)(93,361,239)(94,362,240)(95,363,226)(96,364,227)(97,365,228)(98,366,229)(99,367,230)(100,368,231)(101,369,232)(102,370,233)(103,371,234)(104,372,235)(105,373,236)(106,389,250)(107,390,251)(108,376,252)(109,377,253)(110,378,254)(111,379,255)(112,380,241)(113,381,242)(114,382,243)(115,383,244)(116,384,245)(117,385,246)(118,386,247)(119,387,248)(120,388,249)(121,397,258)(122,398,259)(123,399,260)(124,400,261)(125,401,262)(126,402,263)(127,403,264)(128,404,265)(129,405,266)(130,391,267)(131,392,268)(132,393,269)(133,394,270)(134,395,256)(135,396,257), (1,102,59)(2,103,60)(3,104,46)(4,105,47)(5,91,48)(6,92,49)(7,93,50)(8,94,51)(9,95,52)(10,96,53)(11,97,54)(12,98,55)(13,99,56)(14,100,57)(15,101,58)(16,112,65)(17,113,66)(18,114,67)(19,115,68)(20,116,69)(21,117,70)(22,118,71)(23,119,72)(24,120,73)(25,106,74)(26,107,75)(27,108,61)(28,109,62)(29,110,63)(30,111,64)(31,134,76)(32,135,77)(33,121,78)(34,122,79)(35,123,80)(36,124,81)(37,125,82)(38,126,83)(39,127,84)(40,128,85)(41,129,86)(42,130,87)(43,131,88)(44,132,89)(45,133,90)(136,235,190)(137,236,191)(138,237,192)(139,238,193)(140,239,194)(141,240,195)(142,226,181)(143,227,182)(144,228,183)(145,229,184)(146,230,185)(147,231,186)(148,232,187)(149,233,188)(150,234,189)(151,241,202)(152,242,203)(153,243,204)(154,244,205)(155,245,206)(156,246,207)(157,247,208)(158,248,209)(159,249,210)(160,250,196)(161,251,197)(162,252,198)(163,253,199)(164,254,200)(165,255,201)(166,269,213)(167,270,214)(168,256,215)(169,257,216)(170,258,217)(171,259,218)(172,260,219)(173,261,220)(174,262,221)(175,263,222)(176,264,223)(177,265,224)(178,266,225)(179,267,211)(180,268,212)(271,361,318)(272,362,319)(273,363,320)(274,364,321)(275,365,322)(276,366,323)(277,367,324)(278,368,325)(279,369,326)(280,370,327)(281,371,328)(282,372,329)(283,373,330)(284,374,316)(285,375,317)(286,376,342)(287,377,343)(288,378,344)(289,379,345)(290,380,331)(291,381,332)(292,382,333)(293,383,334)(294,384,335)(295,385,336)(296,386,337)(297,387,338)(298,388,339)(299,389,340)(300,390,341)(301,395,348)(302,396,349)(303,397,350)(304,398,351)(305,399,352)(306,400,353)(307,401,354)(308,402,355)(309,403,356)(310,404,357)(311,405,358)(312,391,359)(313,392,360)(314,393,346)(315,394,347), (1,42,23)(2,43,24)(3,44,25)(4,45,26)(5,31,27)(6,32,28)(7,33,29)(8,34,30)(9,35,16)(10,36,17)(11,37,18)(12,38,19)(13,39,20)(14,40,21)(15,41,22)(46,89,74)(47,90,75)(48,76,61)(49,77,62)(50,78,63)(51,79,64)(52,80,65)(53,81,66)(54,82,67)(55,83,68)(56,84,69)(57,85,70)(58,86,71)(59,87,72)(60,88,73)(91,134,108)(92,135,109)(93,121,110)(94,122,111)(95,123,112)(96,124,113)(97,125,114)(98,126,115)(99,127,116)(100,128,117)(101,129,118)(102,130,119)(103,131,120)(104,132,106)(105,133,107)(136,166,160)(137,167,161)(138,168,162)(139,169,163)(140,170,164)(141,171,165)(142,172,151)(143,173,152)(144,174,153)(145,175,154)(146,176,155)(147,177,156)(148,178,157)(149,179,158)(150,180,159)(181,219,202)(182,220,203)(183,221,204)(184,222,205)(185,223,206)(186,224,207)(187,225,208)(188,211,209)(189,212,210)(190,213,196)(191,214,197)(192,215,198)(193,216,199)(194,217,200)(195,218,201)(226