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G = C5×C23.32D6  order 480 = 25·3·5

Direct product of C5 and C23.32D6

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

Aliases: C5×C23.32D6, C30.39C42, (C2×C60)⋊16C4, (C2×C12)⋊3C20, C6.5(C4×C20), (C2×C20)⋊8Dic3, C30.61(C4⋊C4), (C2×C30).21Q8, (C2×Dic3)⋊2C20, (C2×C10).52D12, (C2×C30).122D4, (C22×C60).2C2, (C22×C20).6S3, C2.5(Dic3×C20), C10.58(D6⋊C4), (C10×Dic3)⋊12C4, C23.32(S3×C10), C22.12(S3×C20), (C22×C12).1C10, C10.29(C4×Dic3), (C2×C10).15Dic6, C22.11(C5×D12), C22.3(C5×Dic6), C10.22(C4⋊Dic3), C15⋊7(C2.C42), (C22×C10).147D6, C30.100(C22⋊C4), C10.26(Dic3⋊C4), (C22×Dic3).1C10, C22.10(C10×Dic3), C10.32(C6.D4), (C22×C30).169C22, C6.9(C5×C4⋊C4), C2.2(C5×D6⋊C4), C3⋊(C5×C2.C42), (C2×C6).4(C5×Q8), (C2×C4)⋊2(C5×Dic3), (C2×C6).32(C5×D4), C2.2(C5×C4⋊Dic3), (C2×C6).13(C2×C20), (C2×C10).84(C4×S3), C6.12(C5×C22⋊C4), C2.2(C5×Dic3⋊C4), (C22×C4).4(C5×S3), (Dic3×C2×C10).9C2, (C2×C30).158(C2×C4), C2.2(C5×C6.D4), C22.16(C5×C3⋊D4), (C2×C10).88(C3⋊D4), (C22×C6).31(C2×C10), (C2×C10).62(C2×Dic3), SmallGroup(480,150)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C5×C23.32D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C22×C30 — Dic3×C2×C10 — C5×C23.32D6
Lower central
C3 — C6 — C5×C23.32D6
Upper central
C1 — C22×C10 — C22×C20

Generators and relations for C5×C23.32D6
 G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=c, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 292 in 152 conjugacy classes, 90 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22 [×3], C22 [×4], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×10], C23, C10 [×3], C10 [×4], Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C15, C22×C4, C22×C4 [×2], C20 [×6], C2×C10 [×3], C2×C10 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×4], C2.C42, C2×C20 [×2], C2×C20 [×10], C22×C10, C22×Dic3 [×2], C22×C12, C5×Dic3 [×4], C60 [×2], C2×C30 [×3], C2×C30 [×4], C22×C20, C22×C20 [×2], C23.32D6, C10×Dic3 [×4], C10×Dic3 [×4], C2×C60 [×2], C2×C60 [×2], C22×C30, C5×C2.C42, Dic3×C2×C10 [×2], C22×C60, C5×C23.32D6
Quotients: C1, C2 [×3], C4 [×6], C22, C5, S3, C2×C4 [×3], D4 [×3], Q8, C10 [×3], Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C5×S3, C2.C42, C2×C20 [×3], C5×D4 [×3], C5×Q8, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C5×Dic3 [×2], S3×C10, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C23.32D6, C5×Dic6, S3×C20 [×2], C5×D12, C10×Dic3, C5×C3⋊D4 [×2], C5×C2.C42, Dic3×C20, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C5×D6⋊C4 [×2], C5×C6.D4, C5×C23.32D6

