metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.3Q8, C4.3Dic30, C20.9Dic6, C12.9Dic10, C4⋊C4.6D15, (C2×C4).41D30, C30.69(C2×Q8), C60⋊5C4.9C2, (C2×C20).137D6, (C2×C12).37D10, C2.8(C2×Dic30), C3⋊5(C4.Dic10), C5⋊5(C4.Dic6), (C2×C60).19C22, (C4×Dic15).2C2, C6.37(C2×Dic10), C10.37(C2×Dic6), C15⋊17(C42.C2), C30.4Q8.4C2, C30.219(C4○D4), C2.4(Q8⋊3D15), C6.97(D4⋊2D5), (C2×C30).287C23, C6.38(Q8⋊2D5), C10.97(D4⋊2S3), C2.12(D4⋊2D15), C10.38(Q8⋊3S3), C22.48(C22×D15), (C2×Dic15).10C22, (C5×C4⋊C4).7S3, (C3×C4⋊C4).7D5, (C15×C4⋊C4).7C2, (C2×C6).283(C22×D5), (C2×C10).282(C22×S3), SmallGroup(480,855)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, Dic5, C20, C20, C2×C10, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, Dic15, C60, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C4.Dic6, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C4.Dic10, C4×Dic15, C30.4Q8, C60⋊5C4, C60⋊5C4, C15×C4⋊C4, C4.Dic30
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, Dic6, C22×S3, D15, C42.C2, Dic10, C22×D5, C2×Dic6, D4⋊2S3, Q8⋊3S3, D30, C2×Dic10, D4⋊2D5, Q8⋊2D5, C4.Dic6, Dic30, C22×D15, C4.Dic10, C2×Dic30, D4⋊2D15, Q8⋊3D15, C4.Dic30
►Regular action on 480 points
Generators in S480
(1 447 314 182)(2 183 315 448)(3 449 316 184)(4 185 317 450)(5 451 318 186)(6 187 319 452)(7 453 320 188)(8 189 321 454)(9 455 322 190)(10 191 323 456)(11 457 324 192)(12 193 325 458)(13 459 326 194)(14 195 327 460)(15 461 328 196)(16 197 329 462)(17 463 330 198)(18 199 331 464)(19 465 332 200)(20 201 333 466)(21 467 334 202)(22 203 335 468)(23 469 336 204)(24 205 337 470)(25 471 338 206)(26 207 339 472)(27 473 340 208)(28 209 341 474)(29 475 342 210)(30 211 343 476)(31 477 344 212)(32 213 345 478)(33 479 346 214)(34 215 347 480)(35 421 348 216)(36 217 349 422)(37 423 350 218)(38 219 351 424)(39 425 352 220)(40 221 353 426)(41 427 354 222)(42 223 355 428)(43 429 356 224)(44 225 357 430)(45 431 358 226)(46 227 359 432)(47 433 360 228)(48 229 301 434)(49 435 302 230)(50 231 303 436)(51 437 304 232)(52 233 305 438)(53 439 306 234)(54 235 307 440)(55 441 308 236)(56 237 309 442)(57 443 310 238)(58 239 311 444)(59 445 312 240)(60 181 313 446)(61 149 371 277)(62 278 372 150)(63 151 373 279)(64 280 374 152)(65 153 375 281)(66 282 376 154)(67 155 377 283)(68 284 378 156)(69 157 379 285)(70 286 380 158)(71 159 381 287)(72 288 382 160)(73 161 383 289)(74 290 384 162)(75 163 385 291)(76 292 386 164)(77 165 387 293)(78 294 388 166)(79 167 389 295)(80 296 390 168)(81 169 391 297)(82 298 392 170)(83 171 393 299)(84 300 394 172)(85 173 395 241)(86 242 396 174)(87 175 397 243)(88 244 398 176)(89 177 399 245)(90 246 400 178)(91 179 401 247)(92 248 402 180)(93 121 403 249)(94 250 404 122)(95 123 405 251)(96 252 406 124)(97 125 407 253)(98 254 408 126)(99 127 409 255)(100 256 410 128)(101 129 411 257)(102 258 412 130)(103 131 413 259)(104 260 414 132)(105 133 415 261)(106 262 416 134)(107 135 417 263)(108 264 418 136)(109 137 419 265)(110 266 420 138)(111 139 361 267)(112 268 362 140)(113 141 363 269)(114 270 364 142)(115 143 365 271)(116 272 366 144)(117 145 367 273)(118 274 368 146)(119 147 369 275)(120 276 370 148)