,260,241)(227,261,242)(228,262,243)(229,263,244)(230,264,245)(231,265,246)(232,266,247)(233,267,248)(234,268,249)(235,269,250)(236,270,251)(237,256,252)(238,257,253)(239,258,254)(240,259,255)(271,303,288)(272,304,289)(273,305,290)(274,306,291)(275,307,292)(276,308,293)(277,309,294)(278,310,295)(279,311,296)(280,312,297)(281,313,298)(282,314,299)(283,315,300)(284,301,286)(285,302,287)(316,348,342)(317,349,343)(318,350,344)(319,351,345)(320,352,331)(321,353,332)(322,354,333)(323,355,334)(324,356,335)(325,357,336)(326,358,337)(327,359,338)(328,360,339)(329,346,340)(330,347,341)(361,397,378)(362,398,379)(363,399,380)(364,400,381)(365,401,382)(366,402,383)(367,403,384)(368,404,385)(369,405,386)(370,391,387)(371,392,388)(372,393,389)(373,394,390)(374,395,376)(375,396,377), (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)>;
G:=Group( (1,280,149)(2,281,150)(3,282,136)(4,283,137)(5,284,138)(6,285,139)(7,271,140)(8,272,141)(9,273,142)(10,274,143)(11,275,144)(12,276,145)(13,277,146)(14,278,147)(15,279,148)(16,290,151)(17,291,152)(18,292,153)(19,293,154)(20,294,155)(21,295,156)(22,296,157)(23,297,158)(24,298,159)(25,299,160)(26,300,161)(27,286,162)(28,287,163)(29,288,164)(30,289,165)(31,301,168)(32,302,169)(33,303,170)(34,304,171)(35,305,172)(36,306,173)(37,307,174)(38,308,175)(39,309,176)(40,310,177)(41,311,178)(42,312,179)(43,313,180)(44,314,166)(45,315,167)(46,329,190)(47,330,191)(48,316,192)(49,317,193)(50,318,194)(51,319,195)(52,320,181)(53,321,182)(54,322,183)(55,323,184)(56,324,185)(57,325,186)(58,326,187)(59,327,188)(60,328,189)(61,342,198)(62,343,199)(63,344,200)(64,345,201)(65,331,202)(66,332,203)(67,333,204)(68,334,205)(69,335,206)(70,336,207)(71,337,208)(72,338,209)(73,339,210)(74,340,196)(75,341,197)(76,348,215)(77,349,216)(78,350,217)(79,351,218)(80,352,219)(81,353,220)(82,354,221)(83,355,222)(84,356,223)(85,357,224)(86,358,225)(87,359,211)(88,360,212)(89,346,213)(90,347,214)(91,374,237)(92,375,238)(93,361,239)(94,362,240)(95,363,226)(96,364,227)(97,365,228)(98,366,229)(99,367,230)(100,368,231)(101,369,232)(102,370,233)(103,371,234)(104,372,235)(105,373,236)(106,389,250)(107,390,251)(108,376,252)(109,377,253)(110,378,254)(111,379,255)(112,380,241)(113,381,242)(114,382,243)(115,383,244)(116,384,245)(117,385,246)(118,386,247)(119,387,248)(120,388,249)(121,397,258)(122,398,259)(123,399,260)(124,400,261)(125,401,262)(126,402,263)(127,403,264)(128,404,265)(129,405,266)(130,391,267)(131,392,268)(132,393,269)(133,394,270)(134,395,256)(135,396,257), (1,102,59)(2,103,60)(3,104,46)(4,105,47)(5,91,48)(6,92,49)(7,93,50)(8,94,51)(9,95,52)(10,96,53)(11,97,54)(12,98,55)(13,99,56)(14,100,57)(15,101,58)(16,112,65)(17,113,66)(18,114,67)(19,115,68)(20,116,69)(21,117,70)(22,118,71)(23,119,72)(24,120,73)(25,106,74)(26,107,75)(27,108,61)(28,109,62)(29,110,63)(30,111,64)(31,134,76)(32,135,77)(33,121,78)(34,122,79)(35,123,80)(36,124,81)(37,125,82)(38,126,83)(39,127,84)(40,128,85)(41,129,86)(42,130,87)(43,131,88)(44,132,89)(45,133,90)(136,235,190)(137,236,191)(138,237,192)(139,238,193)(140,239,194)(141,240,195)(142,226,181)(143,227,182)