direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C5⋊2C32, C5⋊2C96, C15⋊4C32, C80.3C6, C48.4D5, C60.12C8, C30.4C16, C20.5C24, C10.2C48, C40.7C12, C240.6C2, C120.17C4, C24.7Dic5, C16.2(C3×D5), C8.3(C3×Dic5), C12.5(C5⋊2C8), C6.2(C5⋊2C16), C2.(C3×C5⋊2C16), C4.2(C3×C5⋊2C8), SmallGroup(480,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×C5⋊2C32 |
Generators and relations for C3×C5⋊2C32
G = < a,b,c | a3=b5=c32=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C5⋊2C32
►Regular action on 480 points
Generators in S480
(1 118 216)(2 119 217)(3 120 218)(4 121 219)(5 122 220)(6 123 221)(7 124 222)(8 125 223)(9 126 224)(10 127 193)(11 128 194)(12 97 195)(13 98 196)(14 99 197)(15 100 198)(16 101 199)(17 102 200)(18 103 201)(19 104 202)(20 105 203)(21 106 204)(22 107 205)(23 108 206)(24 109 207)(25 110 208)(26 111 209)(27 112 210)(28 113 211)(29 114 212)(30 115 213)(31 116 214)(32 117 215)(33 363 94)(34 364 95)(35 365 96)(36 366 65)(37 367 66)(38 368 67)(39 369 68)(40 370 69)(41 371 70)(42 372 71)(43 373 72)(44 374 73)(45 375 74)(46 376 75)(47 377 76)(48 378 77)(49 379 78)(50 380 79)(51 381 80)(52 382 81)(53 383 82)(54 384 83)(55 353 84)(56 354 85)(57 355 86)(58 356 87)(59 357 88)(60 358 89)(61 359 90)(62 360 91)(63 361 92)(64 362 93)(129 179 248)(130 180 249)(131 181 250)(132 182 251)(133 183 252)(134 184 253)(135 185 254)(136 186 255)(137 187 256)(138 188 225)(139 189 226)(140 190 227)(141 191 228)(142 192 229)(143 161 230)(144 162 231)(145 163 232)(146 164 233)(147 165 234)(148 166 235)(149 167 236)(150 168 237)(151 169 238)(152 170 239)(153 171 240)(154 172 241)(155 173 242)(156 174 243)(157 175 244)(158 176 245)(159 177 246)(160 178 247)(257 308 332)(258 309 333)(259 310 334)(260 311 335)(261 312 336)(262 313 337)(263 314 338)(264 315 339)(265 316 340)(266 317 341)(267 318 342)(268 319 343)(269 320 344)(270 289 345)(271 290 346)(272 291 347)(273 292 348)(274 293 349)(275 294 350)(276 295 351)(277 296 352)(278 297 321)(279 298 322)(280 299 323)(281 300 324)(282 301 325)(283 302 326)(284 303 327)(285 304 328)(286 305 329)(287 306 330)(288 307 331)(385 447 463)(386 448 464)(387 417 465)(388 418 466)(389 419 467)(390 420 468)(391 421 469)(392 422 470)(393 423 471)(394 424 472)(395 425 473)(396 426 474)(397 427 475)(398 428 476)(399 429 477)(400 430 478)(401 431 479)(402 432 480)(403 433 449)(404 434 450)(405 435 451)(406 436 452)(407 437 453)(408 438 454)(409 439 455)(410 440 456)(411 441 457)(412 442 458)(413 443 459)(414 444 460)(415 445 461)(416 446 462)
(1 79 298 140 411)(2 412 141 299 80)(3 81 300 142 413)(4 414 143 301 82)(5 83 302 144 415)(6 416 145 303 84)(7 85 304 146 385)(8 386 147 305 86)(9 87 306 148 387)(10 388 149 307 88)(11 89 308 150 389)(12 390 151 309 90)(13 91 310 152 391)(14 392 153 311 92)(15 93 312 154 393)(16 394 155 313 94)(17 95 314 156 395)(18 396 157 315 96)(19 65 316 158 397)(20 398 159 317 66)(21 67 318 160 399)(22 400 129 319 68)(23 69 320 130 401)(24 402 131 289 70)(25 71 290 132 403)(26 404 133 291 72)(27 73 292 134 405)(28 406 135 293 74)(29 75 294 136 407)(30 408 137 295 76)(31 77 296 138 409)(32 410 139 