(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 77 31 107)(2 386 32 416)(3 75 33 105)(4 384 34 414)(5 73 35 103)(6 382 36 412)(7 71 37 101)(8 380 38 410)(9 69 39 99)(10 378 40 408)(11 67 41 97)(12 376 42 406)(13 65 43 95)(14 374 44 404)(15 63 45 93)(16 372 46 402)(17 61 47 91)(18 370 48 400)(19 119 49 89)(20 368 50 398)(21 117 51 87)(22 366 52 396)(23 115 53 85)(24 364 54 394)(25 113 55 83)(26 362 56 392)(27 111 57 81)(28 420 58 390)(29 109 59 79)(30 418 60 388)(62 359 92 329)(64 357 94 327)(66 355 96 325)(68 353 98 323)(70 351 100 321)(72 349 102 319)(74 347 104 317)(76 345 106 315)(78 343 108 313)(80 341 110 311)(82 339 112 309)(84 337 114 307)(86 335 116 305)(88 333 118 303)(90 331 120 301)(121 461 151 431)(122 195 152 225)(123 459 153 429)(124 193 154 223)(125 457 155 427)(126 191 156 221)(127 455 157 425)(128 189 158 219)(129 453 159 423)(130 187 160 217)(131 451 161 421)(132 185 162 215)(133 449 163 479)(134 183 164 213)(135 447 165 477)(136 181 166 211)(137 445 167 475)(138 239 168 209)(139 443 169 473)(140 237 170 207)(141 441 171 471)(142 235 172 205)(143 439 173 469)(144 233 174 203)(145 437 175 467)(146 231 176 201)(147 435 177 465)(148 229 178 199)(149 433 179 463)(150 227 180 197)(182 293 212 263)(184 291 214 261)(186 289 216 259)(188 287 218 257)(190 285 220 255)(192 283 222 253)(194 281 224 251)(196 279 226 249)(198 277 228 247)(200 275 230 245)(202 273 232 243)(204 271 234 241)(206 269 236 299)(208 267 238 297)(210 265 240 295)(242 468 272 438)(244 466 274 436)(246 464 276 434)(248 462 278 432)(250 460 280 430)(252 458 282 428)(254 456 284 426)(256 454 286 424)(258 452 288 422)(260 450 290 480)(262 448 292 478)(264 446 294 476)(266 444 296 474)(268 442 298 472)(270 440 300 470)(302 399 332 369)(304 397 334 367)(306 395 336 365)(308 393 338 363)(310 391 340 361)(312 389 342 419)(314 387 344 417)(316 385 346 415)(318 383 348 413)(320 381 350 411)(322 379 352 409)(324 377 354 407)(326 375 356 405)(328 373 358 403)(330 371 360 401)
G:=sub<Sym(480)| (1,447,314,182)(2,183,315,448)(3,449,316,184)(4,185,317,450)(5,451,318,186)(6,187,319,452)(7,453,320,188)(8,189,321,454)(9,455,322,190)(10,191,323,456)(11,457,324,192)(12,193,325,458)(13,459,326,194)(14,195,327,460)(15,461,328,196)(16,197,329,462)(17,463,330,198)(18,199,331,464)(19,465,332,200)(20,201,333,466)(21,467,334,202)(22,203,335,468)(23,469,336,204)(24,205,337,470)(25,471,338,206)(26,207,339,472)(27,473,340,208)(28,209,341,474)(29,475,342,210)(30,211,343,476)(31,477,344,212)(32,213,345,478)(33,479,346,214)(34,215,347,480)(35