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## G = C4.Dic30order 480 = 25·3·5

### 3rd non-split extension by C4 of 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=a2b-1 >

Subgroups: 516 in 112 conjugacy classes, 55 normal (33 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×6], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C10 [×3], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4, C4⋊C4 [×5], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×Dic3 [×4], C2×C12, C2×C12 [×2], C30 [×3], C42.C2, C2×Dic5 [×4], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3 [×3], C3×C4⋊C4, Dic15 [×4], C60 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5 [×3], C5×C4⋊C4, C4.Dic6, C2×Dic15 [×2], C2×Dic15 [×2], C2×C60, C2×C60 [×2], C4.Dic10, C4×Dic15, C30.4Q8 [×2], C605C4, C605C4 [×2], C15×C4⋊C4, C4.Dic30
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, C4○D4 [×2], D10 [×3], Dic6 [×2], C22×S3, D15, C42.C2, Dic10 [×2], C22×D5, C2×Dic6, D42S3, Q83S3, D30 [×3], C2×Dic10, D42D5, Q82D5, C4.Dic6, Dic30 [×2], C22×D15, C4.Dic10, C2×Dic30, D42D15, Q83D15, C4.Dic30

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

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

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

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

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

Matrix representation of C4.Dic30 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 24 9 0 0 24 37
,
 55 49 0 0 45 39 0 0 0 0 1 59 0 0 0 60
,
 52 20 0 0 2 9 0 0 0 0 20 38 0 0 20 41
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,24,24,0,0,9,37],[55,45,0,0,49,39,0,0,0,0,1,0,0,0,59,60],[52,2,0,0,20,9,0,0,0,0,20,20,0,0,38,41] >;

C4.Dic30 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,855);
// by ID

G=gap.SmallGroup(480,855);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,590,219,58,2693,18822]);
// 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=a^2*b^-1>;
// generators/relations

׿
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