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G = C5×C3⋊Q32order 480 = 25·3·5

Direct product of C5 and C3⋊Q32

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C3⋊Q32, C159Q32, C40.59D6, C30.56D8, C60.139D4, C120.66C22, Dic12.2C10, C3⋊C16.C10, C32(C5×Q32), Q16.(C5×S3), C8.7(S3×C10), C6.11(C5×D8), C12.6(C5×D4), C24.5(C2×C10), (C5×Q16).2S3, (C15×Q16).3C2, (C3×Q16).1C10, C10.27(D4⋊S3), C20.69(C3⋊D4), (C5×Dic12).4C2, (C5×C3⋊C16).2C2, C2.7(C5×D4⋊S3), C4.4(C5×C3⋊D4), SmallGroup(480,148)

Series: Derived Chief Lower central Upper central

C1C24 — C5×C3⋊Q32
C1C3C6C12C24C120C5×Dic12 — C5×C3⋊Q32
C3C6C12C24 — C5×C3⋊Q32
C1C10C20C40C5×Q16

Generators and relations for C5×C3⋊Q32
 G = < a,b,c,d | a5=b3=c16=1, d2=c8, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

4C4
12C4
2Q8
6Q8
4C12
4Dic3
4C20
12C20
3Q16
3C16
2C3×Q8
2Dic6
2C5×Q8
6C5×Q8
4C60
4C5×Dic3
3Q32
3C80
3C5×Q16
2C5×Dic6
2Q8×C15
3C5×Q32

