Copied to
clipboard

G = C33×C15order 405 = 34·5

Abelian group of type [3,3,3,15]

direct product, abelian, monomial, 3-elementary

Aliases: C33×C15, SmallGroup(405,16)

Series: Derived Chief Lower central Upper central

C1 — C33×C15
C1C5C15C3×C15C32×C15 — 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 [×40], C5, C32 [×130], C15 [×40], C33 [×40], C3×C15 [×130], C34, C32×C15 [×40], C33×C15
Quotients: C1, C3 [×40], C5, C32 [×130], C15 [×40], C33 [×40], C3×C15 [×130], C34, C32×C15 [×40], C33×C15

Smallest permutation representation of C33×C15
Regular action on 405 points
Generators in S405
(1 278 147)(2 279 148)(3 280 149)(4 281 150)(5 282 136)(6 283 137)(7 284 138)(8 285 139)(9 271 140)(10 272 141)(11 273 142)(12 274 143)(13 275 144)(14 276 145)(15 277 146)(16 294 155)(17 295 156)(18 296 157)(19 297 158)(20 298 159)(21 299 160)(22 300 161)(23 286 162)(24 287 163)(25 288 164)(26 289 165)(27 290 151)(28 291 152)(29 292 153)(30 293 154)(31 301 177)(32 302 178)(33 303 179)(34 304 180)(35 305 166)(36 306 167)(37 307 168)(38 308 169)(39 309 170)(40 310 171)(41 311 172)(42 312 173)(43 313 174)(44 314 175)(45 315 176)(46 329 192)(47 330 193)(48 316 194)(49 317 195)(50 318 181)(51 319 182)(52 320 183)(53 321 184)(54 322 185)(55 323 186)(56 324 187)(57 325 188)(58 326 189)(59 327 190)(60 328 191)(61 335 202)(62 336 203)(63 337 204)(64 338 205)(65 339 206)(66 340 207)(67 341 208)(68 342 209)(69 343 210)(70 344 196)(71 345 197)(72 331 198)(73 332 199)(74 333 200)(75 334 201)(76 355 211)(77 356 212)(78 357 213)(79 358 214)(80 359 215)(81 360 216)(82 346 217)(83 347 218)(84 348 219)(85 349 220)(86 350 221)(87 351 222)(88 352 223)(89 353 224)(90 354 225)(91 374 230)(92 375 231)(93 361 232)(94 362 233)(95 363 234)(96 364 235)(97 365 236)(98 366 237)(99 367 238)(100 368 239)(101 369 240)(102 370 226)(103 371 227)(104 372 228)(105 373 229)(106 376 245)(107 377 246)(108 378 247)(109 379 248)(110 380 249)(111 381 250)(112 382 251)(113 383 252)(114 384 253)(115 385 254)(116 386 255)(117 387 241)(118 388 242)(119 389 243)(120 390 244)(121 391 269)(122 392 270)(123 393 256)(124 394 257)(125 395 258)(126 396 259)(127 397 260)(128 398 261)(129 399 262)(130 400 263)(131 401 264)(132 402 265)(133 403 266)(134 404 267)(135 405 268)
(1 102 46)(2 103 47)(3 104 48)(4 105 49)(5 91 50)(6 92 51)(7 93 52)(8 94 53)(9 95 54)(10 96 55)(11 97 56)(12 98 57)(13 99 58)(14 100 59)(15 101 60)(16 110 63)(17 111 64)(18 112 65)(19 113 66)(20 114 67)(21 115 68)(22 116 69)(23 117 70)(24 118 71)(25 119 72)(26 120 73)(27 106 74)(28 107 75)(29 108 61)(30 109 62)(31 121 76)(32 122 77)(33 123 78)(34 124 79)(35 125 80)(36 126 81)(37 127 82)(38 128 83)(39 129 84)(40 130 85)(41 131 86)(42 132 87)(43 133 88)(44 134 89)(45 135 90)(136 230 181)(137 231 182)(138 232 183)(139 233 184)(140 234 185)(141 235 186)(142 236 187)(143 237 188)(144 238 189)(145 239 190)(146 240 191)(147 226 192)(148 227 193)(149 228 194)(150 229 195)(151 245 200)(152 246 201)(153 247 202)(154 248 203)(155 249 204)(156 250 205)(157 251 206)(158 252 207)(159 253 208)(160 254 209)(161 