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## G = C2×Dic15⋊5C4order 480 = 25·3·5

### Direct product of C2 and Dic15⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×Dic15⋊5C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic15⋊5C4 — C2×Dic15⋊5C4
 Lower central C15 — C30 — C2×Dic15⋊5C4
 Upper central C1 — C23

Generators and relations for C2×Dic155C4
G = < a,b,c,d | a2=b30=d4=1, c2=b15, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b11, dcd-1=b15c >

Subgroups: 668 in 184 conjugacy classes, 92 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, Dic3⋊C4, C22×Dic3, C22×Dic3, C22×C12, C5×Dic3, C3×Dic5, Dic15, C2×C30, C2×C30, C10.D4, C22×Dic5, C22×Dic5, C22×C20, C2×Dic3⋊C4, C6×Dic5, C6×Dic5, C10×Dic3, C10×Dic3, C2×Dic15, C22×C30, C2×C10.D4, Dic155C4, C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C2×Dic155C4
Quotients:

Smallest permutation representation of C2×Dic155C4
Regular action on 480 points
Generators in S480
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(121 231)(122 232)(123 233)(124 234)(125 235)(126 236)(127 237)(128 238)(129 239)(130 240)(131 211)(132 212)(133 213)(134 214)(135 215)(136 216)(137 217)(138 218)(139 219)(140 220)(141 221)(142 222)(143 223)(144 224)(145 225)(146 226)(147 227)(148 228)(149 229)(150 230)(151 206)(152 207)(153 208)(154 209)(155 210)(156 181)(157 182)(158 183)(159 184)(160 185)(161 186)(162 187)(163 188)(164 189)(165 190)(166 191)(167 192)(168 193)(169 194)(170 195)(171 196)(172 197)(173 198)(174 199)(175 200)(176 201)(177 202)(178 203)(179 204)(180 205)(241 340)(242 341)(243 342)(244 343)(245 344)(246 345)(247 346)(248 347)(249 348)(250 349)(251 350)(252 351)(253 352)(254 353)(255 354)(256 355)(257 356)(258 357)(259 358)(260 359)(261 360)(262 331)(263 332)(264 333)(265 334)(266 335)(267 336)(268 337)(269 338)(270 339)(271 309)(272 310)(273 311)(274 312)(275 313)(276 314)(277 315)(278 316)(279 317)(280 318)(281 319)(282 320)(283 321)(284 322)(285 323)(286 324)(287 325)(288 326)(289 327)(290 328)(291 329)(292 330)(293 301)(294 302)(295 303)(296 304)(297 305)(298 306)(299 307)(300 308)(361 466)(362 467)(363 468)(364 469)(365 470)(366 471)(367 472)(368 473)(369 474)(370 475)(371 476)(372 477)(373 478)(374 479)(375 480)(376 451)(377 452)(378 453)(379 454)(380 455)(381 456)(382 457)(383 458)(384 459)(385 460)(386 461)(387 462)(388 463)(389 464)(390 465)(391 449)(392 450)(393 421)(394 422)(395 423)(396 424)(397 425)(398 426)(399 427)(400 428)(401 429)(402 430)(403 431)(404 432)(405 433)(406 434)(407 435)(408 436)(409 437)(410 438)(411 439)(412 440)(413 441)(414 442)(415 443)(416 444)(417 445)(418 446)(419 447)(420 448)
(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)
(1 376 16 361)(2 375 17 390)(3 374 18 389)(4 373 19 388)(5 372 20 387)(6 371 21 386)(7 370 22 385)(8 369 23 384)(9 368 24 383)(10 367 25 382)(11 366 26 381)(12 365 27 380)(13 364 28 379)(14 363 29 378)(15 362 30 377)(31 406 46 391)(32 405 47 420)(33 404 48 419)(34 403 49 418)(35 402 50 417)(36 401 51 416)(37 400 52 415)(38 399 53 414)(39 398 54 413)(40 397 55 412)(41 396 56 411)(42 395 57 410)(43 394 58 409)(44 393 59 408)(45 392 60 407)(61 434 76 449)(62 433 77 448)(63 432 78 447)(64 431 79 446)(65 430 80 445)(66 429 81 444)(67 428 82 443)(68 427 83 442)(69 426 84 441)(70 425 85 440)(71 424 86 439)(72 423 87 438)(73 422 88 437)(74 421 89 436)(75 450 90 435)(91 471 106 456)(92 470 107 455)(93 469 108 454)(94 468 109 453)(95 467 110 452)(96 466 111 451)(97 465 112 480)(98 464 113 479)(99 463 114 478)(100 462 115 477)(101 461 116 476)(102 460 117 475)(103 459 118 474)(104 458 119 473)(105 457 120 472)(121 273 136 288)(122 272 137 287)(123 271 138 286)(124 300 139 285)(125 299 140 284)(126 298 141 283)(127 297 142 282)(128 296 143 281)(129 295 144 280)(130 294 145 279)(131 293 146 278)(132 292 147 277)(133 291 148 276)(134 290 149 275)(135 289 150 274)(151 255 166 270)(152 254 167 269)(153 253 168 268)(154 252 169 267)(155 251 170 266)(156 250 171 265)(157 249 172 264)(158 248 173 263)(159 247 174 262)(160 246 175 261)(161 245 176 260)(162 244 177 259)(163 243 178 258)(164 242 179 257)(165 241 180 256)(181 349 196 334)(182 348 197 333)(183 347 198 332)(184 346 199 331)(185 345 200 360)(186 344 201 359)(187 343 202 358)(188 342 203 357)(189 341 204 356)(190 340 205 355)(191 339 206 354)(192 338 207 353)(193 337 208 352)(194 336 209 351)(195 335 210 350)(211 301 226 316)(212 330 227 315)(213 329 228 314)(214 328 229 313)(215 327 230 312)(216 326 231 311)(217 325 232 310)(218 324 233 309)(219 323 234 308)(220 322 235 307)(221 321 236 306)(222 320 237 305)(223 319 238 304)(224 318 239 303)(225 317 240 302)
(1 231 46 186)(2 212 47 197)(3 223 48 208)(4 234 49 189)(5 215 50 200)(6 226 51 181)(7 237 52 192)(8 218 53 203)(9 229 54 184)(10 240 55 195)(11 221 56 206)(12 232 57 187)(13 213 58 198)(14 224 59 209)(15 235 60 190)(16 216 31 201)(17 227 32 182)(18 238 33 193)(19 219 34 204)(20 230 35 185)(21 211 36 196)(22 222 37 207)(23 233 38 188)(24 214 39 199)(25 225 40 210)(26 236 41 191)(27 217 42 202)(28 228 43 183)(29 239 44 194)(30 220 45 205)(61 176 96 136)(62 157 97 147)(63 168 98 128)(64 179 99 139)(65 160 100 150)(66 171 101 131)(67 152 102 142)(68 163 103 123)(69 174 104 134)(70 155 105 145)(71 166 106 126)(72 177 107 137)(73 158 108 148)(74 169 109 129)(75 180 110 140)(76 161 111 121)(77 172 112 132)(78 153 113 143)(79 164 114 124)(80 175 115 135)(81 156 116 146)(82 167 117 127)(83 178 118 138)(84 159 119 149)(85 170 120 130)(86 151 91 141)(87 162 92 122)(88 173 93 133)(89 154 94 144)(90 165 95 125)(241 452 299 450)(242 463 300 431)(243 474 271 442)(244 455 272 423)(245 466 273 434)(246 477 274 445)(247 458 275 426)(248 469 276 437)(249 480 277 448)(250 461 278 429)(251 472 279 440)(252 453 280 421)(253 464 281 432)(254 475 282 443)(255 456 283 424)(256 467 284 435)(257 478 285 446)(258 459 286 427)(259 470 287 438)(260 451 288 449)(261 462 289 430)(262 473 290 441)(263 454 291 422)(264 465 292 433)(265 476 293 444)(266 457 294 425)(267 468 295 436)(268 479 296 447)(269 460 297 428)(270 471 298 439)(301 416 334 371)(302 397 335 382)(303 408 336 363)(304 419 337 374)(305 400 338 385)(306 411 339 366)(307 392 340 377)(308 403 341 388)(309 414 342 369)(310 395 343 380)(311 406 344 361)(312 417 345 372)(313 398 346 383)(314 409 347 364)(315 420 348 375)(316 401 349 386)(317 412 350 367)(318 393 351 378)(319 404 352 389)(320 415 353 370)(321 396 354 381)(322 407 355 362)(323 418 356 373)(324 399 357 384)(325 410 358 365)(326 391 359 376)(327 402 360 387)(328 413 331 368)(329 394 332 379)(330 405 333 390)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A ··· 6G 10A ··· 10N 12A ··· 12H 15A 15B 20A ··· 20P 30A ··· 30N order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 15 20 ··· 20 30 ··· 30 size 1 1 ··· 1 2 6 6 6 6 10 10 10 10 30 30 30 30 2 2 2 ··· 2 2 ··· 2 10 ··· 10 4 4 6 ··· 6 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + + + + - - + + - - + image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D5 D6 D6 D10 D10 Dic6 C4×S3 C3⋊D4 Dic10 C4×D5 C5⋊D4 S3×D5 D30.C2 C15⋊D4 C15⋊Q8 C2×S3×D5 kernel C2×Dic15⋊5C4 Dic15⋊5C4 C2×C6×Dic5 Dic3×C2×C10 C22×Dic15 C2×Dic15 C22×Dic5 C2×C30 C2×C30 C22×Dic3 C2×Dic5 C22×C10 C2×Dic3 C22×C6 C2×C10 C2×C10 C2×C10 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C22 # reps 1 4 1 1 1 8 1 2 2 2 2 1 4 2 4 4 4 8 8 8 2 4 4 4 2

