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## G = C4⋊Dic30order 480 = 25·3·5

### The semidirect product of C4 and Dic30 acting via Dic30/Dic15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C4⋊Dic30
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C4×Dic15 — C4⋊Dic30
 Lower central C15 — C2×C30 — C4⋊Dic30
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊Dic30
G = < a,b,c | a4=b60=1, c2=b30, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 708 in 136 conjugacy classes, 59 normal (33 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×8], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10 [×3], Dic3 [×6], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4, C4⋊C4 [×3], C2×Q8 [×2], Dic5 [×6], C20 [×2], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C30 [×3], C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, C2×Dic6 [×2], Dic15 [×4], Dic15 [×2], C60 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C2×Dic10 [×2], C12⋊Q8, Dic30 [×4], C2×Dic15 [×2], C2×Dic15 [×2], C2×C60, C2×C60 [×2], C20⋊Q8, C4×Dic15, C30.4Q8 [×2], C605C4, C15×C4⋊C4, C2×Dic30 [×2], C4⋊Dic30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], Dic6 [×2], C22×S3, D15, C4⋊Q8, Dic10 [×2], C22×D5, C2×Dic6, S3×D4, S3×Q8, D30 [×3], C2×Dic10, D4×D5, Q8×D5, C12⋊Q8, Dic30 [×2], C22×D15, C20⋊Q8, C2×Dic30, D4×D15, Q8×D15, C4⋊Dic30