255 210)(162 241 196)(163 242 197)(164 243 198)(165 244 199)(166 258 215)(167 259 216)(168 260 217)(169 261 218)(170 262 219)(171 263 220)(172 264 221)(173 265 222)(174 266 223)(175 267 224)(176 268 225)(177 269 211)(178 270 212)(179 256 213)(180 257 214)(271 363 322)(272 364 323)(273 365 324)(274 366 325)(275 367 326)(276 368 327)(277 369 328)(278 370 329)(279 371 330)(280 372 316)(281 373 317)(282 374 318)(283 375 319)(284 361 320)(285 362 321)(286 387 344)(287 388 345)(288 389 331)(289 390 332)(290 376 333)(291 377 334)(292 378 335)(293 379 336)(294 380 337)(295 381 338)(296 382 339)(297 383 340)(298 384 341)(299 385 342)(300 386 343)(301 391 355)(302 392 356)(303 393 357)(304 394 358)(305 395 359)(306 396 360)(307 397 346)(308 398 347)(309 399 348)(310 400 349)(311 401 350)(312 402 351)(313 403 352)(314 404 353)(315 405 354)
(1 44 23)(2 45 24)(3 31 25)(4 32 26)(5 33 27)(6 34 28)(7 35 29)(8 36 30)(9 37 16)(10 38 17)(11 39 18)(12 40 19)(13 41 20)(14 42 21)(15 43 22)(46 89 70)(47 90 71)(48 76 72)(49 77 73)(50 78 74)(51 79 75)(52 80 61)(53 81 62)(54 82 63)(55 83 64)(56 84 65)(57 85 66)(58 86 67)(59 87 68)(60 88 69)(91 123 106)(92 124 107)(93 125 108)(94 126 109)(95 127 110)(96 128 111)(97 129 112)(98 130 113)(99 131 114)(100 132 115)(101 133 116)(102 134 117)(103 135 118)(104 121 119)(105 122 120)(136 179 151)(137 180 152)(138 166 153)(139 167 154)(140 168 155)(141 169 156)(142 170 157)(143 171 158)(144 172 159)(145 173 160)(146 174 161)(147 175 162)(148 176 163)(149 177 164)(150 178 165)(181 213 200)(182 214 201)(183 215 202)(184 216 203)(185 217 204)(186 218 205)(187 219 206)(188 220 207)(189 221 208)(190 222 209)(191 223 210)(192 224 196)(193 225 197)(194 211 198)(195 212 199)(226 267 241)(227 268 242)(228 269 243)(229 270 244)(230 256 245)(231 257 246)(232 258 247)(233 259 248)(234 260 249)(235 261 250)(236 262 251)(237 263 252)(238 264 253)(239 265 254)(240 266 255)(271 307 294)(272 308 295)(273 309 296)(274 310 297)(275 311 298)(276 312 299)(277 313 300)(278 314 286)(279 315 287)(280 301 288)(281 302 289)(282 303 290)(283 304 291)(284 305 292)(285 306 293)(316 355 331)(317 356 332)(318 357 333)(319 358 334)(320 359 335)(321 360 336)(322 346 337)(323 347 338)(324 348 339)(325 349 340)(326 350 341)(327 351 342)(328 352 343)(329 353 344)(330 354 345)(361 395 378)(362 396 379)(363 397 380)(364 398 381)(365 399 382)(366 400 383)(367 401 384)(368 402 385)(369 403 386)(370 404 387)(371 405 388)(372 391 389)(373 392 390)(374 393 376)(375 394 377)
(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)| (1,278,147)(2,279,148)(3,280,149)(4,281,150)(5,282,136)(6,283,137)(7,284,138)(8,285,139)(9,271,140)(10,272,141)(11,273,142)(12,274,143)(13,275,144)(14,276,145)(15,277,146)(16,294,155)(17,295,156)(18,296,157)(19,297,158)(20,298,159)(21,299,160)(22,300,161)(23,286,162)(24,287,163)(25,288,164)(26,289,165)(27,290,151)(28,291,152)(29,292,153)(30,293,154)(31,301,177)(32,302,178)(33,303,179)(34,304,180)(35,305,166)(36,306,167)(37,307,168)(38,308,169)(39,309,170)(40,310,171)(41,311,172)(42,312,173)(43,313,174)(44,314