297 78)(33 101 424 173 337)(34 338 174 425 102)(35 103 426 175 339)(36 340 176 427 104)(37 105 428 177 341)(38 342 178 429 106)(39 107 430 179 343)(40 344 180 431 108)(41 109 432 181 345)(42 346 182 433 110)(43 111 434 183 347)(44 348 184 435 112)(45 113 436 185 349)(46 350 186 437 114)(47 115 438 187 351)(48 352 188 439 116)(49 117 440 189 321)(50 322 190 441 118)(51 119 442 191 323)(52 324 192 443 120)(53 121 444 161 325)(54 326 162 445 122)(55 123 446 163 327)(56 328 164 447 124)(57 125 448 165 329)(58 330 166 417 126)(59 127 418 167 331)(60 332 168 419 128)(61 97 420 169 333)(62 334 170 421 98)(63 99 422 171 335)(64 336 172 423 100)(193 466 236 288 357)(194 358 257 237 467)(195 468 238 258 359)(196 360 259 239 469)(197 470 240 260 361)(198 362 261 241 471)(199 472 242 262 363)(200 364 263 243 473)(201 474 244 264 365)(202 366 265 245 475)(203 476 246 266 367)(204 368 267 247 477)(205 478 248 268 369)(206 370 269 249 479)(207 480 250 270 371)(208 372 271 251 449)(209 450 252 272 373)(210 374 273 253 451)(211 452 254 274 375)(212 376 275 255 453)(213 454 256 276 377)(214 378 277 225 455)(215 456 226 278 379)(216 380 279 227 457)(217 458 228 280 381)(218 382 281 229 459)(219 460 230 282 383)(220 384 283 231 461)(221 462 232 284 353)(222 354 285 233 463)(223 464 234 286 355)(224 356 287 235 465)
(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)(417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)(449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)
G:=sub<Sym(480)| (1,118,216)(2,119,217)(3,120,218)(4,121,219)(5,122,220)(6,123,221)(7,124,222)(8,125,223)(9,126,224)(10,127,193)(11,128,194)(12,97,195)(13,98,196)(14,99,197)(15,100,198)(16,101,199)(17,102,200)(18,103,201)(19,104,202)(20,105,203)(21,106,204)(22,107,205)(23,108,206)(24,109,207)(25,110,208)(26,111,209)(27,112,210)(28,113,211)(29,114,212)(30,115,213)(31,116,214)(32,117,215)(33,363,94)(34,364,95)(35,365,96)(36,366,65)(37,367,66)(38,368,67)(39,369,68)(40,370,69)(41,371,70)(42,372,71)(43,373,72)(44,374,73)(45,375,74)(46,376,75)(47,377,76)(48,378,77)(49,379,78)(50,380,79)(51,381,80)(52,382,81)(53,383,82)(54,384,83)(55,353,84)(56,354,85)(57,355,86)(58,356,87)(59,357,88)(60,358,89)(61,359,90)(62,360,91)(63,361,92)(64,362,93)(129,179,248)(130,180,249)(131,181,250)(132,182,251)(133,183,252)(134,184,253)(135,185,254)(136,186,255)(137,187,256)(138,188,225)(139,189,226)(140,190,227)(141,191,228)(142,192,229)(143,161,230)(144,162,231)(145,163,232)(146,164,233)(147,165,234)(148,166,235)(149,167,236)(150,168,237)(151,169,238)(152,170,239)(153,171,240)(154,172,241)(155,173,242)(156,174,243)(157,175,244)(158,176,245)(159,177,246)(160,178,247)(257,308,332)(258,309,333)(259,310,334)(260,311,335)(261,312,336)(262,313,337)(263,314,338)(264,315,339)(265,316,340)(266,317,341)(267,318,342)(268,319,343)(269,320,344)(270,289,345)(271,290,346)(272,291,347)(273,292,348)(274,293,349)(275,294,350)(276,295,351)(277,296,352)(278,297,321)(279,298,322)(280,299,323)(281,300,324)(282,301,325)(283,302,326)(284,303,327)(285,304,328)(286,305,329)(287,306,330)(288,307,331)(385,447,463)(386,448,464)(387,417,465