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G = C5×C3⋊C32order 480 = 25·3·5

Direct product of C5 and C3⋊C32

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5×C3⋊C32, C3⋊C160, C6.C80, C155C32, 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

C1C3 — C5×C3⋊C32
C1C3C6C12C24C48C240 — C5×C3⋊C32
C3 — C5×C3⋊C32
C1C80

Generators and relations for C5×C3⋊C32
 G = < a,b,c | a5=b3=c32=1, ab=ba, ac=ca, cbc-1=b-1 >

3C32
3C160

Smallest permutation representation of C5×C3⋊C32
Regular action on 480 points
Generators in S480
(1 330 399 216 309)(2 331 400 217 310)(3 332 401 218 311)(4 333 402 219 312)(5 334 403 220 313)(6 335 404 221 314)(7 336 405 222 315)(8 337 406 223 316)(9 338 407 224 317)(10 339 408 193 318)(11 340 409 194 319)(12 341 410 195 320)(13 342 411 196 289)(14 343 412 197 290)(15 344 413 198 291)(16 345 414 199 292)(17 346 415 200 293)(18 347 416 201 294)(19 348 385 202 295)(20 349 386 203 296)(21 350 387 204 297)(22 351 388 205 298)(23 352 389 206 299)(24 321 390 207 300)(25 322 391 208 301)(26 323 392 209 302)(27 324 393 210 303)(28 325 394 211 304)(29 326 395 212 305)(30 327 396 213 306)(31 328 397 214 307)(32 329 398 215 308)(33 267 152 362 456)(34 268 153 363 457)(35 269 154 364 458)(36 270 155 365 459)(37 271 156 366 460)(38 272 157 367 461)(39 273 158 368 462)(40 274 159 369 463)(41 275 160 370 464)(42 276 129 371 465)(43 277 130 372 466)(44 278 131 373 467)(45 279 132 374 468)(46 280 133 375 469)(47 281 134 376 470)(48 282 135 377 471)(49 283 136 378 472)(50 284 137 379 473)(51 285 138 380 474)(52 286 139 381 475)(53 287 140 382 476)(54 288 141 383 477)(55 257 142 384 478)(56 258 143 353 479)(57 259 144 354 480)(58 260 145 355 449)(59 261 146 356 450)(60 262 147 357 451)(61 263 148 358 452)(62 264 149 359 453)(63 265 150 360 454)(64 266 151 361 455)(65 255 121 176 432)(66 256 122 177 433)(67 225 123 178 434)(68 226 124 179 435)(69 227 125 180 436)(70 228 126 181 437)(71 229 127 182 438)(72 230 128 183 439)(73 231 97 184 440)(74 232 98 185 441)(75 233 99 186 442)(76 234 100 187 443)(77 235 101 188 444)(78 236 102 189 445)(79 237 103 190 446)(80 238 104 191 447)(81 239 105 192 448)(82 240 106 161 417)(83 241 107 162 418)(84 242 108 163 419)(85 243 109 164 420)(86 244 110 165 421)(87 245 111 166 422)(88 246 112 167 423)(89 247 113 168 424)(90 248 114 169 425)(91 249 115 170 426)(92 250 116 171 427)(93 251 117 172 428)(94 252 118 173 429)(95 253 119 174 430)(96 254 120 175 431)
(1 143 106)(2 107 144)(3 145 108)(4 109 146)(5 147 110)(6 111 148)(7 149 112)(8 113 150)(9 151 114)(10 115 152)(11 153 116)(12 117 154)(13 155 118)(14 119 156)(15 157 120)(16 121 158)(17 159 122)(18 123 160)(19 129 124)(20 125 130)(21 131 126)(22 127 132)(23 133 128)(24 97 134)(25 135 98)(26 99 136)(27 137 100)(28 101 138)(29 139 102)(30 103 140)(31 141 104)(32 105 142)(33 193 91)(34 92 194)(35 195 93)(36 94 196)(37 197 95)(38 96 198)(39 199 65)(40 66 200)(41 201 67)(42 68 202)(43 203 69)(44 70 204)(45 205 71)(46 72 206)(47 207 73)(48 74 208)(49 209 75)(50 76 210)(51 211 77)(52 78 212)(53 213 79)(54 80 214)(55 215 81)(56 82 216)(57 217 83)(58 84 218)(59 219 85)(60 86 220)(61 221 87)(62 88 222)(63 223 89)(64 90 224)(161 330 353)(162 354 331)(163 332 355)(164 356 333)(165 334 357)(166 358 335)(167 336 359)(168 360 337)(169 338 361)(170 362 339)(171 340 363)(172 364 341)(173 342 365)(174 366 343)(175 344 367)(176 368 345)(177 346 369)(178 370 347)(179 348 371)(180 372 349)(181 350 373)(182 374 351)(183 352 375)(184 376 321)(185 322 377)(186 378 323)(187 324 379)(188 380 325)(189 326 381)(190 382 327)(191 328 383)(192 384 329)(225 275 294)(226 295 276)(227 277 296)(228 297 278)(229 279 298)(230 299 280)(231 281 300)(232 301 282)(233 283 302)(234 303 284)(235 285 304)(236 305 286)(237 287 306)(238 307 288)(239 257 308)(240 309 258)(241 259 310)(242 311 260)(243 261 312)(244 313 262)(245 263 314)(246 315 264)(247 265 316)(248 317 266)(249 267 318)(250 319 268)(251 269 320)(252 289 270)(253 271 290)(254 291 272)(255 273 292)(256 293 274)(385 465 435)(386 436 466)(387 467 437)(388 438 468)(389 469 439)(390 440 470)(391 471 441)(392 442 472)(393 473 443)(394 444 474)(395 475 445)(396 446 476)(397 477 447)(398 448 478)(399 479 417)(400 418 480)(401 449 419)(402 420 450)(403 451 421)(404 422 452)(405 453 423)(406 424 454)(407 455 425)(408 426 456)(409 457 427)(410 428 458)(411 459 429)(412 430 460)(413 461 431)(414 432 462)(415 463 433)(416 434 464)
(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)| (1,330,399,216,309)(2,331,400,217,310)(3,332,401,218,311)(4,333,402,219,312)(5,334,403,220,313)(6,335,404,221,314)(7,336,405,222,315)(8,337,406,223,316)(9,338,407,224,317)(10,339,408,193,318)(11,340,409,194,319)(12,341,410,195,320)(13,342,411,196,289)(14,343,412,197,290)(15,344,413,198,291)(16,345,414,199,292)(17,346,415,200,293)(18,347,416,201,294)(19,348,385,202,295)(20,349,386,203,296)(21,350,387,204,297)(22,351,388,205,298)(23,352,389,206,299)(24,321,390,207,300)(25,322,391,208,301)(26,323,392,209,302)(27,324,393,210,303)(28,325,394,211,304)(29,326,395,212,305)(30,327,396,213,306)(31,328,397,214,307)(32,329,398,215,308)(33,267,152,362,456)(34,268,153,363,457)(35,269,154,364,458)(36,270,155,365,459)(37,271,156,366,460)(38,272,157,367,461)(39,273,158,368,462)(40,274,159,369,463)(41,275,160,370,464)(42,276,129,371,465)(43,277,130,372,466)(44,278,131,373,467)(45,279,132,374,468)(46,280,133,375,469)(47,281,134,376,470)(48,282,135,377,471)(49,283,136,378,472)(50,284,137,379,473)(51,285,138,380,474)(52,286,139,381,475)(53,287,140,382,476)(54,288,141,383,477)(55,257,142,384,478)(56,258,143,353,479)(57,259,144,354,480)(58,260,145,355,449)(59,261,146,356,450)(60,262,147,357,451)(61,263,148,358,452)(62,264,149,359,453)(63,265,150,360,454)(64,266,151,361,455)(65,255,121,176,432)(66,256,122,177,433)(67,225,123,178,434)(68,226,124,179,435)(69,227,125,180,436)(70,228,126,181,437)(71,229,127,182,438)(72,230,128,183,439)(73,231,97,184,440)(74,232,98,185,441)(75,233,99,186,442)(76,234,100,187,443)(77,235,101,188,444)(78,236,102,189,445)(79,237,103,190,446)(80,238,104,191,447)(81,239,105,192,448)(82,240,106,161,417)(83,241,107,162,418)(84,242,108,163,419)(85,243,109,164,420)(86,244,110,165,421)(87,245,111,166,422)(88,246,112,167,423)(89,247,113,168,424)(90,248,114,169,425)(91,249,115,170,426)(92,250,116,171,427)(93,251,117,172,428)(94,252,118,173,429)(95,253,119,174,430)(96,254,120,175,431), (1,143,106)(2,107,144)(3,145,108)(4,109,146)(5,147,110)(6,111,148)(7,149,112)(8,113,150)(9,151,114)(10,115,152)(11,153,116)(12,117,154)(13,155,118)(14,119,156)(15,157,120)(16,121,158)(17,159,122)(18,123,160)(19,129,124)(20,125,130)(21,131,126)(22,127,132)(23,133,128)(24,97,134)(25,135,98)(26,99,136)(27,137,100)(28,101,138)(29,139,102)(30,103,140)(31,141,104)(32,105,142)(33,193,91)(34,92,194)(35,195,93)(36,94,196)(37,197,95)(38,96,198)(39,199,65)(40,66,200)(41,201,67)(42,68,202)(43,203,69)(44,70,204)(45,205,71)(46,72,206)(47,207,73)(48,74,208)(49,209,75)(50,76,210)(51,211,77)(52,78,212)(53,213,79)(54,80,214)(55,215,81)(56,82,216)(57,217,83)(58,84,218)(59,219,85)(60,86,220)(61,221,87)(62,88,222)(63,223,89)(64,90,224)(161,330,353)(162,354,331)(163,332,355)(164,356,333)(165,334,357)(166,358,335)(167,336,359)(168,360,337)(169,338,361)(170,362,339)(171,340,363)(172,364,341)(173,342,365)(174,366,343)(175,344,367)(176,368,345)(177,346,369)(178,370,347)(179,348,371)(180,372,349)(181,350,373)(182,374,351)(183,352,375)(184,376,321)(185,322,377)(186,378,323)(187,324,379)(188,380,325)(189,326,381)(190,382,327)(191,328,383)(192,384,329)(225,275,294)(226,295,276)(227,277,296)(228,297,278)(229,279,298)(230,299,280)(231,281,300)(232,301,282)(233,283,302)(234,303,284)(235,285,304)(236,305,286)(237,287,306)(238,307,288)(239,257,308)(240,309,258)(241,259,310)(242,311,260)(243,261,312)(244,313,262)(245,263,314)(246,315,264)(247,265,316)(248,317,266)(249,267,318)(250,319,268)(251,269,320)(252,289,270)(253,271,290)(254,291,272)(255,273,292)(256,293,274)(385,465,435)(386,436,466)(387,467,437)(388,438,468)(389,469,439)(390,440,470)(391,471,441)(392,442,472)(393,473,443)(394,444,474)(395,475,445)(396,446,476)(397,477,447)(398,448,478)(399,479,417)(400,418,480)(401,449,419)(402,420,450)(403,451,421)(404,422,452)(405,453,423)(406,424,454)(407,455,425)(408,426,456)(409,457,427)(410,428,458)(411,459,429)(412,430,460)(413,461,431)(414,432,462)(415,463,433)(416,434,464), (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( (1,330,399,216,309)(2,331,400,217,310)(3,332,401,218,311)(4,333,402,219,312)(5,334,403,220,313)(6,335,404,221,314)(7,336,405,222,315)(8,337,406,223,316)(9,338,407,224,317)(10,339,408,193,318)(11,340,409,194,319)(12,341,410,195,320)(13,342,411,196,289)(14,343,412,197,290)(15,344,413,198,291)(16,345,414,199,292)(17,346,415,200,293)(18,347,416,201,294)(19,348,385,202,295)(20,349,386,203,296)(21,350,387,204,297)(22,351,388,205,298)(23,352,389,206,299)(24,321,390,207,300)(25,322,391,208,301)(26,323,392,209,302)(27,324,393,210,303)(28,325,394,211,304)(29,326,395,212,305)(30,327,396,213,306)(31,328,397,214,307)(32,329,398,215,308)(33,267,152,362,456)(34,268,153,363,457)(35,269,154,364,458)(36,270,155,365,459)(37,271,156,366,460)(38,272,157,367,461)(39,273,158,368,462)(40,274,159,369,463)(41,275,160,370,464)(42,276,129,371,465)(43,277,130,372,466)(44,278,131,373,467)(45,279,132,374,468)(46,280,133,375,469)(47,281,134,376,470)(48,282,135,377,471)(49,283,136,378,472)(50,284,137,379,473)(51,285,138,380,474)(52,286,139,381,475)(53,287,140,382,476)(54,288,141,383,477)(55,257,142,384,478)(56,258,143,353,479)(57,259,144,354,480)(58,260,145,355,449)(59,261,146,356,450)(60,262,147,357,451)(61,263,148,358,452)(62,264,149,359,453)(63,265,150,360,454)(64,266,151,361,455)(65,255,121,176,432)(66,256,122,177,433)(67,225,123,178,434)(68,226,124,179,435)(69,227,125,180,436)(70,228,126,181,437)(71,229,127,182,438)(72,230,128,183,439)(73,231,97,184,440)(74,232,98,185,441)(75,233,99,186,442)(76,234,100,187,443)(77,235,101,188,444)(78,236,102,189,445)(79,237,103,190,446)(80,238,104,191,447)(81,239,105,192,448)(82,240,106,161,417)(83,241,107,162,418)(84,242,108,163,419)(85,243,109,164,420)(86,244,110,165,421)(87,245,111,166,422)(88,246,112,167,423)(89,247,113,168,424)(90,248,114,169,425)(91,249,115,170,426)(92,250,116,171,427)(93,251,117,172,428)(94,252,118,173,429)(95,253,119,174,430)(96,254,120,175,431), (1,143,106)(2,107,144)(3,145,108)(4,109,146)(5,147,110)(6,111,148)(7,149,112)(8,113,150)(9,151,114)(10,115,152)(11,153,116)(12,117,154)(13,155,118)(14,119,156)(15,157,120)(16,121,158)(17,159,122)(18,123,160)(19,129,124)(20,125,130)(21,131,126)(22,127,132)(23,133,128)(24,97,134)(25,135,98)(26,99,136)(27,137,100)(28,101,138)(29,139,102)(30,103,140)(31,141,104)(32,105,142)(33,193,91)(34,92,194)(35,195,93)(36,94,196)(37,197,95)(38,96,198)(39,199,65)(40,66,200)(41,201,67)(42,68,202)(43,203,69)(44,70,204)(45,205,71)(46,72,206)(47,207,73)(48,74,208)(49,209,75)(50,76,210)(51,211,77)(52,78,212)(53,213,79)(54,80,214)(55,215,81)(56,82,216)(57,217,83)(58,84,218)(59,219,85)(60,86,220)(61,221,87)(62,88,222)(63,223,89)(64,90,224)(161,330,353)(162,354,331)(163,332,355)(164,356,333)(165,334,357)(166,358,335)(167,336,359)(168,360,337)(169,338,361)(170,362,339)(171,340,363)(172,364,341)(173,342,365)(174,366,343)(175,344,367)(176,368,345)(177,346,369)(178,370,347)(179,348,371)(180,372,349)(181,350,373)(182,374,351)(183,352,375)(184,376,321)(185,322,377)(186,378,323)(187,324,379)(188,380,325)(189,326,381)(190,382,327)(191,328,383)(192,384,329)(225,275,294)(226,295,276)(227,277,296)(228,297,278)(229,279,298)(230,299,280)(231,281,300)(232,301,282)(233,283,302)(234,303,284)(235,285,304)(236,305,286)(237,287,306)(238,307,288)(239,257,308)(240,309,258)(241,259,310)(242,311,260)(243,261,312)(244,313,262)(245,263,314)(246,315,264)(247,265,316)(248,317,266)(249,267,318)(250,319,268)(251,269,320)(252,289,270)(253,271,290)(254,291,272)(255,273,292)(256,293,274)(385,465,435)(386,436,466)(387,467,437)(388,438,468)(389,469,439)(390,440,470)(391,471,441)(392,442,472)(393,473,443)(394,444,474)(395,475,445)(396,446,476)(397,477,447)(398,448,478)(399,479,417)(400,418,480)(401,449,419)(402,420,450)(403,451,421)(404,422,452)(405,453,423)(406,424,454)(407,455,425)(408,426,456)(409,457,427)(410,428,458)(411,459,429)(412,430,460)(413,461,431)(414,432,462)(415,463,433)(416,434,464), (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([(1,330,399,216,309),(2,331,400,217,310),(3,332,401,218,311),(4,333,402,219,312),(5,334,403,220,313),(6,335,404,221,314),(7,336,405,222,315),(8,337,406,223,316),(9,338,407,224,317),(10,339,408,193,318),(11,340,409,194,319),(12,341,410,195,320),(13,342,411,196,289),(14,343,412,197,290),(15,344,413,198,291),(16,345,414,199,292),(17,346,415,200,293),(18,347,416,201,294),(19,348,385,202,295),(20,349,386,203,296),(21,350,387,204,297),(22,351,388,205,298),(23,352,389,206,299),(24,321,390,207,300),(25,322,391,208,301),(26,323,392,209,302),(27,324,393,210,303),(28,325,394,211,304),(29,326,395,212,305),(30,327,396,213,306),(31,328,397,214,307),(32,329,398,215,308),(33,267,152,362,456),(34,268,153,363,457),(35,269,154,364,458),(36,270,155,365,459),(37,271,156,366,460),(38,272,157,367,461),(39,273,158,368,462),(40,274,159,369,463),(41,275,160,370,464),(42,276,129,371,465),(43,277,130,372,466),(44,278,131,373,467),(45,279,132,374,468),(46,280,133,375,469),(47,281,134,376,470),(48,282,135,377,471),(49,283,136,378,472),(50,284,137,379,473),(51,285,138,380,474),(52,286,139,381,475),(53,287,140,382,476),(54,288,141,383,477),(55,257,142,384,478),(56,258,143,353,479),(57,259,144,354,480),(58,260,145,355,449),(59,261,146,356,450),(60,262,147,357,451),(61,263,148,358,452),(62,264,149,359,453),(63,265,150,360,454),(64,266,151,361,455),(65,255,121,176,432),(66,256,122,177,433),(67,225,123,178,434),(68,226,124,179,435),(69,227,125,180,436),(70,228,126,181,437),(71,229,127,182,438),(72,230,128,183,439),(73,231,97,184,440),(74,232,98,185,441),(75,233,99,186,442),(76,234,100,187,443),(77,235,101,188,444),(78,236,102,189,445),(79,237,103,190,446),(80,238,104,191,447),(81,239,105,192,448),(82,240,106,161,417),(83,241,107,162,418),(84,242,108,163,419),(85,243,109,164,420),(86,244,110,165,421),(87,245,111,166,422),(88,246,112,167,423),(89,247,113,168,424),(90,248,114,169,425),(91,249,115,170,426),(92,250,116,171,427),(93,251,117,172,428),(94,252,118,173,429),(95,253,119,174,430),(96,254,120,175,431)], [(1,143,106),(2,107,144),(3,145,108),(4,109,146),(5,147,110),(6,111,148),(7,149,112),(8,113,150),(9,151,114),(10,115,152),(11,153,116),(12,117,154),(13,155,118),(14,119,156),(15,157,120),(16,121,158),(17,159,122),(18,123,160),(19,129,124),(20,125,130),(21,131,126),(22,127,132),(23,133,128),(24,97,134),(25,135,98),(26,99,136),(27,137,100),(28,101,138),(29,139,102),(30,103,140),(31,141,104),(32,105,142),(33,193,91),(34,92,194),(35,195,93),(36,94,196),(37,197,95),(38,96,198),(39,199,65),(40,66,200),(41,201,67),(42,68,202),(43,203,69),(44,70,204),(45,205,71),(46,72,206),(47,207,73),(48,74,208),(49,209,75),(50,76,210),(51,211,77),(52,78,212),(53,213,79),(54,80,214),(55,215,81),(56,82,216),(57,217,83),(58,84,218),(59,219,85),(60,86,220),(61,221,87),(62,88,222),(63,223,89),(64,90,224),(161,330,353),(162,354,331),(163,332,355),(164,356,333),(165,334,357),(166,358,335),(167,336,359),(168,360,337),(169,338,361),(170,362,339),(171,340,363),(172,364,341),(173,342,365),(174,366,343),(175,344,367),(176,368,345),(177,346,369),(178,370,347),(179,348,371),(180,372,349),(181,350,373),(182,374,351),(183,352,375),(184,376,321),(185,322,377),(186,378,323),(187,324,379),(188,380,325),(189,326,381),(190,382,327),(191,328,383),(192,384,329),(225,275,294),(226,295,276),(227,277,296),(228,297,278),(229,279,298),(230,299,280),(231,281,300),(232,301,282),(233,283,302),(234,303,284),(235,285,304),(236,305,286),(237,287,306),(238,307,288),(239,257,308),(240,309,258),(241,259,310),(242,311,260),(243,261,312),(244,313,262),(245,263,314),(246,315,264),(247,265,316),(248,317,266),(249,267,318),(250,319,268),(251,269,320),(252,289,270),(253,271,290),(254,291,272),(255,273,292),(256,293,274),(385,465,435),(386,436,466),(387,467,437),(388,438,468),(389,469,439),(390,440,470),(391,471,441),(392,442,472),(393,473,443),(394,444,474),(395,475,445),(396,446,476),(397,477,447),(398,448,478),(399,479,417),(400,418,480),(401,449,419),(402,420,450),(403,451,421),(404,422,452),(405,453,423),(406,424,454),(407,455,425),(408,426,456),(409,457,427),(410,428,458),(411,459,429),(412,430,460),(413,461,431),(414,432,462),(415,463,433),(416,434,464)], [(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 4A4B5A5B5C5D 6 8A8B8C8D10A10B10C10D12A12B15A15B15C15D16A···16H20A···20H24A24B24C24D30A30B30C30D32A···32P40A···40P48A···48H60A···60H80A···80AF120A···120P160A···160BL240A···240AF
order123445555688881010101012121515151516···1620···20242424243030303032···3240···4048···4860···6080···80120···120160···160240···240
size1121111112111111112222221···11···1222222223···31···12···22···21···12···23···32···2

240 irreducible representations

dim1111111111112222222222
type+++-
imageC1C2C4C5C8C10C16C20C32C40C80C160S3Dic3C3⋊C8C5×S3C3⋊C16C5×Dic3C3⋊C32C5×C3⋊C8C5×C3⋊C16C5×C3⋊C32
kernelC5×C3⋊C32C240C120C3⋊C32C60C48C30C24C15C12C6C3C80C40C20C16C10C8C5C4C2C1
# reps1124448816163264112444881632

Matrix representation of C5×C3⋊C32 in GL2(𝔽3361) generated by

30290
03029
,
03360
13360
,
6111438
20492750
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

Export

Subgroup lattice of C5×C3⋊C32 in TeX

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