direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×C3⋊C32, C3⋊C160, C6.C80, C15⋊5C32, C80.4S3, C60.14C8, C24.3C20, C12.2C40, C240.7C2, C48.3C10, C30.5C16, C120.20C4, C40.12Dic3, C20.8(C3⋊C8), C16.2(C5×S3), C10.3(C3⋊C16), C8.3(C5×Dic3), C2.(C5×C3⋊C16), C4.2(C5×C3⋊C8), SmallGroup(480,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C5×C3⋊C32 |
Generators and relations for C5×C3⋊C32
G = < a,b,c | a5=b3=c32=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C3⋊C32
►Regular action on 480 points
Generators in S480
(1 399 45 193 300)(2 400 46 194 301)(3 401 47 195 302)(4 402 48 196 303)(5 403 49 197 304)(6 404 50 198 305)(7 405 51 199 306)(8 406 52 200 307)(9 407 53 201 308)(10 408 54 202 309)(11 409 55 203 310)(12 410 56 204 311)(13 411 57 205 312)(14 412 58 206 313)(15 413 59 207 314)(16 414 60 208 315)(17 415 61 209 316)(18 416 62 210 317)(19 385 63 211 318)(20 386 64 212 319)(21 387 33 213 320)(22 388 34 214 289)(23 389 35 215 290)(24 390 36 216 291)(25 391 37 217 292)(26 392 38 218 293)(27 393 39 219 294)(28 394 40 220 295)(29 395 41 221 296)(30 396 42 222 297)(31 397 43 223 298)(32 398 44 224 299)(65 431 135 256 356)(66 432 136 225 357)(67 433 137 226 358)(68 434 138 227 359)(69 435 139 228 360)(70 436 140 229 361)(71 437 141 230 362)(72 438 142 231 363)(73 439 143 232 364)(74 440 144 233 365)(75 441 145 234 366)(76 442 146 235 367)(77 443 147 236 368)(78 444 148 237 369)(79 445 149 238 370)(80 446 150 239 371)(81 447 151 240 372)(82 448 152 241 373)(83 417 153 242 374)(84 418 154 243 375)(85 419 155 244 376)(86 420 156 245 377)(87 421 157 246 378)(88 422 158 247 379)(89 423 159 248 380)(90 424 160 249 381)(91 425 129 250 382)(92 426 130 251 383)(93 427 131 252 384)(94 428 132 253 353)(95 429 133 254 354)(96 430 134 255 355)(97 331 468 190 273)(98 332 469 191 274)(99 333 470 192 275)(100 334 471 161 276)(101 335 472 162 277)(102 336 473 163 278)(103 337 474 164 279)(104 338 475 165 280)(105 339 476 166 281)(106 340 477 167 282)(107 341 478 168 283)(108 342 479 169 284)(109 343 480 170 285)(110 344 449 171 286)(111 345 450 172 287)(112 346 451 173 288)(113 347 452 174 257)(114 348 453 175 258)(115 349 454 176 259)(116 350 455 177 260)(117 351 456 178 261)(118 352 457 179 262)(119 321 458 180 263)(120 322 459 181 264)(121 323 460 182 265)(122 324 461 183 266)(123 325 462 184 267)(124 326 463 185 268)(125 327 464 186 269)(126 328 465 187 270)(127 329 466 188 271)(128 330 467 189 272)
(1 241 281)(2 282 242)(3 243 283)(4 284 244)(5 245 285)(6 286 246)(7 247 287)(8 288 248)(9 249 257)(10 258 250)(11 251 259)(12 260 252)(13 253 261)(14 262 254)(15 255 263)(16 264 256)(17 225 265)(18 266 226)(19 227 267)(20 268 228)(21 229 269)(22 270 230)(23 231 271)(24 272 232)(25 233 273)(26 274 234)(27 235 275)(28 276 236)(29 237 277)(30 278 238)(31 239 279)(32 280 240)(33 70 327)(34 328 71)(35 72 329)(36 330 73)(37 74 331)(38 332 75)(39 76 333)(40 334 77)(41 78 335)(42 336 79)(43 80 337)(44 338 81)(45 82 339)(46 340 83)(47 84 341)(48 342 85)(49 86 343)(50 344 87)(51 88 