Matrix representation of C2×Dic155C4 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 44 1 0 0 0 0 60 0 0 0 0 0 0 0 44 1 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 60 1
,
 24 53 0 0 0 0 34 37 0 0 0 0 0 0 37 8 0 0 0 0 27 24 0 0 0 0 0 0 18 52 0 0 0 0 9 43
,
 29 54 0 0 0 0 7 32 0 0 0 0 0 0 47 16 0 0 0 0 45 14 0 0 0 0 0 0 0 60 0 0 0 0 60 0

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[44,60,0,0,0,0,1,0,0,0,0,0,0,0,44,60,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,1,1],[24,34,0,0,0,0,53,37,0,0,0,0,0,0,37,27,0,0,0,0,8,24,0,0,0,0,0,0,18,9,0,0,0,0,52,43],[29,7,0,0,0,0,54,32,0,0,0,0,0,0,47,45,0,0,0,0,16,14,0,0,0,0,0,0,0,60,0,0,0,0,60,0] >;

C2×Dic155C4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{15}\rtimes_5C_4
% in TeX

G:=Group("C2xDic15:5C4");
// GroupNames label

G:=SmallGroup(480,620);
// by ID

G=gap.SmallGroup(480,620);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,64,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=d^4=1,c^2=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^15*c>;
// generators/relations

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