(144,228,183)(145,229,184)(146,230,185)(147,231,186)(148,232,187)(149,233,188)(150,234,189)(151,241,202)(152,242,203)(153,243,204)(154,244,205)(155,245,206)(156,246,207)(157,247,208)(158,248,209)(159,249,210)(160,250,196)(161,251,197)(162,252,198)(163,253,199)(164,254,200)(165,255,201)(166,269,213)(167,270,214)(168,256,215)(169,257,216)(170,258,217)(171,259,218)(172,260,219)(173,261,220)(174,262,221)(175,263,222)(176,264,223)(177,265,224)(178,266,225)(179,267,211)(180,268,212)(271,361,318)(272,362,319)(273,363,320)(274,364,321)(275,365,322)(276,366,323)(277,367,324)(278,368,325)(279,369,326)(280,370,327)(281,371,328)(282,372,329)(283,373,330)(284,374,316)(285,375,317)(286,376,342)(287,377,343)(288,378,344)(289,379,345)(290,380,331)(291,381,332)(292,382,333)(293,383,334)(294,384,335)(295,385,336)(296,386,337)(297,387,338)(298,388,339)(299,389,340)(300,390,341)(301,395,348)(302,396,349)(303,397,350)(304,398,351)(305,399,352)(306,400,353)(307,401,354)(308,402,355)(309,403,356)(310,404,357)(311,405,358)(312,391,359)(313,392,360)(314,393,346)(315,394,347), (1,42,23)(2,43,24)(3,44,25)(4,45,26)(5,31,27)(6,32,28)(7,33,29)(8,34,30)(9,35,16)(10,36,17)(11,37,18)(12,38,19)(13,39,20)(14,40,21)(15,41,22)(46,89,74)(47,90,75)(48,76,61)(49,77,62)(50,78,63)(51,79,64)(52,80,65)(53,81,66)(54,82,67)(55,83,68)(56,84,69)(57,85,70)(58,86,71)(59,87,72)(60,88,73)(91,134,108)(92,135,109)(93,121,110)(94,122,111)(95,123,112)(96,124,113)(97,125,114)(98,126,115)(99,127,116)(100,128,117)(101,129,118)(102,130,119)(103,131,120)(104,132,106)(105,133,107)(136,166,160)(137,167,161)(138,168,162)(139,169,163)(140,170,164)(141,171,165)(142,172,151)(143,173,152)(144,174,153)(145,175,154)(146,176,155)(147,177,156)(148,178,157)(149,179,158)(150,180,159)(181,219,202)(182,220,203)(183,221,204)(184,222,205)(185,223,206)(186,224,207)(187,225,208)(188,211,209)(189,212,210)(190,213,196)(191,214,197)(192,215,198)(193,216,199)(194,217,200)(195,218,201)(226,260,241)(227,261,242)(228,262,243)(229,263,244)(230,264,245)(231,265,246)(232,266,247)(233,267,248)(234,268,249)(235,269,250)(236,270,251)(237,256,252)(238,257,253)(239,258,254)(240,259,255)(271,303,288)(272,304,289)(273,305,290)(274,306,291)(275,307,292)(276,308,293)(277,309,294)(278,310,295)(279,311,296)(280,312,297)(281,313,298)(282,314,299)(283,315,300)(284,301,286)(285,302,287)(316,348,342)(317,349,343)(318,350,344)(319,351,345)(320,352,331)(321,353,332)(322,354,333)(323,355,334)(324,356,335)(325,357,336)(326,358,337)(327,359,338)(328,360,339)(329,346,340)(330,347,341)(361,397,378)(362,398,379)(363,399,380)(364,400,381)(365,401,382)(366,402,383)(367,403,384)(368,404,385)(369,405,386)(370,391,387)(371,392,388)(372,393,389)(373,394,390)(374,395,376)(375,396,377), (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) );