345)(52 346 89)(53 90 347)(54 348 91)(55 92 349)(56 350 93)(57 94 351)(58 352 95)(59 96 321)(60 322 65)(61 66 323)(62 324 67)(63 68 325)(64 326 69)(97 391 365)(98 366 392)(99 393 367)(100 368 394)(101 395 369)(102 370 396)(103 397 371)(104 372 398)(105 399 373)(106 374 400)(107 401 375)(108 376 402)(109 403 377)(110 378 404)(111 405 379)(112 380 406)(113 407 381)(114 382 408)(115 409 383)(116 384 410)(117 411 353)(118 354 412)(119 413 355)(120 356 414)(121 415 357)(122 358 416)(123 385 359)(124 360 386)(125 387 361)(126 362 388)(127 389 363)(128 364 390)(129 309 175)(130 176 310)(131 311 177)(132 178 312)(133 313 179)(134 180 314)(135 315 181)(136 182 316)(137 317 183)(138 184 318)(139 319 185)(140 186 320)(141 289 187)(142 188 290)(143 291 189)(144 190 292)(145 293 191)(146 192 294)(147 295 161)(148 162 296)(149 297 163)(150 164 298)(151 299 165)(152 166 300)(153 301 167)(154 168 302)(155 303 169)(156 170 304)(157 305 171)(158 172 306)(159 307 173)(160 174 308)(193 448 476)(194 477 417)(195 418 478)(196 479 419)(197 420 480)(198 449 421)(199 422 450)(200 451 423)(201 424 452)(202 453 425)(203 426 454)(204 455 427)(205 428 456)(206 457 429)(207 430 458)(208 459 431)(209 432 460)(210 461 433)(211 434 462)(212 463 435)(213 436 464)(214 465 437)(215 438 466)(216 467 439)(217 440 468)(218 469 441)(219 442 470)(220 471 443)(221 444 472)(222 473 445)(223 446 474)(224 475 447)
(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)| (1,399,45,193,300)(2,400,46,194,301)(3,401,47,195,302)(4,402,48,196,303)(5,403,49,197,304)(6,404,50,198,305)(7,405,51,199,306)(8,406,52,200,307)(9,407,53,201,308)(10,408,54,202,309)(11,409,55,203,310)(12,410,56,204,311)(13,411,57,205,312)(14,412,58,206,313)(15,413,59,207,314)(16,414,60,208,315)(17,415,61,209,316)(18,416,62,210,317)(19,385,63,211,318)(20,386,64,212,319)(21,387,33,213,320)(22,388,34,214,289)(23,389,35,215,290)(24,390,36,216,291)(25,391,37,217,292)(26,392,38,218,293)(27,393,39,219,294)(28,394,40,220,295)(29,395,41,221,296)(30,396,42,222,297)(31,397,43,223,298)(32,398,44,224,299)(65,431,135,256,356)(66,432,136,225,357)(67,433,137,226,358)(68,434,138,227,359)(69,435,139,228,360)(70,436,140,229,361)(71,437,141,230,362)(72,438,142,231,363)(73,439,143,232,364)(74,440,144,233,365)(75,441,145,234,366)(76,442,146,235,367)(77,443,147,236,368)(78,444,148,237,369)(79,445,149,238,370)(80,446,150,239,371)(81,447,151,240,372)(82,448,152,241,373)(83,417,153,242,374)(84,418,154,243,375)(85,419,155,244,376)(86,420,156,245,377)(87,421,157,246,378)(88,422,158,247,379)(89,423,159,248,380)(90,424,160,249,381)(91,425,129,250,382)(92,426,130,251,383)(93,427,131,252,384)(94,428,132,253,353)(95,429,133,254,354)(96,430,134,255,355)(97,331,468,190,273)(98,332,469,191,274)(99,333,470,192,275)(100,334,471,161,276)(101,335,472,162,277)(102,336,473,163,278)(103,337,474,164,279)(104,338,475,165,280)(105,339,476,166,281)(106,340,477,167,282)(107,341,478,168,283)(108,342,479,169,284)(109,343,480,170,285)(110,344,449,171,286)(111,345,450,172,287)(112,346,451,