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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 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×14], C23, C10 [×3], C10 [×4], Dic3 [×6], C12 [×2], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×6], C20 [×2], C2×C10, C2×C10 [×6], C2×Dic3 [×2], C2×Dic3 [×8], C2×C12 [×4], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×Dic5 [×2], C2×Dic5 [×8], C2×C20 [×4], C22×C10, Dic3⋊C4 [×4], C22×Dic3, C22×Dic3, C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×4], C2×C30, C2×C30 [×6], C10.D4 [×4], C22×Dic5, C22×Dic5, C22×C20, C2×Dic3⋊C4, C6×Dic5 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C10×Dic3 [×2], C2×Dic15 [×6], C22×C30, C2×C10.D4, Dic155C4 [×4], C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C2×Dic155C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, S3×D5, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×Dic3⋊C4, D30.C2 [×2], C15⋊D4 [×2], C15⋊Q8 [×2], C2×S3×D5, C2×C10.D4, Dic155C4 [×4], C2×D30.C2, C2×C15⋊D4, C2×C15⋊Q8, C2×Dic155C4

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

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

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

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

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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