G=PermutationGroup([[(1,280,149),(2,281,150),(3,282,136),(4,283,137),(5,284,138),(6,285,139),(7,271,140),(8,272,141),(9,273,142),(10,274,143),(11,275,144),(12,276,145),(13,277,146),(14,278,147),(15,279,148),(16,290,151),(17,291,152),(18,292,153),(19,293,154),(20,294,155),(21,295,156),(22,296,157),(23,297,158),(24,298,159),(25,299,160),(26,300,161),(27,286,162),(28,287,163),(29,288,164),(30,289,165),(31,301,168),(32,302,169),(33,303,170),(34,304,171),(35,305,172),(36,306,173),(37,307,174),(38,308,175),(39,309,176),(40,310,177),(41,311,178),(42,312,179),(43,313,180),(44,314,166),(45,315,167),(46,329,190),(47,330,191),(48,316,192),(49,317,193),(50,318,194),(51,319,195),(52,320,181),(53,321,182),(54,322,183),(55,323,184),(56,324,185),(57,325,186),(58,326,187),(59,327,188),(60,328,189),(61,342,198),(62,343,199),(63,344,200),(64,345,201),(65,331,202),(66,332,203),(67,333,204),(68,334,205),(69,335,206),(70,336,207),(71,337,208),(72,338,209),(73,339,210),(74,340,196),(75,341,197),(76,348,215),(77,349,216),(78,350,217),(79,351,218),(80,352,219),(81,353,220),(82,354,221),(83,355,222),(84,356,223),(85,357,224),(86,358,225),(87,359,211),(88,360,212),(89,346,213),(90,347,214),(91,374,237),(92,375,238),(93,361,239),(94,362,240),(95,363,226),(96,364,227),(97,365,228),(98,366,229),(99,367,230),(100,368,231),(101,369,232),(102,370,233),(103,371,234),(104,372,235),(105,373,236),(106,389,250),(107,390,251),(108,376,252),(109,377,253),(110,378,254),(111,379,255),(112,380,241),(113,381,242),(114,382,243),(115,383,244),(116,384,245),(117,385,246),(118,386,247),(119,387,248),(120,388,249),(121,397,258),(122,398,259),(123,399,260),(124,400,261),(125,401,262),(126,402,263),(127,403,264),(128,404,265),(129,405,266),(130,391,267),(131,392,268),(132,393,269),(133,394,270),(134,395,256),(135,396,257)], [(1,102,59),(2,103,60),(3,104,46),(4,105,47),(5,91,48),(6,92,49),(7,93,50),(8,94,51),(9,95,52),(10,96,53),(11,97,54),(12,98,55),(13,99,56),(14,100,57),(15,101,58),(16,112,65),(17,113,66),(18,114,67),(19,115,68),(20,116,69),(21,117,70),(22,118,71),(23,119,72),(24,120,73),(25,106,74),(26,107,75),(27,108,61),(28,109,62),(29,110,63),(30,111,64),(31,134,76),(32,135,77),(33,121,78),(34,122,79),(35,123,80),(36,124,81),(37,125,82),(38,126,83),(39,127,84),(40,128,85),(41,129,86),(42,130,87),(43,131,88),(44,132,89),(45,133,90),(136,235,190),(137,236,191),(138,237,192),(139,238,193),(140,239,194),(141,240,195),(142,226,181),(143,227,182),(144,228,183),(145,229,184),(146,230,185),(147,231,186),(148,232,187),(149,233,188),(150,234,189),(151,241,202),(152,242,203),(153,243,204),(154,244,205),(155,245,206),(156,246,207),(157,247,208),(158,248,209),(159,249,210),(160,250,196),(161,251,197),(162,252,198),(163,253,199),(164,254,200),(165,255,201),(166,269,213),(167,270,214),(168,256,215),(169,257,216),(170,258,217),(171,259,218),(172,260,219),(173,261,220),(174,262,221),(175,263,222),(176,264,223),(177,265,224),(178,266,225),(179,267,211),(180,268,212),(271,361,318),(272,362,319),(273,363,320),(274,364,321),(275,365,322),(276,366,323),(277,367,324),(278,368,325),(279,369,326),(280,370,327),(281,371,328),(282,372,329),(283,373,330),(284,374,316),(285,375,317),(286,376,342),(287,377,343),(288,378,344),(289,379,345),(290,380,331),(291,381,332),(292,382,333),(293,383,334),(294,384,335),(295,385,336),(296,386,337),(297,387,338),(298,388,339),(299,389,340),(300,390,341