173,288)(113,347,452,174,257)(114,348,453,175,258)(115,349,454,176,259)(116,350,455,177,260)(117,351,456,178,261)(118,352,457,179,262)(119,321,458,180,263)(120,322,459,181,264)(121,323,460,182,265)(122,324,461,183,266)(123,325,462,184,267)(124,326,463,185,268)(125,327,464,186,269)(126,328,465,187,270)(127,329,466,188,271)(128,330,467,189,272), (1,241,281)(2,282,242)(3,243,283)(4,284,244)(5,245,285)(6,286,246)(7,247,287)(8,288,248)(9,249,257)(10,258,250)(11,251,259)(12,260,252)(13,253,261)(14,262,254)(15,255,263)(16,264,256)(17,225,265)(18,266,226)(19,227,267)(20,268,228)(21,229,269)(22,270,230)(23,231,271)(24,272,232)(25,233,273)(26,274,234)(27,235,275)(28,276,236)(29,237,277)(30,278,238)(31,239,279)(32,280,240)(33,70,327)(34,328,71)(35,72,329)(36,330,73)(37,74,331)(38,332,75)(39,76,333)(40,334,77)(41,78,335)(42,336,79)(43,80,337)(44,338,81)(45,82,339)(46,340,83)(47,84,341)(48,342,85)(49,86,343)(50,344,87)(51,88,345)(52,346,89)(53,90,347)(54,348,91)(55,92,349)(56,350,93)(57,94,351)(58,352,95)(59,96,321)(60,322,65)(61,66,323)(62,324,67)(63,68,325)(64,326,69)(97,391,365)(98,366,392)(99,393,367)(100,368,394)(101,395,369)(102,370,396)(103,397,371)(104,372,398)(105,399,373)(106,374,400)(107,401,375)(108,376,402)(109,403,377)(110,378,404)(111,405,379)(112,380,406)(113,407,381)(114,382,408)(115,409,383)(116,384,410)(117,411,353)(118,354,412)(119,413,355)(120,356,414)(121,415,357)(122,358,416)(123,385,359)(124,360,386)(125,387,361)(126,362,388)(127,389,363)(128,364,390)(129,309,175)(130,176,310)(131,311,177)(132,178,312)(133,313,179)(134,180,314)(135,315,181)(136,182,316)(137,317,183)(138,184,318)(139,319,185)(140,186,320)(141,289,187)(142,188,290)(143,291,189)(144,190,292)(145,293,191)(146,192,294)(147,295,161)(148,162,296)(149,297,163)(150,164,298)(151,299,165)(152,166,300)(153,301,167)(154,168,302)(155,303,169)(156,170,304)(157,305,171)(158,172,306)(159,307,173)(160,174,308)(193,448,476)(194,477,417)(195,418,478)(196,479,419)(197,420,480)(198,449,421)(199,422,450)(200,451,423)(201,424,452)(202,453,425)(203,426,454)(204,455,427)(205,428,456)(206,457,429)(207,430,458)(208,459,431)(209,432,460)(210,461,433)(211,434,462)(212,463,435)(213,436,464)(214,465,437)(215,438,466)(216,467,439)(217,440,468)(218,469,441)(219,442,470)(220,471,443)(221,444,472)(222,473,445)(223,446,474)(224,475,447), (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( (1,399,45,193,300)(2,400,46,194,301)(3,401,47,195,302)(4,402,48,196,303)(5,403,49,197,304)(6,404,50,198,305)(7,405,51,199,306)(8,406,52,200,307)(9,407,53,201,308)(10,408,54,202,309)(11,409,55,203,310)(12,410,56,204,311)(13,411,57,205,312)(14,412,58,206,313)(15,413,59,207,314)(16,414,60,208,315)(17,415,61,209,316)(18,416,62,210,317)(19,385,63,211,318)(20,386,64,212,319)(21,387,33,213,320)(22,388,34,214,289)(23,389,35,215,290)(24,390,36,216,291)(25,391,37,217,292)(26,392,38,218,293)(27,393,39,219,294)(28,394,40,220,295)(29,395,41,221,296)(30,396,42,222,297)(31,397,43,223,298)(32,398,44,224,299)(65,431,135,256,356)(66,432,136,225,357)