,421,348,216)(36,217,349,422)(37,423,350,218)(38,219,351,424)(39,425,352,220)(40,221,353,426)(41,427,354,222)(42,223,355,428)(43,429,356,224)(44,225,357,430)(45,431,358,226)(46,227,359,432)(47,433,360,228)(48,229,301,434)(49,435,302,230)(50,231,303,436)(51,437,304,232)(52,233,305,438)(53,439,306,234)(54,235,307,440)(55,441,308,236)(56,237,309,442)(57,443,310,238)(58,239,311,444)(59,445,312,240)(60,181,313,446)(61,149,371,277)(62,278,372,150)(63,151,373,279)(64,280,374,152)(65,153,375,281)(66,282,376,154)(67,155,377,283)(68,284,378,156)(69,157,379,285)(70,286,380,158)(71,159,381,287)(72,288,382,160)(73,161,383,289)(74,290,384,162)(75,163,385,291)(76,292,386,164)(77,165,387,293)(78,294,388,166)(79,167,389,295)(80,296,390,168)(81,169,391,297)(82,298,392,170)(83,171,393,299)(84,300,394,172)(85,173,395,241)(86,242,396,174)(87,175,397,243)(88,244,398,176)(89,177,399,245)(90,246,400,178)(91,179,401,247)(92,248,402,180)(93,121,403,249)(94,250,404,122)(95,123,405,251)(96,252,406,124)(97,125,407,253)(98,254,408,126)(99,127,409,255)(100,256,410,128)(101,129,411,257)(102,258,412,130)(103,131,413,259)(104,260,414,132)(105,133,415,261)(106,262,416,134)(107,135,417,263)(108,264,418,136)(109,137,419,265)(110,266,420,138)(111,139,361,267)(112,268,362,140)(113,141,363,269)(114,270,364,142)(115,143,365,271)(116,272,366,144)(117,145,367,273)(118,274,368,146)(119,147,369,275)(120,276,370,148), (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), (1,77,31,107)(2,386,32,416)(3,75,33,105)(4,384,34,414)(5,73,35,103)(6,382,36,412)(7,71,37,101)(8,380,38,410)(9,69,39,99)(10,378,40,408)(11,67,41,97)(12,376,42,406)(13,65,43,95)(14,374,44,404)(15,63,45,93)(16,372,46,402)(17,61,47,91)(18,370,48,400)(19,119,49,89)(20,368,50,398)(21,117,51,87)(22,366,52,396)(23,115,53,85)(24,364,54,394)(25,113,55,83)(26,362,56,392)(27,111,57,81)(28,420,58,390)(29,109,59,79)(30,418,60,388)(62,359,92,329)(64,357,94,327)(66,355,96,325)(68,353,98,323)(70,351,100,321)(72,349,102,319)(74,347,104,317)(76,345,106,315)(78,343,108,313)(80,341,110,311)(82,339,112,309)(84,337,114,307)(86,335,116,305)(88,333,118,303)(90,331,120,301)(121,461,151,431)(122,195,152,225)(123,459,153,429)(124,193,154,223)(125,457,155,427)(126,191,156,221)(127,455,157,425)(128,189,158,219)(129,453,159,423)(130,187,160,217)(131,451,161,421)(132,185,162,215)(133,449,163,479)(134,183,164,213)(135,447,165,477)(136,181,166,211)(137,445,167,475)(138,239,168,209)(139,443,169,473)(140,237,170,207)(141,441,171,471)(142,235,172,205)(143,439,173,469)(144,233,174,203)(145,437,175,467)(146,231,176,201)(147,435,177,465)(148,229,178,199)(149,433,179,463)(150,227,180,197)(182,293,212,263)(184,291,214,261)(186,289,216,259)(188,287,218,257)(190,285,220,255)(192,283,222,253)(194,281,224,251)(196,279,226,249)(198,277,228,247)(200,275,230,245)(202,273,232,243)(204,271,234,241)(206,269,236