,175)(45,315,176)(46,329,192)(47,330,193)(48,316,194)(49,317,195)(50,318,181)(51,319,182)(52,320,183)(53,321,184)(54,322,185)(55,323,186)(56,324,187)(57,325,188)(58,326,189)(59,327,190)(60,328,191)(61,335,202)(62,336,203)(63,337,204)(64,338,205)(65,339,206)(66,340,207)(67,341,208)(68,342,209)(69,343,210)(70,344,196)(71,345,197)(72,331,198)(73,332,199)(74,333,200)(75,334,201)(76,355,211)(77,356,212)(78,357,213)(79,358,214)(80,359,215)(81,360,216)(82,346,217)(83,347,218)(84,348,219)(85,349,220)(86,350,221)(87,351,222)(88,352,223)(89,353,224)(90,354,225)(91,374,230)(92,375,231)(93,361,232)(94,362,233)(95,363,234)(96,364,235)(97,365,236)(98,366,237)(99,367,238)(100,368,239)(101,369,240)(102,370,226)(103,371,227)(104,372,228)(105,373,229)(106,376,245)(107,377,246)(108,378,247)(109,379,248)(110,380,249)(111,381,250)(112,382,251)(113,383,252)(114,384,253)(115,385,254)(116,386,255)(117,387,241)(118,388,242)(119,389,243)(120,390,244)(121,391,269)(122,392,270)(123,393,256)(124,394,257)(125,395,258)(126,396,259)(127,397,260)(128,398,261)(129,399,262)(130,400,263)(131,401,264)(132,402,265)(133,403,266)(134,404,267)(135,405,268), (1,102,46)(2,103,47)(3,104,48)(4,105,49)(5,91,50)(6,92,51)(7,93,52)(8,94,53)(9,95,54)(10,96,55)(11,97,56)(12,98,57)(13,99,58)(14,100,59)(15,101,60)(16,110,63)(17,111,64)(18,112,65)(19,113,66)(20,114,67)(21,115,68)(22,116,69)(23,117,70)(24,118,71)(25,119,72)(26,120,73)(27,106,74)(28,107,75)(29,108,61)(30,109,62)(31,121,76)(32,122,77)(33,123,78)(34,124,79)(35,125,80)(36,126,81)(37,127,82)(38,128,83)(39,129,84)(40,130,85)(41,131,86)(42,132,87)(43,133,88)(44,134,89)(45,135,90)(136,230,181)(137,231,182)(138,232,183)(139,233,184)(140,234,185)(141,235,186)(142,236,187)(143,237,188)(144,238,189)(145,239,190)(146,240,191)(147,226,192)(148,227,193)(149,228,194)(150,229,195)(151,245,200)(152,246,201)(153,247,202)(154,248,203)(155,249,204)(156,250,205)(157,251,206)(158,252,207)(159,253,208)(160,254,209)(161,255,210)(162,241,196)(163,242,197)(164,243,198)(165,244,199)(166,258,215)(167,259,216)(168,260,217)(169,261,218)(170,262,219)(171,263,220)(172,264,221)(173,265,222)(174,266,223)(175,267,224)(176,268,225)(177,269,211)(178,270,212)(179,256,213)(180,257,214)(271,363,322)(272,364,323)(273,365,324)(274,366,325)(275,367,326)(276,368,327)(277,369,328)(278,370,329)(279,371,330)(280,372,316)(281,373,317)(282,374,318)(283,375,319)(284,361,320)(285,362,321)(286,387,344)(287,388,345)(288,389,331)(289,390,332)(290,376,333)(291,377,334)(292,378,335)(293,379,336)(294,380,337)(295,381,338)(296,382,339)(297,383,340)(298,384,341)(299,385,342)(300,386,343)(301,391,355)(302,392,356)(303,393,357)(304,394,358)(305,395,359)(306,396,360)(307,397,346)(308,398,347)(309,399,348)(310,400,349)(311,401,350)(312,402,351)(313,403,352)(314,404,353)(315,405,354), (1,44,23)(2,45,24)(3,31,25)(4,32,26)(5,33,27)(6,34,28)(7,35,29)(8,36,30)(9,37,16)(10,38,17)(11,39,18)(12,40,19)(13,41,20)(14,42,21)(15,43,22)(46,89,70)(47,90,71)(48,76,72)(49,77,73)(50,78,74)(51,79,75)(52,80,61)(53,81,62)(54,82,63)(55,83,64)(56,84,65)(57,85,66)(58,86,67)