Smallest permutation representation of C5×C3⋊Q32
Regular action on 480 points
Generators in S480
(1 344 354 377 60)(2 345 355 378 61)(3 346 356 379 62)(4 347 357 380 63)(5 348 358 381 64)(6 349 359 382 49)(7 350 360 383 50)(8 351 361 384 51)(9 352 362 369 52)(10 337 363 370 53)(11 338 364 371 54)(12 339 365 372 55)(13 340 366 373 56)(14 341 367 374 57)(15 342 368 375 58)(16 343 353 376 59)(17 406 434 199 236)(18 407 435 200 237)(19 408 436 201 238)(20 409 437 202 239)(21 410 438 203 240)(22 411 439 204 225)(23 412 440 205 226)(24 413 441 206 227)(25 414 442 207 228)(26 415 443 208 229)(27 416 444 193 230)(28 401 445 194 231)(29 402 446 195 232)(30 403 447 196 233)(31 404 448 197 234)(32 405 433 198 235)(33 212 172 427 327)(34 213 173 428 328)(35 214 174 429 329)(36 215 175 430 330)(37 216 176 431 331)(38 217 161 432 332)(39 218 162 417 333)(40 219 163 418 334)(41 220 164 419 335)(42 221 165 420 336)(43 222 166 421 321)(44 223 167 422 322)(45 224 168 423 323)(46 209 169 424 324)(47 210 170 425 325)(48 211 171 426 326)(65 273 147 475 116)(66 274 148 476 117)(67 275 149 477 118)(68 276 150 478 119)(69 277 151 479 120)(70 278 152 480 121)(71 279 153 465 122)(72 280 154 466 123)(73 281 155 467 124)(74 282 156 468 125)(75 283 157 469 126)(76 284 158 470 127)(77 285 159 471 128)(78 286 160 472 113)(79 287 145 473 114)(80 288 146 474 115)(81 464 317 262 251)(82 449 318 263 252)(83 450 319 264 253)(84 451 320 265 254)(85 452 305 266 255)(86 453 306 267 256)(87 454 307 268 241)(88 455 308 269 242)(89 456 309 270 243)(90 457 310 271 244)(91 458 311 272 245)(92 459 312 257 246)(93 460 313 258 247)(94 461 314 259 248)(95 462 315 260 249)(96 463 316 261 250)(97 386 179 136 293)(98 387 180 137 294)(99 388 181 138 295)(100 389 182 139 296)(101 390 183 140 297)(102 391 184 141 298)(103 392 185 142 299)(104 393 186 143 300)(105 394 187 144 301)(106 395 188 129 302)(107 396 189 130 303)(108 397 190 131 304)(109 398 191 132 289)(110 399 192 133 290)(111 400 177 134 291)(112 385 178 135 292)
(1 215 112)(2 97 216)(3 217 98)(4 99 218)(5 219 100)(6 101 220)(7 221 102)(8 103 222)(9 223 104)(10 105 224)(11 209 106)(12 107 210)(13 211 108)(14 109 212)(15 213 110)(16 111 214)(17 152 88)(18 89 153)(19 154 90)(20 91 155)(21 156 92)(22 93 157)(23 158 94)(24 95 159)(25 160 96)(26 81 145)(27 146 82)(28 83 147)(29 148 84)(30 85 149)(31 150 86)(32 87 151)(33 57 289)(34 290 58)(35 59 291)(36 292 60)(37 61 293)(38 294 62)(39 63 295)(40 296 64)(41 49 297)(42 298 50)(43 51 299)(44 300 52)(45 53 301)(46 302 54)(47 55 303)(48 304 56)(65 194 264)(66 265 195)(67 196 266)(68 267 197)(69 198 268)(70 269 199)(71 200 270)(72 271 201)(73 202 272)(74 257 203)(75 204 258)(76 259 205)(77 206 260)(78 261 207)(79 208 262)(80 263 193)(113 316 442)(114 443 317)(115 318 444)(116 445 319)(117 320 446)(118 447 305)(119 306 448)(120 433 307)(121 308 434)(122 435 309)(123 310 436)(124 437 311)(125 312 438)(126 439 313)(127 314 440)(128 441 315)(129 371 324)(130 325 372)(131 373 326)(132 327 374)(133 375 328)(134 329 376)(135 377 330)(136 331 378)(137 379 332)(138 333 380)(139 381 334)(140 335 382)(141 383 336)(142 321 384)(143 369 322)(144 323 370)(161 387 346)(162 347 388)(163 389 348)(164 349 390)(165 391 350)(166 351 392)(167 393 352)(168 337 394)(169 395 338)(170 339 396)(171 397 340)(172 341 398)(173 399 342)(174 343 400)(175 385 344)(176 345 386)(177 429 353)(178 354 430)(179 431 355)(180 356 432)(181 417 357)(182 358 418)(183 419 359)(184 360 420)(185 421 361)(186 362 422)(187 423 363)(188 364 424)(189 425 365)(190 366 426)(191 427 367)(192 368 428)(225 247 283)(226 284 248)(227 249 285)(228 286 250)(229 251 287)(230 288 252)(231 253 273)(232 274 254)(233 255 275)(234 276 256)(235 241 277)(236 278 242)(237 243 279)(238 280 244)(239 245 281)(240 282 246)(401 450 475)(402 476 451)(403 452 477)(404 478 453)(405 454 479)(406 480 455)(407 456 465)(408 466 457)(409 458 467)(410 468 459)(411 460 469)(412 470 461)(413 462 471)(414 472 463)(415 464 473)(416 474 449)
(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)
(1 24 9 32)(2 23 10 31)(3 22 11 30)(4 21 12 29)(5 20 13 28)(6 19 14 27)(7 18 15 26)(8 17 16 25)(33 252 41 244)(34 251 42 243)(35 250 43 242)(36 249 44 241)(37 248 45 256)(38 247 46 255)(39 246 47 254)(40 245 48 253)(49 238 57 230)(50 237 58 229)(51 236 59 228)(52 235 60 227)(53 234 61 226)(54 233 62 225)(55 232 63 240)(56 231 64 239)(65 139 73 131)(66 138 74 130)(67 137 75 129)(68 136 76 144)(69 135 77 143)(70 134 78 142)(71 133 79 141)(72 132 80 140)(81 221 89 213)(82 220 90 212)(83 219 91 211)(84 218 92 210)(85 217 93 209)(86 216 94 224)(87 215 95 223)(88 214 96 222)(97 158 105 150)(98 157 106 149)(99 156 107 148)(100 155 108 147)(101 154 109 146)(102 153 110 145)(103 152 111 160)(104 151 112 159)(113 185 121 177)(114 184 122 192)(115 183 123 191)(116 182 124 190)(117 181 125 189)(118 180 126 188)(119 179 127 187)(120 178 128 186)(161 460 169 452)(162 459 170 451)(163 458 171 450)(164 457 172 449)(165 456 173 464)(166 455 174 463)(167 454 175 462)(168 453 176 461)(193 382 201 374)(194 381 202 373)(195 380 203 372)(196 379 204 371)(197 378 205 370)(198 377 206 369)(199 376 207 384)(200 375 208 383)(257 325 265 333)(258 324 266 332)(259 323 267 331)(260 322 268 330)(261 321 269 329)(262 336 270 328)(263 335 271 327)(264 334 272 326)(273 296 281 304)(274 295 282 303)(275 294 283 302)(276 293 284 301)(277 292 285 300)(278 291 286 299)(279 290 287 298)(280 289 288 297)(305 432 313 424)(306 431 314 423)(307 430 315 422)(308 429 316 421)(309 428 317 420)(310 427 318 419)(311 426 319 418)(312 425 320 417)(337 404 345 412)(338 403 346 411)(339 402 347 410)(340 401 348 409)(341 416 349 408)(342 415 350 407)(343 414 351 406)(344 413 352 405)(353 442 361 434)(354 441 362 433)(355 440 363 448)(356 439 364 447)(357 438 365 446)(358 437 366 445)(359 436 367 444)(360 435 368 443)(385 471 393 479)(386 470 394 478)(387 469 395 477)(388 468 396 476)(389 467 397 475)(390 466 398 474)(391 465 399 473)(392 480 400 472)

