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G = C5×Dic6⋊C4  order 480 = 25·3·5

Direct product of C5 and Dic6⋊C4

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

Aliases: C5×Dic6⋊C4, Dic6⋊5C20, C3⋊2(Q8×C20), C15⋊20(C4×Q8), C4.4(S3×C20), C20.79(C4×S3), Dic3⋊3(C5×Q8), C6.10(Q8×C10), C10.47(S3×Q8), C60.180(C2×C4), C12.10(C2×C20), (C5×Dic6)⋊17C4, (C5×Dic3)⋊10Q8, (C2×C20).353D6, C6.8(C22×C20), C30.108(C2×Q8), Dic3⋊C4.5C10, Dic3.2(C2×C20), (C2×Dic6).7C10, (C4×Dic3).1C10, C30.248(C4○D4), (C2×C60).350C22, C30.199(C22×C4), (C2×C30).407C23, (Dic3×C20).10C2, (C10×Dic6).17C2, C10.113(D4⋊2S3), (C10×Dic3).239C22, C2.1(C5×S3×Q8), C4⋊C4.7(C5×S3), C2.10(S3×C2×C20), (C3×C4⋊C4).4C10, (C5×C4⋊C4).14S3, C10.135(S3×C2×C4), C6.23(C5×C4○D4), (C15×C4⋊C4).18C2, (C2×C4).41(S3×C10), C2.3(C5×D4⋊2S3), C22.15(S3×C2×C10), (C2×C12).56(C2×C10), (C5×Dic3⋊C4).14C2, (C2×C6).28(C22×C10), (C5×Dic3).38(C2×C4), (C2×C10).341(C22×S3), (C2×Dic3).23(C2×C10), SmallGroup(480,766)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C5×Dic6⋊C4
Chief series
C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C20 — C5×Dic6⋊C4
Lower central
C3 — C6 — C5×Dic6⋊C4
Upper central
C1 — C2×C10 — C5×C4⋊C4

Generators and relations for C5×Dic6⋊C4
 G = < a,b,c,d | a5=b12=d4=1, c2=b6, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b7, cd=dc >

Subgroups: 244 in 140 conjugacy classes, 86 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, C2×Q8, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4×Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, C5×Dic3, C5×Dic3, C60, C60, C2×C30, C4×C20, C5×C4⋊C4, C5×C4⋊C4, Q8×C10, Dic6⋊C4, C5×Dic6, C10×Dic3, C10×Dic3, C2×C60, C2×C60, Q8×C20, Dic3×C20, Dic3×C20, C5×Dic3⋊C4, C15×C4⋊C4, C10×Dic6, C5×Dic6⋊C4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, Q8, C23, C10, D6, C22×C4, C2×Q8, C4○D4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C4×Q8, C2×C20, C5×Q8, C22×C10, S3×C2×C4, D4⋊2S3, S3×Q8, S3×C10, C22×C20, Q8×C10, C5×C4○D4, Dic6⋊C4, S3×C20, S3×C2×C10, Q8×C20, S3×C2×C20, C5×D4⋊2S3, C5×S3×Q8, C5×Dic6⋊C4