(59,87,68)(60,88,69)(91,123,106)(92,124,107)(93,125,108)(94,126,109)(95,127,110)(96,128,111)(97,129,112)(98,130,113)(99,131,114)(100,132,115)(101,133,116)(102,134,117)(103,135,118)(104,121,119)(105,122,120)(136,179,151)(137,180,152)(138,166,153)(139,167,154)(140,168,155)(141,169,156)(142,170,157)(143,171,158)(144,172,159)(145,173,160)(146,174,161)(147,175,162)(148,176,163)(149,177,164)(150,178,165)(181,213,200)(182,214,201)(183,215,202)(184,216,203)(185,217,204)(186,218,205)(187,219,206)(188,220,207)(189,221,208)(190,222,209)(191,223,210)(192,224,196)(193,225,197)(194,211,198)(195,212,199)(226,267,241)(227,268,242)(228,269,243)(229,270,244)(230,256,245)(231,257,246)(232,258,247)(233,259,248)(234,260,249)(235,261,250)(236,262,251)(237,263,252)(238,264,253)(239,265,254)(240,266,255)(271,307,294)(272,308,295)(273,309,296)(274,310,297)(275,311,298)(276,312,299)(277,313,300)(278,314,286)(279,315,287)(280,301,288)(281,302,289)(282,303,290)(283,304,291)(284,305,292)(285,306,293)(316,355,331)(317,356,332)(318,357,333)(319,358,334)(320,359,335)(321,360,336)(322,346,337)(323,347,338)(324,348,339)(325,349,340)(326,350,341)(327,351,342)(328,352,343)(329,353,344)(330,354,345)(361,395,378)(362,396,379)(363,397,380)(364,398,381)(365,399,382)(366,400,383)(367,401,384)(368,402,385)(369,403,386)(370,404,387)(371,405,388)(372,391,389)(373,392,390)(374,393,376)(375,394,377), (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( (1,278,147)(2,279,148)(3,280,149)(4,281,150)(5,282,136)(6,283,137)(7,284,138)(8,285,139)(9,271,140)(10,272,141)(11,273,142)(12,274,143)(13,275,144)(14,276,145)(15,277,146)(16,294,155)(17,295,156)(18,296,157)(19,297,158)(20,298,159)(21,299,160)(22,300,161)(23,286,162)(24,287,163)(25,288,164)(26,289,165)(27,290,151)(28,291,152)(29,292,153)(30,293,154)(31,301,177)(32,302,178)(33,303,179)(34,304,180)(35,305,166)(36,306,167)(37,307,168)(38,308,169)(39,309,170)(40,310,171)(41,311,172)(42,312,173)(43,313,174)(44,314,175)(45,315,176)(46,329,192)(47,330,193)(48,316,194)(49,317,195)(50,318,181)(51,319,182)(52,320,183)(53,321,184)(54,322,185)(55,323,186)(56,324,187)(57,325,188)(58,326,189)(59,327,190)(60,328,191)(61,335,202)(62,336,203)(63,337,204)(64,338,205)(65,339,206)(66,340,207)(67,341,208)(68,342,209)(69,343,210)(70,344,196)(71,345,197)(72,331,198)(73,332,199)(74,333,200)(75,334,201)(76,355,211)(77,356,212)(78,357,213)(79,358,214)(80,359,215)(81,360,216)(82,346,217)(83,347,218)(84,348,219)(85,349,220)(86,350,221)(87,351,222)(88,352,223)(89,353,224)(90,354,225)(91,374,230)(92,375,231)(93,361,232)(94,362,233)(95,363,234)(96,364,235)(97,365,236)(98,366,237)(99,367,238)(100,368,239)(101,369,240)(102,370,226)(103,371,227)(104,372,228)(105,373,229)(106,376,245)(107,377,246)(108,378,247)(109,379,248)(110,380,249)(111,381,250)(112,382,251)(113,383,252)(114,384,253)(115,385,254)(116,386,255)(117,387,241)(118,388,242)(119,389,243)(120,390,244)(121,391,269)(122,392,270)(123,393,256)(124,394,257)(125,395,258)(126,396,259)(127,397,260)(128,398,261)(129,399,262)(130,400,263)(131,401,264)(132,402,265)(133,403,266)(134