(67,433,137,226,358)(68,434,138,227,359)(69,435,139,228,360)(70,436,140,229,361)(71,437,141,230,362)(72,438,142,231,363)(73,439,143,232,364)(74,440,144,233,365)(75,441,145,234,366)(76,442,146,235,367)(77,443,147,236,368)(78,444,148,237,369)(79,445,149,238,370)(80,446,150,239,371)(81,447,151,240,372)(82,448,152,241,373)(83,417,153,242,374)(84,418,154,243,375)(85,419,155,244,376)(86,420,156,245,377)(87,421,157,246,378)(88,422,158,247,379)(89,423,159,248,380)(90,424,160,249,381)(91,425,129,250,382)(92,426,130,251,383)(93,427,131,252,384)(94,428,132,253,353)(95,429,133,254,354)(96,430,134,255,355)(97,331,468,190,273)(98,332,469,191,274)(99,333,470,192,275)(100,334,471,161,276)(101,335,472,162,277)(102,336,473,163,278)(103,337,474,164,279)(104,338,475,165,280)(105,339,476,166,281)(106,340,477,167,282)(107,341,478,168,283)(108,342,479,169,284)(109,343,480,170,285)(110,344,449,171,286)(111,345,450,172,287)(112,346,451,173,288)(113,347,452,174,257)(114,348,453,175,258)(115,349,454,176,259)(116,350,455,177,260)(117,351,456,178,261)(118,352,457,179,262)(119,321,458,180,263)(120,322,459,181,264)(121,323,460,182,265)(122,324,461,183,266)(123,325,462,184,267)(124,326,463,185,268)(125,327,464,186,269)(126,328,465,187,270)(127,329,466,188,271)(128,330,467,189,272), (1,241,281)(2,282,242)(3,243,283)(4,284,244)(5,245,285)(6,286,246)(7,247,287)(8,288,248)(9,249,257)(10,258,250)(11,251,259)(12,260,252)(13,253,261)(14,262,254)(15,255,263)(16,264,256)(17,225,265)(18,266,226)(19,227,267)(20,268,228)(21,229,269)(22,270,230)(23,231,271)(24,272,232)(25,233,273)(26,274,234)(27,235,275)(28,276,236)(29,237,277)(30,278,238)(31,239,279)(32,280,240)(33,70,327)(34,328,71)(35,72,329)(36,330,73)(37,74,331)(38,332,75)(39,76,333)(40,334,77)(41,78,335)(42,336,79)(43,80,337)(44,338,81)(45,82,339)(46,340,83)(47,84,341)(48,342,85)(49,86,343)(50,344,87)(51,88,345)(52,346,89)(53,90,347)(54,348,91)(55,92,349)(56,350,93)(57,94,351)(58,352,95)(59,96,321)(60,322,65)(61,66,323)(62,324,67)(63,68,325)(64,326,69)(97,391,365)(98,366,392)(99,393,367)(100,368,394)(101,395,369)(102,370,396)(103,397,371)(104,372,398)(105,399,373)(106,374,400)(107,401,375)(108,376,402)(109,403,377)(110,378,404)(111,405,379)(112,380,406)(113,407,381)(114,382,408)(115,409,383)(116,384,410)(117,411,353)(118,354,412)(119,413,355)(120,356,414)(121,415,357)(122,358,416)(123,385,359)(124,360,386)(125,387,361)(126,362,388)(127,389,363)(128,364,390)(129,309,175)(130,176,310)(131,311,177)(132,178,312)(133,313,179)(134,180,314)(135,315,181)(136,182,316)(137,317,183)(138,184,318)(139,319,185)(140,186,320)(141,289,187)(142,188,290)(143,291,189)(144,190,292)(145,293,191)(146,192,294)(147,295,161)(148,162,296)(149,297,163)(150,164,298)(151,299,165)(152,166,300)(153,301,167)(154,168,302)(155,303,169)(156,170,304)(157,305,171)(158,172,306)(159,307,173)(160,174,308)(193,448,476)(194,477,417)(195,418,478)(196,479,419)(197,420,480)(198,449,421)(199,422,450)(200,451,423)(201,424,452)(202,453,425)(203,426,454)(204,455,427)(205,428,456)(206,457,429)(207,430