)(388,418,466)(389,419,467)(390,420,468)(391,421,469)(392,422,470)(393,423,471)(394,424,472)(395,425,473)(396,426,474)(397,427,475)(398,428,476)(399,429,477)(400,430,478)(401,431,479)(402,432,480)(403,433,449)(404,434,450)(405,435,451)(406,436,452)(407,437,453)(408,438,454)(409,439,455)(410,440,456)(411,441,457)(412,442,458)(413,443,459)(414,444,460)(415,445,461)(416,446,462), (1,79,298,140,411)(2,412,141,299,80)(3,81,300,142,413)(4,414,143,301,82)(5,83,302,144,415)(6,416,145,303,84)(7,85,304,146,385)(8,386,147,305,86)(9,87,306,148,387)(10,388,149,307,88)(11,89,308,150,389)(12,390,151,309,90)(13,91,310,152,391)(14,392,153,311,92)(15,93,312,154,393)(16,394,155,313,94)(17,95,314,156,395)(18,396,157,315,96)(19,65,316,158,397)(20,398,159,317,66)(21,67,318,160,399)(22,400,129,319,68)(23,69,320,130,401)(24,402,131,289,70)(25,71,290,132,403)(26,404,133,291,72)(27,73,292,134,405)(28,406,135,293,74)(29,75,294,136,407)(30,408,137,295,76)(31,77,296,138,409)(32,410,139,297,78)(33,101,424,173,337)(34,338,174,425,102)(35,103,426,175,339)(36,340,176,427,104)(37,105,428,177,341)(38,342,178,429,106)(39,107,430,179,343)(40,344,180,431,108)(41,109,432,181,345)(42,346,182,433,110)(43,111,434,183,347)(44,348,184,435,112)(45,113,436,185,349)(46,350,186,437,114)(47,115,438,187,351)(48,352,188,439,116)(49,117,440,189,321)(50,322,190,441,118)(51,119,442,191,323)(52,324,192,443,120)(53,121,444,161,325)(54,326,162,445,122)(55,123,446,163,327)(56,328,164,447,124)(57,125,448,165,329)(58,330,166,417,126)(59,127,418,167,331)(60,332,168,419,128)(61,97,420,169,333)(62,334,170,421,98)(63,99,422,171,335)(64,336,172,423,100)(193,466,236,288,357)(194,358,257,237,467)(195,468,238,258,359)(196,360,259,239,469)(197,470,240,260,361)(198,362,261,241,471)(199,472,242,262,363)(200,364,263,243,473)(201,474,244,264,365)(202,366,265,245,475)(203,476,246,266,367)(204,368,267,247,477)(205,478,248,268,369)(206,370,269,249,479)(207,480,250,270,371)(208,372,271,251,449)(209,450,252,272,373)(210,374,273,253,451)(211,452,254,274,375)(212,376,275,255,453)(213,454,256,276,377)(214,378,277,225,455)(215,456,226,278,379)(216,380,279,227,457)(217,458,228,280,381)(218,382,281,229,459)(219,460,230,282,383)(220,384,283,231,461)(221,462,232,284,353)(222,354,285,233,463)(223,464,234,286,355)(224,356,287,235,465), (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)(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)(449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)>;
G:=Group( (1,118,216)(2,119,217)(3,120,218)(4,121,219)(5,122,220)(6,123,221)(7,124,222)(8,125,223)(9,126,224)(10,127,193)(11,128,194)(12,97,195)(13,98,196)(14,99,197)(15,100,198)(16,101,199)(17,102,200)(18,103,201)(19,104,202)(20,105,203)(21,106,204)(22,107,205)(23,108,206)(24,109,207)(25,110,208)(26,111,209)(27,112,210)(28,113,211)(29,114,212)(30,115,213)(31,116,214)(32,117,215)(33,363,94)(34,364,95)(35,365,96)(36,366,65)(37,367,66)(38,368,67)(39,369,68)(40,370,69)(41,371,70)(42,372,71)(43,373,72)(44,374,73)(45,375,74)(46,376,75)(47,377,76)(48,378,77)(49,379,78)(50,380,79)(51,381,80)(52,382,81)(53,383,82)(54,384,83)(55,353,84)