,299)(208,267,238,297)(210,265,240,295)(242,468,272,438)(244,466,274,436)(246,464,276,434)(248,462,278,432)(250,460,280,430)(252,458,282,428)(254,456,284,426)(256,454,286,424)(258,452,288,422)(260,450,290,480)(262,448,292,478)(264,446,294,476)(266,444,296,474)(268,442,298,472)(270,440,300,470)(302,399,332,369)(304,397,334,367)(306,395,336,365)(308,393,338,363)(310,391,340,361)(312,389,342,419)(314,387,344,417)(316,385,346,415)(318,383,348,413)(320,381,350,411)(322,379,352,409)(324,377,354,407)(326,375,356,405)(328,373,358,403)(330,371,360,401)>;
G:=Group( (1,447,314,182)(2,183,315,448)(3,449,316,184)(4,185,317,450)(5,451,318,186)(6,187,319,452)(7,453,320,188)(8,189,321,454)(9,455,322,190)(10,191,323,456)(11,457,324,192)(12,193,325,458)(13,459,326,194)(14,195,327,460)(15,461,328,196)(16,197,329,462)(17,463,330,198)(18,199,331,464)(19,465,332,200)(20,201,333,466)(21,467,334,202)(22,203,335,468)(23,469,336,204)(24,205,337,470)(25,471,338,206)(26,207,339,472)(27,473,340,208)(28,209,341,474)(29,475,342,210)(30,211,343,476)(31,477,344,212)(32,213,345,478)(33,479,346,214)(34,215,347,480)(35,421,348,216)(36,217,349,422)(37,423,350,218)(38,219,351,424)(39,425,352,220)(40,221,353,426)(41,427,354,222)(42,223,355,428)(43,429,356,224)(44,225,357,430)(45,431,358,226)(46,227,359,432)(47,433,360,228)(48,229,301,434)(49,435,302,230)(50,231,303,436)(51,437,304,232)(52,233,305,438)(53,439,306,234)(54,235,307,440)(55,441,308,236)(56,237,309,442)(57,443,310,238)(58,239,311,444)(59,445,312,240)(60,181,313,446)(61,149,371,277)(62,278,372,150)(63,151,373,279)(64,280,374,152)(65,153,375,281)(66,282,376,154)(67,155,377,283)(68,284,378,156)(69,157,379,285)(70,286,380,158)(71,159,381,287)(72,288,382,160)(73,161,383,289)(74,290,384,162)(75,163,385,291)(76,292,386,164)(77,165,387,293)(78,294,388,166)(79,167,389,295)(80,296,390,168)(81,169,391,297)(82,298,392,170)(83,171,393,299)(84,300,394,172)(85,173,395,241)(86,242,396,174)(87,175,397,243)(88,244,398,176)(89,177,399,245)(90,246,400,178)(91,179,401,247)(92,248,402,180)(93,121,403,249)(94,250,404,122)(95,123,405,251)(96,252,406,124)(97,125,407,253)(98,254,408,126)(99,127,409,255)(100,256,410,128)(101,129,411,257)(102,258,412,130)(103,131,413,259)(104,260,414,132)(105,133,415,261)(106,262,416,134)(107,135,417,263)(108,264,418,136)(109,137,419,265)(110,266,420,138)(111,139,361,267)(112,268,362,140)(113,141,363,269)(114,270,364,142)(115,143,365,271)(116,272,366,144)(117,145,367,273)(118,274,368,146)(119,147,369,275)(120,276,370,148), (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), (1,77,31,107)(2,386,32,416)(3,75,33,105)(4,384,34,414)(5,73,35,103)(6,382,36,412)(7,71,37,101)(8,380,38,410)(9,69,39,99)(10,378,40,408)(11,67,41,97)(12,376,42,406)(13,65,43,95)(14,374,44,404)(15,63,45,93)(16,372,46,402)(17,61,47,91)(18,370,48,400)(19,119,49,89)(20,368,50,398)(21