,404,267)(135,405,268), (1,102,46)(2,103,47)(3,104,48)(4,105,49)(5,91,50)(6,92,51)(7,93,52)(8,94,53)(9,95,54)(10,96,55)(11,97,56)(12,98,57)(13,99,58)(14,100,59)(15,101,60)(16,110,63)(17,111,64)(18,112,65)(19,113,66)(20,114,67)(21,115,68)(22,116,69)(23,117,70)(24,118,71)(25,119,72)(26,120,73)(27,106,74)(28,107,75)(29,108,61)(30,109,62)(31,121,76)(32,122,77)(33,123,78)(34,124,79)(35,125,80)(36,126,81)(37,127,82)(38,128,83)(39,129,84)(40,130,85)(41,131,86)(42,132,87)(43,133,88)(44,134,89)(45,135,90)(136,230,181)(137,231,182)(138,232,183)(139,233,184)(140,234,185)(141,235,186)(142,236,187)(143,237,188)(144,238,189)(145,239,190)(146,240,191)(147,226,192)(148,227,193)(149,228,194)(150,229,195)(151,245,200)(152,246,201)(153,247,202)(154,248,203)(155,249,204)(156,250,205)(157,251,206)(158,252,207)(159,253,208)(160,254,209)(161,255,210)(162,241,196)(163,242,197)(164,243,198)(165,244,199)(166,258,215)(167,259,216)(168,260,217)(169,261,218)(170,262,219)(171,263,220)(172,264,221)(173,265,222)(174,266,223)(175,267,224)(176,268,225)(177,269,211)(178,270,212)(179,256,213)(180,257,214)(271,363,322)(272,364,323)(273,365,324)(274,366,325)(275,367,326)(276,368,327)(277,369,328)(278,370,329)(279,371,330)(280,372,316)(281,373,317)(282,374,318)(283,375,319)(284,361,320)(285,362,321)(286,387,344)(287,388,345)(288,389,331)(289,390,332)(290,376,333)(291,377,334)(292,378,335)(293,379,336)(294,380,337)(295,381,338)(296,382,339)(297,383,340)(298,384,341)(299,385,342)(300,386,343)(301,391,355)(302,392,356)(303,393,357)(304,394,358)(305,395,359)(306,396,360)(307,397,346)(308,398,347)(309,399,348)(310,400,349)(311,401,350)(312,402,351)(313,403,352)(314,404,353)(315,405,354), (1,44,23)(2,45,24)(3,31,25)(4,32,26)(5,33,27)(6,34,28)(7,35,29)(8,36,30)(9,37,16)(10,38,17)(11,39,18)(12,40,19)(13,41,20)(14,42,21)(15,43,22)(46,89,70)(47,90,71)(48,76,72)(49,77,73)(50,78,74)(51,79,75)(52,80,61)(53,81,62)(54,82,63)(55,83,64)(56,84,65)(57,85,66)(58,86,67)(59,87,68)(60,88,69)(91,123,106)(92,124,107)(93,125,108)(94,126,109)(95,127,110)(96,128,111)(97,129,112)(98,130,113)(99,131,114)(100,132,115)(101,133,116)(102,134,117)(103,135,118)(104,121,119)(105,122,120)(136,179,151)(137,180,152)(138,166,153)(139,167,154)(140,168,155)(141,169,156)(142,170,157)(143,171,158)(144,172,159)(145,173,160)(146,174,161)(147,175,162)(148,176,163)(149,177,164)(150,178,165)(181,213,200)(182,214,201)(183,215,202)(184,216,203)(185,217,204)(186,218,205)(187,219,206)(188,220,207)(189,221,208)(190,222,209)(191,223,210)(192,224,196)(193,225,197)(194,211,198)(195,212,199)(226,267,241)(227,268,242)(228,269,243)(229,270,244)(230,256,245)(231,257,246)(232,258,247)(233,259,248)(234,260,249)(235,261,250)(236,262,251)(237,263,252)(238,264,253)(239,265,254)(240,266,255)(271,307,294)(272,308,295)(273,309,296)(274,310,297)(275,311,298)(276,312,299)(277,313,300)(278,314,286)(279,315,287)(280,301,288)(281,302,289)(282,303,290)(283,304,291)(284,305,292)(285,306,293)(316,355,331)(317,356,332)(318,357,333)(319,358,334)(320,359,335)(321,360,336)(322,346,337)(323,347,338)(324,348,339)(325,349,