,458)(208,459,431)(209,432,460)(210,461,433)(211,434,462)(212,463,435)(213,436,464)(214,465,437)(215,438,466)(216,467,439)(217,440,468)(218,469,441)(219,442,470)(220,471,443)(221,444,472)(222,473,445)(223,446,474)(224,475,447), (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([[(1,399,45,193,300),(2,400,46,194,301),(3,401,47,195,302),(4,402,48,196,303),(5,403,49,197,304),(6,404,50,198,305),(7,405,51,199,306),(8,406,52,200,307),(9,407,53,201,308),(10,408,54,202,309),(11,409,55,203,310),(12,410,56,204,311),(13,411,57,205,312),(14,412,58,206,313),(15,413,59,207,314),(16,414,60,208,315),(17,415,61,209,316),(18,416,62,210,317),(19,385,63,211,318),(20,386,64,212,319),(21,387,33,213,320),(22,388,34,214,289),(23,389,35,215,290),(24,390,36,216,291),(25,391,37,217,292),(26,392,38,218,293),(27,393,39,219,294),(28,394,40,220,295),(29,395,41,221,296),(30,396,42,222,297),(31,397,43,223,298),(32,398,44,224,299),(65,431,135,256,356),(66,432,136,225,357),(67,433,137,226,358),(68,434,138,227,359),(69,435,139,228,360),(70,436,140,229,361),(71,437,141,230,362),(72,438,142,231,363),(73,439,143,232,364),(74,440,144,233,365),(75,441,145,234,366),(76,442,146,235,367),(77,443,147,236,368),(78,444,148,237,369),(79,445,149,238,370),(80,446,150,239,371),(81,447,151,240,372),(82,448,152,241,373),(83,417,153,242,374),(84,418,154,243,375),(85,419,155,244,376),(86,420,156,245,377),(87,421,157,246,378),(88,422,158,247,379),(89,423,159,248,380),(90,424,160,249,381),(91,425,129,250,382),(92,426,130,251,383),(93,427,131,252,384),(94,428,132,253,353),(95,429,133,254,354),(96,430,134,255,355),(97,331,468,190,273),(98,332,469,191,274),(99,333,470,192,275),(100,334,471,161,276),(101,335,472,162,277),(102,336,473,163,278),(103,337,474,164,279),(104,338,475,165,280),(105,339,476,166,281),(106,340,477,167,282),(107,341,478,168,283),(108,342,479,169,284),(109,343,480,170,285),(110,344,449,171,286),(111,345,450,172,287),(112,346,451,173,288),(113,347,452,174,257),(114,348,453,175,258),(115,349,454,176,259),(116,350,455,177,260),(117,351,456,178,261),(118,352,457,179,262),(119,321,458,180,263),(120,322,459,181,264),(121,323,460,182,265),(122,324,461,183,266),(123,325,462,184,267),(124,326,463,185,268),(125,327,464,186,269),(126,328,465,187,270),(127,329,466,188,271),(128,330,467,189,272)], [(1,241,281),(2,282,242),(3,243,283),(4,284,244),(5,245,285),(6,286,246),(7,247,287),(8,288,248),(9,249,257),(10,258,250),(11,251,259),(12,260,252),(13,253,261),(14,262,254),(15,255,263),(16,264,256),(17,225,265),(18,266,226),(19,227,267),(20,268,228),(21,229,269),(22,270,230),(23,231,271),(24,272,232),(25,233,273),(26,274,234),(27,235,275),(28,276,236),(29,237,277),(30,278,238),(31,239,279),(32,280,240),(33,70,327),(34,328,71),(35,72,329),(36,330,73),(37,74,331),(38,332,75),(39,76,333),(40,334,77),(41,78,335),(42,336,79),(43,80,337),(44,338,81),(45,82,339),(46,340,83),(47,84,341),(48,342,85),(49,86,343),(50,344,87),(51,88,345),(52,346,89),(53,90,347),(54,348,91),(55,92,349),(56,350,93),(57,94,351),(58,352,95),(59