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

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

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

90 conjugacy classes

class 1  2  3 4A4B4C5A5B5C5D 6 8A8B10A10B10C10D12A12B12C15A15B15C15D16A16B16C16D20A20B20C20D20E20F20G20H20I20J20K20L24A24B30A30B30C30D40A···40H60A60B60C60D60E···60L80A···80P120A···120H
order123444555568810101010121212151515151616161620202020202020202020202024243030303040···406060606060···6080···80120···120
size1122824111122211114882222666622228888242424244422222···244448···86···64···4

90 irreducible representations

dim111111112222222222224444
type++++++++-+-
imageC1C2C2C2C5C10C10C10S3D4D6D8C3⋊D4C5×S3Q32C5×D4S3×C10C5×D8C5×C3⋊D4C5×Q32D4⋊S3C3⋊Q32C5×D4⋊S3C5×C3⋊Q32
kernelC5×C3⋊Q32C5×C3⋊C16C5×Dic12C15×Q16C3⋊Q32C3⋊C16Dic12C3×Q16C5×Q16C60C40C30C20Q16C15C12C8C6C4C3C10C5C2C1
# reps1111444411122444488161248

Matrix representation of C5×C3⋊Q32 in GL6(𝔽241)

9800000
0980000
001000
000100
000010
000001
,
100000
010000
001000
000100
00000240
00001240
,
112300000
11110000
001125400
002145800
00007189
0000160170
,
1381130000
1131030000
0020416500
002403700
00002400
00000240

G:=sub<GL(6,GF(241))| [98,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[11,11,0,0,0,0,230,11,0,0,0,0,0,0,112,214,0,0,0,0,54,58,0,0,0,0,0,0,71,160,0,0,0,0,89,170],[138,113,0,0,0,0,113,103,0,0,0,0,0,0,204,240,0,0,0,0,165,37,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;

C5×C3⋊Q32 in GAP, Magma, Sage, TeX

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

G:=Group("C5xC3:Q32");
// GroupNames label

G:=SmallGroup(480,148);
// by ID

G=gap.SmallGroup(480,148);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,560,309,568,1683,850,192,4204,2111,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of C5×C3⋊Q32 in TeX

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