340)(326,350,341)(327,351,342)(328,352,343)(329,353,344)(330,354,345)(361,395,378)(362,396,379)(363,397,380)(364,398,381)(365,399,382)(366,400,383)(367,401,384)(368,402,385)(369,403,386)(370,404,387)(371,405,388)(372,391,389)(373,392,390)(374,393,376)(375,394,377), (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([(1,278,147),(2,279,148),(3,280,149),(4,281,150),(5,282,136),(6,283,137),(7,284,138),(8,285,139),(9,271,140),(10,272,141),(11,273,142),(12,274,143),(13,275,144),(14,276,145),(15,277,146),(16,294,155),(17,295,156),(18,296,157),(19,297,158),(20,298,159),(21,299,160),(22,300,161),(23,286,162),(24,287,163),(25,288,164),(26,289,165),(27,290,151),(28,291,152),(29,292,153),(30,293,154),(31,301,177),(32,302,178),(33,303,179),(34,304,180),(35,305,166),(36,306,167),(37,307,168),(38,308,169),(39,309,170),(40,310,171),(41,311,172),(42,312,173),(43,313,174),(44,314,175),(45,315,176),(46,329,192),(47,330,193),(48,316,194),(49,317,195),(50,318,181),(51,319,182),(52,320,183),(53,321,184),(54,322,185),(55,323,186),(56,324,187),(57,325,188),(58,326,189),(59,327,190),(60,328,191),(61,335,202),(62,336,203),(63,337,204),(64,338,205),(65,339,206),(66,340,207),(67,341,208),(68,342,209),(69,343,210),(70,344,196),(71,345,197),(72,331,198),(73,332,199),(74,333,200),(75,334,201),(76,355,211),(77,356,212),(78,357,213),(79,358,214),(80,359,215),(81,360,216),(82,346,217),(83,347,218),(84,348,219),(85,349,220),(86,350,221),(87,351,222),(88,352,223),(89,353,224),(90,354,225),(91,374,230),(92,375,231),(93,361,232),(94,362,233),(95,363,234),(96,364,235),(97,365,236),(98,366,237),(99,367,238),(100,368,239),(101,369,240),(102,370,226),(103,371,227),(104,372,228),(105,373,229),(106,376,245),(107,377,246),(108,378,247),(109,379,248),(110,380,249),(111,381,250),(112,382,251),(113,383,252),(114,384,253),(115,385,254),(116,386,255),(117,387,241),(118,388,242),(119,389,243),(120,390,244),(121,391,269),(122,392,270),(123,393,256),(124,394,257),(125,395,258),(126,396,259),(127,397,260),(128,398,261),(129,399,262),(130,400,263),(131,401,264),(132,402,265),(133,403,266),(134,404,267),(135,405,268)], [(1,102,46),(2,103,47),(3,104,48),(4,105,49),(5,91,50),(6,92,51),(7,93,52),(8,94,53),(9,95,54),(10,96,55),(11,97,56),(12,98,57),(13,99,58),(14,100,59),(15,101,60),(16,110,63),(17,111,64),(18,112,65),(19,113,66),(20,114,67),(21,115,68),(22,116,69),(23,117,70),(24,118,71),(25,119,72),(26,120,73),(27,106,74),(28,107,75),(29,108,61),(30,109,62),(31,121,76),(32,122,77),(33,123,78),(34,124,79),(35,125,80),(36,126,81),(37,127,82),(38,128,83),(39,129,84),(40,130,85),(41,131,86),(42,132,87),(43,133,88),(44,134,89),(45,135,90),(136,230,181),(137,231,182),(138,232,183),(139,233,184),(140,234,185),(141,235,186),(142,236,187),(143,237,188),(144,238,189),(145,239,190),(146,240,191),(147,226,192),(148,227,193),(149,228,194),(150,229,195),(151,245,200),(152,246,201),(153,247,202),(154,248,203),(155,249,204),(156,250,205),(157,251,206),(158,252,207),(159,253,208),(160,254,209),(161,255,210),(162,241,196),(163,242,197),(164,243,198),(165,244,199),(166,258,