(56,354,85)(57,355,86)(58,356,87)(59,357,88)(60,358,89)(61,359,90)(62,360,91)(63,361,92)(64,362,93)(129,179,248)(130,180,249)(131,181,250)(132,182,251)(133,183,252)(134,184,253)(135,185,254)(136,186,255)(137,187,256)(138,188,225)(139,189,226)(140,190,227)(141,191,228)(142,192,229)(143,161,230)(144,162,231)(145,163,232)(146,164,233)(147,165,234)(148,166,235)(149,167,236)(150,168,237)(151,169,238)(152,170,239)(153,171,240)(154,172,241)(155,173,242)(156,174,243)(157,175,244)(158,176,245)(159,177,246)(160,178,247)(257,308,332)(258,309,333)(259,310,334)(260,311,335)(261,312,336)(262,313,337)(263,314,338)(264,315,339)(265,316,340)(266,317,341)(267,318,342)(268,319,343)(269,320,344)(270,289,345)(271,290,346)(272,291,347)(273,292,348)(274,293,349)(275,294,350)(276,295,351)(277,296,352)(278,297,321)(279,298,322)(280,299,323)(281,300,324)(282,301,325)(283,302,326)(284,303,327)(285,304,328)(286,305,329)(287,306,330)(288,307,331)(385,447,463)(386,448,464)(387,417,465)(388,418,466)(389,419,467)(390,420,468)(391,421,469)(392,422,470)(393,423,471)(394,424,472)(395,425,473)(396,426,474)(397,427,475)(398,428,476)(399,429,477)(400,430,478)(401,431,479)(402,432,480)(403,433,449)(404,434,450)(405,435,451)(406,436,452)(407,437,453)(408,438,454)(409,439,455)(410,440,456)(411,441,457)(412,442,458)(413,443,459)(414,444,460)(415,445,461)(416,446,462), (1,79,298,140,411)(2,412,141,299,80)(3,81,300,142,413)(4,414,143,301,82)(5,83,302,144,415)(6,416,145,303,84)(7,85,304,146,385)(8,386,147,305,86)(9,87,306,148,387)(10,388,149,307,88)(11,89,308,150,389)(12,390,151,309,90)(13,91,310,152,391)(14,392,153,311,92)(15,93,312,154,393)(16,394,155,313,94)(17,95,314,156,395)(18,396,157,315,96)(19,65,316,158,397)(20,398,159,317,66)(21,67,318,160,399)(22,400,129,319,68)(23,69,320,130,401)(24,402,131,289,70)(25,71,290,132,403)(26,404,133,291,72)(27,73,292,134,405)(28,406,135,293,74)(29,75,294,136,407)(30,408,137,295,76)(31,77,296,138,409)(32,410,139,297,78)(33,101,424,173,337)(34,338,174,425,102)(35,103,426,175,339)(36,340,176,427,104)(37,105,428,177,341)(38,342,178,429,106)(39,107,430,179,343)(40,344,180,431,108)(41,109,432,181,345)(42,346,182,433,110)(43,111,434,183,347)(44,348,184,435,112)(45,113,436,185,349)(46,350,186,437,114)(47,115,438,187,351)(48,352,188,439,116)(49,117,440,189,321)(50,322,190,441,118)(51,119,442,191,323)(52,324,192,443,120)(53,121,444,161,325)(54,326,162,445,122)(55,123,446,163,327)(56,328,164,447,124)(57,125,448,165,329)(58,330,166,417,126)(59,127,418,167,331)(60,332,168,419,128)(61,97,420,169,333)(62,334,170,421,98)(63,99,422,171,335)(64,336,172,423,100)(193,466,236,288,357)(194,358,257,237,467)(195,468,238,258,359)(196,360,259,239,469)(197,470,240,260,361)(198,362,261,241,471)(199,472,242,262,363)(200,364,263,243,473)(201,474,244,264,365)(202,366,265,245,475)(203,476,246,266,367)(204,368,267,247,477)(205,478,248,268,369)(206,370,269,249,479)(207,480,250,270,371)(208,372,271,251,449)(209,450,252,272,373)(210,374,273,253,451)(211,452,254,274,375)(212,376,275,255,453)(213,454,256,276,377)(214,378,277,225,455)(215,456,226,278,379)(216,380,279,227,457)(217,458,228,280,381)(218,382,281,229,459)(219,460,230,282,383)(220,384,283,231,461)(221,462,232,284,353)(222,354,285,233,463)(223,464,234,286,355)(224,356,287,235,465), (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)(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)(449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480) );