Smallest permutation representation of C5×C23.32D6
►Regular action on 480 points
Generators in S480
(1 87 192 359 374)(2 88 181 360 375)(3 89 182 349 376)(4 90 183 350 377)(5 91 184 351 378)(6 92 185 352 379)(7 93 186 353 380)(8 94 187 354 381)(9 95 188 355 382)(10 96 189 356 383)(11 85 190 357 384)(12 86 191 358 373)(13 142 221 328 167)(14 143 222 329 168)(15 144 223 330 157)(16 133 224 331 158)(17 134 225 332 159)(18 135 226 333 160)(19 136 227 334 161)(20 137 228 335 162)(21 138 217 336 163)(22 139 218 325 164)(23 140 219 326 165)(24 141 220 327 166)(25 174 368 283 433)(26 175 369 284 434)(27 176 370 285 435)(28 177 371 286 436)(29 178 372 287 437)(30 179 361 288 438)(31 180 362 277 439)(32 169 363 278 440)(33 170 364 279 441)(34 171 365 280 442)(35 172 366 281 443)(36 173 367 282 444)(37 105 315 296 431)(38 106 316 297 432)(39 107 317 298 421)(40 108 318 299 422)(41 97 319 300 423)(42 98 320 289 424)(43 99 321 290 425)(44 100 322 291 426)(45 101 323 292 427)(46 102 324 293 428)(47 103 313 294 429)(48 104 314 295 430)(49 78 458 272 414)(50 79 459 273 415)(51 80 460 274 416)(52 81 461 275 417)(53 82 462 276 418)(54 83 463 265 419)(55 84 464 266 420)(56 73 465 267 409)(57 74 466 268 410)(58 75 467 269 411)(59 76 468 270 412)(60 77 457 271 413)(61 345 124 238 148)(62 346 125 239 149)(63 347 126 240 150)(64 348 127 229 151)(65 337 128 230 152)(66 338 129 231 153)(67 339 130 232 154)(68 340 131 233 155)(69 341 132 234 156)(70 342 121 235 145)(71 343 122 236 146)(72 344 123 237 147)(109 200 400 450 244)(110 201 401 451 245)(111 202 402 452 246)(112 203 403 453 247)(113 204 404 454 248)(114 193 405 455 249)(115 194 406 456 250)(116 195 407 445 251)(117 196 408 446 252)(118 197 397 447 241)(119 198 398 448 242)(120 199 399 449 243)(205 390 257 472 311)(206 391 258 473 312)(207 392 259 474 301)(208 393 260 475 302)(209 394 261 476 303)(210 395 262 477 304)(211 396 263 478 305)(212 385 264 479 306)(213 386 253 480 307)(214 387 254 469 308)(215 388 255 470 309)(216 389 256 471 310)

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

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

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

(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)

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

180 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L5A5B5C5D6A···6G10A···10AB12A···12H15A15B15C15D20A···20P20Q···20AV30A···30AB60A···60AF
order12···2344444···455556···610···1012···121515151520···2020···2030···3060···60
size11···1222226···611112···21···12···222222···26···62···22···2

180 irreducible representations

dim1111111111222222222222222222
type+++++--+-+
imageC1C2C2C4C4C5C10C10C20C20S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4C5×S3C5×D4C5×Q8C5×Dic3S3×C10C5×Dic6S3×C20C5×D12C5×C3⋊D4
kernelC5×C23.32D6Dic3×C2×C10C22×C60C10×Dic3C2×C60C23.32D6C22×Dic3C22×C12C2×Dic3C2×C12C22×C20C2×C30C2×C30C2×C20C22×C10C2×C10C2×C10C2×C10C2×C10C22×C4C2×C6C2×C6C2×C4C23C22C22C22C22
# reps121844843216131212424412484816816

Matrix representation of C5×C23.32D6 ►in GL5(𝔽61)

10000
09000
00900
000200
000020
,
600000
060000
006000
00010
00001
,
600000
01000
00100
00010
00001
,
10000
060000
006000
000600
000060
,
500000
00100
01000
0005252
000943
,
600000
0483000
0311300
000011
000110

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,20,0,0,0,0,0,20],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[50,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,52,9,0,0,0,52,43],[60,0,0,0,0,0,48,31,0,0,0,30,13,0,0,0,0,0,0,11,0,0,0,11,0] >;
 

C5×C23.32D6 in GAP, Magma, Sage, TeX 

C_5\times C_2^3._{32}D_6
 
% in TeX
 
G:=Group("C5xC2^3.32D6");
 
// GroupNames label
 
G:=SmallGroup(480,150);
 
// by ID
 
G=gap.SmallGroup(480,150);
 
# by ID
 
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,1149,288,15686]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^6=c,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^5>;
 
// generators/relations
 

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