Smallest permutation representation of C4⋊Dic30
Regular action on 480 points
Generators in S480
(1 180 67 408)(2 409 68 121)(3 122 69 410)(4 411 70 123)(5 124 71 412)(6 413 72 125)(7 126 73 414)(8 415 74 127)(9 128 75 416)(10 417 76 129)(11 130 77 418)(12 419 78 131)(13 132 79 420)(14 361 80 133)(15 134 81 362)(16 363 82 135)(17 136 83 364)(18 365 84 137)(19 138 85 366)(20 367 86 139)(21 140 87 368)(22 369 88 141)(23 142 89 370)(24 371 90 143)(25 144 91 372)(26 373 92 145)(27 146 93 374)(28 375 94 147)(29 148 95 376)(30 377 96 149)(31 150 97 378)(32 379 98 151)(33 152 99 380)(34 381 100 153)(35 154 101 382)(36 383 102 155)(37 156 103 384)(38 385 104 157)(39 158 105 386)(40 387 106 159)(41 160 107 388)(42 389 108 161)(43 162 109 390)(44 391 110 163)(45 164 111 392)(46 393 112 165)(47 166 113 394)(48 395 114 167)(49 168 115 396)(50 397 116 169)(51 170 117 398)(52 399 118 171)(53 172 119 400)(54 401 120 173)(55 174 61 402)(56 403 62 175)(57 176 63 404)(58 405 64 177)(59 178 65 406)(60 407 66 179)(181 328 472 270)(182 271 473 329)(183 330 474 272)(184 273 475 331)(185 332 476 274)(186 275 477 333)(187 334 478 276)(188 277 479 335)(189 336 480 278)(190 279 421 337)(191 338 422 280)(192 281 423 339)(193 340 424 282)(194 283 425 341)(195 342 426 284)(196 285 427 343)(197 344 428 286)(198 287 429 345)(199 346 430 288)(200 289 431 347)(201 348 432 290)(202 291 433 349)(203 350 434 292)(204 293 435 351)(205 352 436 294)(206 295 437 353)(207 354 438 296)(208 297 439 355)(209 356 440 298)(210 299 441 357)(211 358 442 300)(212 241 443 359)(213 360 444 242)(214 243 445 301)(215 302 446 244)(216 245 447 303)(217 304 448 246)(218 247 449 305)(219 306 450 248)(220 249 451 307)(221 308 452 250)(222 251 453 309)(223 310 454 252)(224 253 455 311)(225 312 456 254)(226 255 457 313)(227 314 458 256)(228 257 459 315)(229 316 460 258)(230 259 461 317)(231 318 462 260)(232 261 463 319)(233 320 464 262)(234 263 465 321)(235 322 466 264)(236 265 467 323)(237 324 468 266)(238 267 469 325)(239 326 470 268)(240 269 471 327)
(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)
(1 433 31 463)(2 432 32 462)(3 431 33 461)(4 430 34 460)(5 429 35 459)(6 428 36 458)(7 427 37 457)(8 426 38 456)(9 425 39 455)(10 424 40 454)(11 423 41 453)(12 422 42 452)(13 421 43 451)(14 480 44 450)(15 479 45 449)(16 478 46 448)(17 477 47 447)(18 476 48 446)(19 475 49 445)(20 474 50 444)(21 473 51 443)(22 472 52 442)(23 471 53 441)(24 470 54 440)(25 469 55 439)(26 468 56 438)(27 467 57 437)(28 466 58 436)(29 465 59 435)(30 464 60 434)(61 208 91 238)(62 207 92 237)(63 206 93 236)(64 205 94 235)(65 204 95 234)(66 203 96 233)(67 202 97 232)(68 201 98 231)(69 200 99 230)(70 199 100 229)(71 198 101 228)(72 197 102 227)(73 196 103 226)(74 195 104 225)(75 194 105 224)(76 193 106 223)(77 192 107 222)(78 191 108 221)(79 190 109 220)(80 189 110 219)(81 188 111 218)(82 187 112 217)(83 186 113 216)(84 185 114 215)(85 184 115 214)(86 183 116 213)(87 182 117 212)(88 181 118 211)(89 240 119 210)(90 239 120 209)(121 348 151 318)(122 347 152 317)(123 346 153 316)(124 345 154 315)(125 344 155 314)(126 343 156 313)(127 342 157 312)(128 341 158 311)(129 340 159 310)(130 339 160 309)(131 338 161 308)(132 337 162 307)(133 336 163 306)(134 335 164 305)(135 334 165 304)(136 333 166 303)(137 332 167 302)(138 331 168 301)(139 330 169 360)(140 329 170 359)(141 328 171 358)(142 327 172 357)(143 326 173 356)(144 325 174 355)(145 324 175 354)(146 323 176 353)(147 322 177 352)(148 321 178 351)(149 320 179 350)(150 319 180 349)(241 368 271 398)(242 367 272 397)(243 366 273 396)(244 365 274 395)(245 364 275 394)(246 363 276 393)(247 362 277 392)(248 361 278 391)(249 420 279 390)(250 419 280 389)(251 418 281 388)(252 417 282 387)(253 416 283 386)(254 415 284 385)(255 414 285 384)(256 413 286 383)(257 412 287 382)(258 411 288 381)(259 410 289 380)(260 409 290 379)(261 408 291 378)(262 407 292 377)(263 406 293 376)(264 405 294 375)(265 404 295 374)(266 403 296 373)(267 402 297 372)(268 401 298 371)(269 400 299 370)(270 399 300 369)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 10A ··· 10F 12A ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 4 4 30 30 30 30 60 60 2 2 2 2 2 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - - + + + - + - + - + - + - + - image C1 C2 C2 C2 C2 C2 S3 D4 Q8 Q8 D5 D6 D10 Dic6 D15 Dic10 D30 Dic30 S3×D4 S3×Q8 D4×D5 Q8×D5 D4×D15 Q8×D15 kernel C4⋊Dic30 C4×Dic15 C30.4Q8 C60⋊5C4 C15×C4⋊C4 C2×Dic30 C5×C4⋊C4 Dic15 Dic15 C60 C3×C4⋊C4 C2×C20 C2×C12 C20 C4⋊C4 C12 C2×C4 C4 C10 C10 C6 C6 C2 C2 # reps 1 1 2 1 1 2 1 2 2 2 2 3 6 4 4 8 12 16 1 1 2 2 4 4

Matrix representation of C4⋊Dic30 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 4 9 0 0 0 0 32 57
,
 57 25 0 0 0 0 36 34 0 0 0 0 0 0 31 25 0 0 0 0 36 33 0 0 0 0 0 0 60 46 0 0 0 0 0 1
,
 29 56 0 0 0 0 22 32 0 0 0 0 0 0 20 8 0 0 0 0 34 41 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,32,0,0,0,0,9,57],[57,36,0,0,0,0,25,34,0,0,0,0,0,0,31,36,0,0,0,0,25,33,0,0,0,0,0,0,60,0,0,0,0,0,46,1],[29,22,0,0,0,0,56,32,0,0,0,0,0,0,20,34,0,0,0,0,8,41,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C4⋊Dic30 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_{30}
% in TeX

G:=Group("C4:Dic30");
// GroupNames label

G:=SmallGroup(480,853);
// by ID

G=gap.SmallGroup(480,853);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,254,219,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^4=b^60=1,c^2=b^30,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
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