G=PermutationGroup([[(1,118,216),(2,119,217),(3,120,218),(4,121,219),(5,122,220),(6,123,221),(7,124,222),(8,125,223),(9,126,224),(10,127,193),(11,128,194),(12,97,195),(13,98,196),(14,99,197),(15,100,198),(16,101,199),(17,102,200),(18,103,201),(19,104,202),(20,105,203),(21,106,204),(22,107,205),(23,108,206),(24,109,207),(25,110,208),(26,111,209),(27,112,210),(28,113,211),(29,114,212),(30,115,213),(31,116,214),(32,117,215),(33,363,94),(34,364,95),(35,365,96),(36,366,65),(37,367,66),(38,368,67),(39,369,68),(40,370,69),(41,371,70),(42,372,71),(43,373,72),(44,374,73),(45,375,74),(46,376,75),(47,377,76),(48,378,77),(49,379,78),(50,380,79),(51,381,80),(52,382,81),(53,383,82),(54,384,83),(55,353,84),(56,354,85),(57,355,86),(58,356,87),(59,357,88),(60,358,89),(61,359,90),(62,360,91),(63,361,92),(64,362,93),(129,179,248),(130,180,249),(131,181,250),(132,182,251),(133,183,252),(134,184,253),(135,185,254),(136,186,255),(137,187,256),(138,188,225),(139,189,226),(140,190,227),(141,191,228),(142,192,229),(143,161,230),(144,162,231),(145,163,232),(146,164,233),(147,165,234),(148,166,235),(149,167,236),(150,168,237),(151,169,238),(152,170,239),(153,171,240),(154,172,241),(155,173,242),(156,174,243),(157,175,244),(158,176,245),(159,177,246),(160,178,247),(257,308,332),(258,309,333),(259,310,334),(260,311,335),(261,312,336),(262,313,337),(263,314,338),(264,315,339),(265,316,340),(266,317,341),(267,318,342),(268,319,343),(269,320,344),(270,289,345),(271,290,346),(272,291,347),(273,292,348),(274,293,349),(275,294,350),(276,295,351),(277,296,352),(278,297,321),(279,298,322),(280,299,323),(281,300,324),(282,301,325),(283,302,326),(284,303,327),(285,304,328),(286,305,329),(287,306,330),(288,307,331),(385,447,463),(386,448,464),(387,417,465),(388,418,466),(389,419,467),(390,420,468),(391,421,469),(392,422,470),(393,423,471),(394,424,472),(395,425,473),(396,426,474),(397,427,475),(398,428,476),(399,429,477),(400,430,478),(401,431,479),(402,432,480),(403,433,449),(404,434,450),(405,435,451),(406,436,452),(407,437,453),(408,438,454),(409,439,455),(410,440,456),(411,441,457),(412,442,458),(413,443,459),(414,444,460),(415,445,461),(416,446,462)], [(1,79,298,140,411),(2,412,141,299,80),(3,81,300,142,413),(4,414,143,301,82),(5,83,302,144,415),(6,416,145,303,84),(7,85,304,146,385),(8,386,147,305,86),(9,87,306,148,387),(10,388,149,307,88),(11,89,308,150,389),(12,390,151,309,90),(13,91,310,152,391),(14,392,153,311,92),(15,93,312,154,393),(16,394,155,313,94),(17,95,314,156,395),(18,396,157,315,96),(19,65,316,158,397),(20,398,159,317,66),(21,67,318,160,399),(22,400,129,319,68),(23,69,320,130,401),(24,402,131,289,70),(25,71,290,132,403),(26,404,133,291,72),(27,73,292,134,405),(28,406,135,293,74),(29,75,294,136,407),(30,408,137,295,76),(31,77,296,138