,117,51,87)(22,366,52,396)(23,115,53,85)(24,364,54,394)(25,113,55,83)(26,362,56,392)(27,111,57,81)(28,420,58,390)(29,109,59,79)(30,418,60,388)(62,359,92,329)(64,357,94,327)(66,355,96,325)(68,353,98,323)(70,351,100,321)(72,349,102,319)(74,347,104,317)(76,345,106,315)(78,343,108,313)(80,341,110,311)(82,339,112,309)(84,337,114,307)(86,335,116,305)(88,333,118,303)(90,331,120,301)(121,461,151,431)(122,195,152,225)(123,459,153,429)(124,193,154,223)(125,457,155,427)(126,191,156,221)(127,455,157,425)(128,189,158,219)(129,453,159,423)(130,187,160,217)(131,451,161,421)(132,185,162,215)(133,449,163,479)(134,183,164,213)(135,447,165,477)(136,181,166,211)(137,445,167,475)(138,239,168,209)(139,443,169,473)(140,237,170,207)(141,441,171,471)(142,235,172,205)(143,439,173,469)(144,233,174,203)(145,437,175,467)(146,231,176,201)(147,435,177,465)(148,229,178,199)(149,433,179,463)(150,227,180,197)(182,293,212,263)(184,291,214,261)(186,289,216,259)(188,287,218,257)(190,285,220,255)(192,283,222,253)(194,281,224,251)(196,279,226,249)(198,277,228,247)(200,275,230,245)(202,273,232,243)(204,271,234,241)(206,269,236,299)(208,267,238,297)(210,265,240,295)(242,468,272,438)(244,466,274,436)(246,464,276,434)(248,462,278,432)(250,460,280,430)(252,458,282,428)(254,456,284,426)(256,454,286,424)(258,452,288,422)(260,450,290,480)(262,448,292,478)(264,446,294,476)(266,444,296,474)(268,442,298,472)(270,440,300,470)(302,399,332,369)(304,397,334,367)(306,395,336,365)(308,393,338,363)(310,391,340,361)(312,389,342,419)(314,387,344,417)(316,385,346,415)(318,383,348,413)(320,381,350,411)(322,379,352,409)(324,377,354,407)(326,375,356,405)(328,373,358,403)(330,371,360,401) );
G=PermutationGroup([[(1,447,314,182),(2,183,315,448),(3,449,316,184),(4,185,317,450),(5,451,318,186),(6,187,319,452),(7,453,320,188),(8,189,321,454),(9,455,322,190),(10,191,323,456),(11,457,324,192),(12,193,325,458),(13,459,326,194),(14,195,327,460),(15,461,328,196),(16,197,329,462),(17,463,330,198),(18,199,331,464),(19,465,332,200),(20,201,333,466),(21,467,334,202),(22,203,335,468),(23,469,336,204),(24,205,337,470),(25,471,338,206),(26,207,339,472),(27,473,340,208),(28,209,341,474),(29,475,342,210),(30,211,343,476),(31,477,344,212),(32,213,345,478),(33,479,346,214),(34,215,347,480),(35,421,348,216),(36,217,349,422),(37,423,350,218),(38,219,351,424),(39,425,352,220),(40,221,353,426),(41,427,354,222),(42,223,355,428),(43,429,356,224),(44,225,357,430),(45,431,358,226),(46,227,359,432),(47,433,360,228),(48,229,301,434),(49,435,302,230),(50,231,303,436),(51,437,304,232),(52,233,305,438),(53,439,306,234),(54,235,307,440),(55,441,308,236),(56,237,309,442),(57,443,310,238),(58,239,311,444),(59,445,312,240),(60,181,313,446),(61,149,371,277),(62,278,372,150),(63,151,373,279),(64,280,374,152),(65,153,375,281),(66,282,376,154),(67,155,377,283),(68,284,378,156),(69,157,379