215),(167,259,216),(168,260,217),(169,261,218),(170,262,219),(171,263,220),(172,264,221),(173,265,222),(174,266,223),(175,267,224),(176,268,225),(177,269,211),(178,270,212),(179,256,213),(180,257,214),(271,363,322),(272,364,323),(273,365,324),(274,366,325),(275,367,326),(276,368,327),(277,369,328),(278,370,329),(279,371,330),(280,372,316),(281,373,317),(282,374,318),(283,375,319),(284,361,320),(285,362,321),(286,387,344),(287,388,345),(288,389,331),(289,390,332),(290,376,333),(291,377,334),(292,378,335),(293,379,336),(294,380,337),(295,381,338),(296,382,339),(297,383,340),(298,384,341),(299,385,342),(300,386,343),(301,391,355),(302,392,356),(303,393,357),(304,394,358),(305,395,359),(306,396,360),(307,397,346),(308,398,347),(309,399,348),(310,400,349),(311,401,350),(312,402,351),(313,403,352),(314,404,353),(315,405,354)], [(1,44,23),(2,45,24),(3,31,25),(4,32,26),(5,33,27),(6,34,28),(7,35,29),(8,36,30),(9,37,16),(10,38,17),(11,39,18),(12,40,19),(13,41,20),(14,42,21),(15,43,22),(46,89,70),(47,90,71),(48,76,72),(49,77,73),(50,78,74),(51,79,75),(52,80,61),(53,81,62),(54,82,63),(55,83,64),(56,84,65),(57,85,66),(58,86,67),(59,87,68),(60,88,69),(91,123,106),(92,124,107),(93,125,108),(94,126,109),(95,127,110),(96,128,111),(97,129,112),(98,130,113),(99,131,114),(100,132,115),(101,133,116),(102,134,117),(103,135,118),(104,121,119),(105,122,120),(136,179,151),(137,180,152),(138,166,153),(139,167,154),(140,168,155),(141,169,156),(142,170,157),(143,171,158),(144,172,159),(145,173,160),(146,174,161),(147,175,162),(148,176,163),(149,177,164),(150,178,165),(181,213,200),(182,214,201),(183,215,202),(184,216,203),(185,217,204),(186,218,205),(187,219,206),(188,220,207),(189,221,208),(190,222,209),(191,223,210),(192,224,196),(193,225,197),(194,211,198),(195,212,199),(226,267,241),(227,268,242),(228,269,243),(229,270,244),(230,256,245),(231,257,246),(232,258,247),(233,259,248),(234,260,249),(235,261,250),(236,262,251),(237,263,252),(238,264,253),(239,265,254),(240,266,255),(271,307,294),(272,308,295),(273,309,296),(274,310,297),(275,311,298),(276,312,299),(277,313,300),(278,314,286),(279,315,287),(280,301,288),(281,302,289),(282,303,290),(283,304,291),(284,305,292),(285,306,293),(316,355,331),(317,356,332),(318,357,333),(319,358,334),(320,359,335),(321,360,336),(322,346,337),(323,347,338),(324,348,339),(325,349,340),(326,350,341),(327,351,342),(328,352,343),(329,353,344),(330,354,345),(361,395,378),(362,396,379),(363,397,380),(364,398,381),(365,399,382),(366,400,383),(367,401,384),(368,402,385),(369,403,386),(370,404,387),(371,405,388),(372,391,389),(373,392,390),(374,393,376),(375,394,377)], [(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···3CB5A5B5C5D15A···15LH
order13···3555515···15
size11···111111···1

405 irreducible representations

dim1111
type+
imageC1C3C5C15
kernelC33×C15C32×C15C34C33
# reps1804320

Matrix representation of C33×C15 in GL4(𝔽31) generated by

5000
02500
0010
0001
,
25000
02500
00250
0005
,
5000
0100
0010
0005
,
25000
0200
0010
00018
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

׿
×
𝔽