),(301,395,348),(302,396,349),(303,397,350),(304,398,351),(305,399,352),(306,400,353),(307,401,354),(308,402,355),(309,403,356),(310,404,357),(311,405,358),(312,391,359),(313,392,360),(314,393,346),(315,394,347)], [(1,42,23),(2,43,24),(3,44,25),(4,45,26),(5,31,27),(6,32,28),(7,33,29),(8,34,30),(9,35,16),(10,36,17),(11,37,18),(12,38,19),(13,39,20),(14,40,21),(15,41,22),(46,89,74),(47,90,75),(48,76,61),(49,77,62),(50,78,63),(51,79,64),(52,80,65),(53,81,66),(54,82,67),(55,83,68),(56,84,69),(57,85,70),(58,86,71),(59,87,72),(60,88,73),(91,134,108),(92,135,109),(93,121,110),(94,122,111),(95,123,112),(96,124,113),(97,125,114),(98,126,115),(99,127,116),(100,128,117),(101,129,118),(102,130,119),(103,131,120),(104,132,106),(105,133,107),(136,166,160),(137,167,161),(138,168,162),(139,169,163),(140,170,164),(141,171,165),(142,172,151),(143,173,152),(144,174,153),(145,175,154),(146,176,155),(147,177,156),(148,178,157),(149,179,158),(150,180,159),(181,219,202),(182,220,203),(183,221,204),(184,222,205),(185,223,206),(186,224,207),(187,225,208),(188,211,209),(189,212,210),(190,213,196),(191,214,197),(192,215,198),(193,216,199),(194,217,200),(195,218,201),(226,260,241),(227,261,242),(228,262,243),(229,263,244),(230,264,245),(231,265,246),(232,266,247),(233,267,248),(234,268,249),(235,269,250),(236,270,251),(237,256,252),(238,257,253),(239,258,254),(240,259,255),(271,303,288),(272,304,289),(273,305,290),(274,306,291),(275,307,292),(276,308,293),(277,309,294),(278,310,295),(279,311,296),(280,312,297),(281,313,298),(282,314,299),(283,315,300),(284,301,286),(285,302,287),(316,348,342),(317,349,343),(318,350,344),(319,351,345),(320,352,331),(321,353,332),(322,354,333),(323,355,334),(324,356,335),(325,357,336),(326,358,337),(327,359,338),(328,360,339),(329,346,340),(330,347,341),(361,397,378),(362,398,379),(363,399,380),(364,400,381),(365,401,382),(366,402,383),(367,403,384),(368,404,385),(369,405,386),(370,391,387),(371,392,388),(372,393,389),(373,394,390),(374,395,376),(375,396,377)], [(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)]])
405 conjugacy classes
| class | 1 | 3A | ··· | 3CB | 5A | 5B | 5C | 5D | 15A | ··· | 15LH |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 |
405 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C5 | C15 |
| kernel | C33×C15 | C32×C15 | C34 | C33 |
| # reps | 1 | 80 | 4 | 320 |
Matrix representation of C33×C15 ►in GL4(𝔽31) generated by
| 5 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 5 |
| 5 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 5 |
| 25 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 18 |
G:=sub<GL(4,GF(31))| [5,0,0,0,0,25,0,0,0,0,1,0,0,0,0,1],[25,0,0,0,0,25,0,0,0,0,25,0,0,0,0,5],[5,0,0,0,0,1,0,0,0,0,1,0,0,0,0,5],[25,0,0,0,0,2,0,0,0,0,1,0,0,0,0,18] >;
C33×C15 in GAP, Magma, Sage, TeX
C_3^3\times C_{15} % in TeX
G:=Group("C3^3xC15"); // GroupNames label
G:=SmallGroup(405,16);
// by ID
G=gap.SmallGroup(405,16);
# by ID
G:=PCGroup([5,-3,-3,-3,-3,-5]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^15=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations