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G = C3×C5⋊C32order 480 = 25·3·5

Direct product of C3 and C5⋊C32

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

Aliases: C3×C5⋊C32, C5⋊C96, C152C32, C10.C48, C60.5C8, C24.9F5, C20.2C24, C40.3C12, C30.2C16, C120.8C4, C8.4(C3×F5), C6.2(C5⋊C16), C12.5(C5⋊C8), C52C16.2C6, C2.(C3×C5⋊C16), C4.2(C3×C5⋊C8), (C3×C52C16).4C2, SmallGroup(480,5)

Series: Derived Chief Lower central Upper central

C1C5 — C3×C5⋊C32
C1C5C10C20C40C52C16C3×C52C16 — C3×C5⋊C32
C5 — C3×C5⋊C32
C1C24

Generators and relations for C3×C5⋊C32
 G = < a,b,c | a3=b5=c32=1, ab=ba, ac=ca, cbc-1=b3 >

5C16
5C32
5C48
5C96

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

120 conjugacy classes

class 1  2 3A3B4A4B 5 6A6B8A8B8C8D 10 12A12B12C12D15A15B16A···16H20A20B24A···24H30A30B32A···32P40A40B40C40D48A···48P60A60B60C60D96A···96AF120A···120H
order12334456688881012121212151516···16202024···24303032···324040404048···486060606096···96120···120
size111111411111141111445···5441···1445···544445···544445···54···4

120 irreducible representations

dim11111111111144444444
type+++-
imageC1C2C3C4C6C8C12C16C24C32C48C96F5C5⋊C8C3×F5C5⋊C16C3×C5⋊C8C5⋊C32C3×C5⋊C16C3×C5⋊C32
kernelC3×C5⋊C32C3×C52C16C5⋊C32C120C52C16C60C40C30C20C15C10C5C24C12C8C6C4C3C2C1
# reps11222448816163211222448

Matrix representation of C3×C5⋊C32 in GL5(𝔽3361)

10000
02468000
00246800
00024680
00002468
,
10000
00003360
01003360
00103360
00013360
,
10660000
0228628412841687
020919712465612
0139089627491896
01674218010753077

G:=sub<GL(5,GF(3361))| [1,0,0,0,0,0,2468,0,0,0,0,0,2468,0,0,0,0,0,2468,0,0,0,0,0,2468],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,3360,3360,3360,3360],[1066,0,0,0,0,0,2286,209,1390,1674,0,284,1971,896,2180,0,1284,2465,2749,1075,0,1687,612,1896,3077] >;

C3×C5⋊C32 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes C_{32}
% in TeX

G:=Group("C3xC5:C32");
// GroupNames label

G:=SmallGroup(480,5);
// by ID

G=gap.SmallGroup(480,5);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-2,-5,42,58,80,102,9414,3156]);
// 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^3>;
// generators/relations

Export

Subgroup lattice of C3×C5⋊C32 in TeX

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