Smallest permutation representation of C5×Dic6⋊C4
►Regular action on 480 points
Generators in S480
(1 336 443 131 83)(2 325 444 132 84)(3 326 433 121 73)(4 327 434 122 74)(5 328 435 123 75)(6 329 436 124 76)(7 330 437 125 77)(8 331 438 126 78)(9 332 439 127 79)(10 333 440 128 80)(11 334 441 129 81)(12 335 442 130 82)(13 115 339 205 450)(14 116 340 206 451)(15 117 341 207 452)(16 118 342 208 453)(17 119 343 209 454)(18 120 344 210 455)(19 109 345 211 456)(20 110 346 212 445)(21 111 347 213 446)(22 112 348 214 447)(23 113 337 215 448)(24 114 338 216 449)(25 465 303 62 293)(26 466 304 63 294)(27 467 305 64 295)(28 468 306 65 296)(29 457 307 66 297)(30 458 308 67 298)(31 459 309 68 299)(32 460 310 69 300)(33 461 311 70 289)(34 462 312 71 290)(35 463 301 72 291)(36 464 302 61 292)(37 142 406 377 419)(38 143 407 378 420)(39 144 408 379 409)(40 133 397 380 410)(41 134 398 381 411)(42 135 399 382 412)(43 136 400 383 413)(44 137 401 384 414)(45 138 402 373 415)(46 139 403 374 416)(47 140 404 375 417)(48 141 405 376 418)(49 281 271 390 245)(50 282 272 391 246)(51 283 273 392 247)(52 284 274 393 248)(53 285 275 394 249)(54 286 276 395 250)(55 287 265 396 251)(56 288 266 385 252)(57 277 267 386 241)(58 278 268 387 242)(59 279 269 388 243)(60 280 270 389 244)(85 356 365 160 262)(86 357 366 161 263)(87 358 367 162 264)(88 359 368 163 253)(89 360 369 164 254)(90 349 370 165 255)(91 350 371 166 256)(92 351 372 167 257)(93 352 361 168 258)(94 353 362 157 259)(95 354 363 158 260)(96 355 364 159 261)(97 222 183 197 478)(98 223 184 198 479)(99 224 185 199 480)(100 225 186 200 469)(101 226 187 201 470)(102 227 188 202 471)(103 228 189 203 472)(104 217 190 204 473)(105 218 191 193 474)(106 219 192 194 475)(107 220 181 195 476)(108 221 182 196 477)(145 430 229 313 176)(146 431 230 314 177)(147 432 231 315 178)(148 421 232 316 179)(149 422 233 317 180)(150 423 234 318 169)(151 424 235 319 170)(152 425 236 320 171)(153 426 237 321 172)(154 427 238 322 173)(155 428 239 323 174)(156 429 240 324 175)

(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)

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

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

150 conjugacy classes

class 1 2A2B2C 3 4A···4F4G4H4I4J4K···4P5A5B5C5D6A6B6C10A···10L12A···12F15A15B15C15D20A···20X20Y···20AN20AO···20BL30A···30L60A···60X
order122234···444444···4555566610···1012···121515151520···2020···2020···2030···3060···60
size111122···233336···611112221···14···422222···23···36···62···24···4

150 irreducible representations

dim11111111111122222222224444
type++++++-+--
imageC1C2C2C2C2C4C5C10C10C10C10C20S3Q8D6C4○D4C4×S3C5×S3C5×Q8S3×C10C5×C4○D4S3×C20D4⋊2S3S3×Q8C5×D4⋊2S3C5×S3×Q8
kernelC5×Dic6⋊C4Dic3×C20C5×Dic3⋊C4C15×C4⋊C4C10×Dic6C5×Dic6Dic6⋊C4C4×Dic3Dic3⋊C4C3×C4⋊C4C2×Dic6Dic6C5×C4⋊C4C5×Dic3C2×C20C30C20C4⋊C4Dic3C2×C4C6C4C10C10C2C2
# reps132118412844321232448128161144

Matrix representation of C5×Dic6⋊C4 ►in GL5(𝔽61)

10000
020000
002000
000200
000020
,
600000
0384300
0432300
000060
000160
,
600000
00100
060000
0003012
0004231
,
110000
00100
060000
000600
000060

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,20],[60,0,0,0,0,0,38,43,0,0,0,43,23,0,0,0,0,0,0,1,0,0,0,60,60],[60,0,0,0,0,0,0,60,0,0,0,1,0,0,0,0,0,0,30,42,0,0,0,12,31],[11,0,0,0,0,0,0,60,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60] >;
 

C5×Dic6⋊C4 in GAP, Magma, Sage, TeX 

C_5\times {\rm Dic}_6\rtimes C_4
 
% in TeX
 
G:=Group("C5xDic6:C4");
 
// GroupNames label
 
G:=SmallGroup(480,766);
 
// by ID
 
G=gap.SmallGroup(480,766);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,288,471,646,15686]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^12=d^4=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^7,c*d=d*c>;
 
// generators/relations
 

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