,409),(32,410,139,297,78),(33,101,424,173,337),(34,338,174,425,102),(35,103,426,175,339),(36,340,176,427,104),(37,105,428,177,341),(38,342,178,429,106),(39,107,430,179,343),(40,344,180,431,108),(41,109,432,181,345),(42,346,182,433,110),(43,111,434,183,347),(44,348,184,435,112),(45,113,436,185,349),(46,350,186,437,114),(47,115,438,187,351),(48,352,188,439,116),(49,117,440,189,321),(50,322,190,441,118),(51,119,442,191,323),(52,324,192,443,120),(53,121,444,161,325),(54,326,162,445,122),(55,123,446,163,327),(56,328,164,447,124),(57,125,448,165,329),(58,330,166,417,126),(59,127,418,167,331),(60,332,168,419,128),(61,97,420,169,333),(62,334,170,421,98),(63,99,422,171,335),(64,336,172,423,100),(193,466,236,288,357),(194,358,257,237,467),(195,468,238,258,359),(196,360,259,239,469),(197,470,240,260,361),(198,362,261,241,471),(199,472,242,262,363),(200,364,263,243,473),(201,474,244,264,365),(202,366,265,245,475),(203,476,246,266,367),(204,368,267,247,477),(205,478,248,268,369),(206,370,269,249,479),(207,480,250,270,371),(208,372,271,251,449),(209,450,252,272,373),(210,374,273,253,451),(211,452,254,274,375),(212,376,275,255,453),(213,454,256,276,377),(214,378,277,225,455),(215,456,226,278,379),(216,380,279,227,457),(217,458,228,280,381),(218,382,281,229,459),(219,460,230,282,383),(220,384,283,231,461),(221,462,232,284,353),(222,354,285,233,463),(223,464,234,286,355),(224,356,287,235,465)], [(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),(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448),(449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)]])
192 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | 30B | 30C | 30D | 32A | ··· | 32P | 40A | ··· | 40H | 48A | ··· | 48P | 60A | ··· | 60H | 80A | ··· | 80P | 96A | ··· | 96AF | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 30 | 30 | 32 | ··· | 32 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 96 | ··· | 96 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
192 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C32 | C48 | C96 | D5 | Dic5 | C3×D5 | C5⋊2C8 | C3×Dic5 | C5⋊2C16 | C3×C5⋊2C8 | C5⋊2C32 | C3×C5⋊2C16 | C3×C5⋊2C32 |
| kernel | C3×C5⋊2C32 | C240 | C5⋊2C32 | C120 | C80 | C60 | C40 | C30 | C20 | C15 | C10 | C5 | C48 | C24 | C16 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 16 | 32 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C3×C5⋊2C32 ►in GL3(𝔽3361) generated by
| 892 | 0 | 0 |
| 0 | 892 | 0 |
| 0 | 0 | 892 |
| 1 | 0 | 0 |
| 0 | 489 | 3360 |
| 0 | 490 | 3360 |
| 1031 | 0 | 0 |
| 0 | 2106 | 286 |
| 0 | 2452 | 1255 |
G:=sub<GL(3,GF(3361))| [892,0,0,0,892,0,0,0,892],[1,0,0,0,489,490,0,3360,3360],[1031,0,0,0,2106,2452,0,286,1255] >;
C3×C5⋊2C32 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes_2C_{32} % in TeX
G:=Group("C3xC5:2C32"); // GroupNames label
G:=SmallGroup(480,2);
// by ID
G=gap.SmallGroup(480,2);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,-2,-2,-5,42,58,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^3=b^5=c^32=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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