,96,321),(60,322,65),(61,66,323),(62,324,67),(63,68,325),(64,326,69),(97,391,365),(98,366,392),(99,393,367),(100,368,394),(101,395,369),(102,370,396),(103,397,371),(104,372,398),(105,399,373),(106,374,400),(107,401,375),(108,376,402),(109,403,377),(110,378,404),(111,405,379),(112,380,406),(113,407,381),(114,382,408),(115,409,383),(116,384,410),(117,411,353),(118,354,412),(119,413,355),(120,356,414),(121,415,357),(122,358,416),(123,385,359),(124,360,386),(125,387,361),(126,362,388),(127,389,363),(128,364,390),(129,309,175),(130,176,310),(131,311,177),(132,178,312),(133,313,179),(134,180,314),(135,315,181),(136,182,316),(137,317,183),(138,184,318),(139,319,185),(140,186,320),(141,289,187),(142,188,290),(143,291,189),(144,190,292),(145,293,191),(146,192,294),(147,295,161),(148,162,296),(149,297,163),(150,164,298),(151,299,165),(152,166,300),(153,301,167),(154,168,302),(155,303,169),(156,170,304),(157,305,171),(158,172,306),(159,307,173),(160,174,308),(193,448,476),(194,477,417),(195,418,478),(196,479,419),(197,420,480),(198,449,421),(199,422,450),(200,451,423),(201,424,452),(202,453,425),(203,426,454),(204,455,427),(205,428,456),(206,457,429),(207,430,458),(208,459,431),(209,432,460),(210,461,433),(211,434,462),(212,463,435),(213,436,464),(214,465,437),(215,438,466),(216,467,439),(217,440,468),(218,469,441),(219,442,470),(220,471,443),(221,444,472),(222,473,445),(223,446,474),(224,475,447)], [(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)]])
240 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | ··· | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 32A | ··· | 32P | 40A | ··· | 40P | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80AF | 120A | ··· | 120P | 160A | ··· | 160BL | 240A | ··· | 240AF |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 32 | ··· | 32 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 160 | ··· | 160 | 240 | ··· | 240 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
240 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 | C4 | C5 | C8 | C10 | C16 | C20 | C32 | C40 | C80 | C160 | S3 | Dic3 | C3⋊C8 | C5×S3 | C3⋊C16 | C5×Dic3 | C3⋊C32 | C5×C3⋊C8 | C5×C3⋊C16 | C5×C3⋊C32 |
| kernel | C5×C3⋊C32 | C240 | C120 | C3⋊C32 | C60 | C48 | C30 | C24 | C15 | C12 | C6 | C3 | C80 | C40 | C20 | C16 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 16 | 32 | 64 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 32 |
Matrix representation of C5×C3⋊C32 ►in GL2(𝔽3361) generated by
| 3029 | 0 |
| 0 | 3029 |
| 0 | 3360 |
| 1 | 3360 |
| 611 | 1438 |
| 2049 | 2750 |
G:=sub<GL(2,GF(3361))| [3029,0,0,3029],[0,1,3360,3360],[611,2049,1438,2750] >;
C5×C3⋊C32 in GAP, Magma, Sage, TeX
C_5\times C_3\rtimes C_{32} % in TeX
G:=Group("C5xC3:C32"); // GroupNames label
G:=SmallGroup(480,1);
// by ID
G=gap.SmallGroup(480,1);
# by ID
G:=PCGroup([7,-2,-5,-2,-2,-2,-2,-3,70,58,80,102,15686]);
// Polycyclic
G:=Group<a,b,c|a^5=b^3=c^32=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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