,285),(70,286,380,158),(71,159,381,287),(72,288,382,160),(73,161,383,289),(74,290,384,162),(75,163,385,291),(76,292,386,164),(77,165,387,293),(78,294,388,166),(79,167,389,295),(80,296,390,168),(81,169,391,297),(82,298,392,170),(83,171,393,299),(84,300,394,172),(85,173,395,241),(86,242,396,174),(87,175,397,243),(88,244,398,176),(89,177,399,245),(90,246,400,178),(91,179,401,247),(92,248,402,180),(93,121,403,249),(94,250,404,122),(95,123,405,251),(96,252,406,124),(97,125,407,253),(98,254,408,126),(99,127,409,255),(100,256,410,128),(101,129,411,257),(102,258,412,130),(103,131,413,259),(104,260,414,132),(105,133,415,261),(106,262,416,134),(107,135,417,263),(108,264,418,136),(109,137,419,265),(110,266,420,138),(111,139,361,267),(112,268,362,140),(113,141,363,269),(114,270,364,142),(115,143,365,271),(116,272,366,144),(117,145,367,273),(118,274,368,146),(119,147,369,275),(120,276,370,148)], [(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)], [(1,77,31,107),(2,386,32,416),(3,75,33,105),(4,384,34,414),(5,73,35,103),(6,382,36,412),(7,71,37,101),(8,380,38,410),(9,69,39,99),(10,378,40,408),(11,67,41,97),(12,376,42,406),(13,65,43,95),(14,374,44,404),(15,63,45,93),(16,372,46,402),(17,61,47,91),(18,370,48,400),(19,119,49,89),(20,368,50,398),(21,117,51,87),(22,366,52,396),(23,115,53,85),(24,364,54,394),(25,113,55,83),(26,362,56,392),(27,111,57,81),(28,420,58,390),(29,109,59,79),(30,418,60,388),(62,359,92,329),(64,357,94,327),(66,355,96,325),(68,353,98,323),(70,351,100,321),(72,349,102,319),(74,347,104,317),(76,345,106,315),(78,343,108,313),(80,341,110,311),(82,339,112,309),(84,337,114,307),(86,335,116,305),(88,333,118,303),(90,331,120,301),(121,461,151,431),(122,195,152,225),(123,459,153,429),(124,193,154,223),(125,457,155,427),(126,191,156,221),(127,455,157,425),(128,189,158,219),(129,453,159,423),(130,187,160,217),(131,451,161,421),(132,185,162,215),(133,449,163,479),(134,183,164,213),(135,447,165,477),(136,181,166,211),(137,445,167,475),(138,239,168,209),(139,443,169,473),(140,237,170,207),(141,441,171,471),(142,235,172,205),(143,439,173,469),(144,233,174,203),(145,437,175,467),(146,231,176,201),(147,435,177,465),(148,229,178,199),(149,433,179,463),(150,227,180,197),(182,293,212,263),(184,291,214,261),(186,289,216,259),(188,287,218,257),(190,285,220,255),(192,283,222,253),(194,281,224,251),(196,279,226,249),(198,277,228,247),(200,275,230,245),(202,273,232,243),(204,271,234,241),(206,269,236,299),(208,267,238,297),(210,265,240,295),(242,468,272,438),(244,466,274,436),(246,464,276,434),(248,462,278,432),(250,460,280,430),(252,458,282,428),(254,456,284,426),(256,454,286,424),(258,452,288,422),(260,450,290,480),(262,448,292,478),(264,446,294,476),(266,444,296,474),(268,442,298,472),(270,440,300,470),(302,399,332,369),(304,397,334,367),(306,395,336,365),(308,393,338,363),(310,391,340,361),(312,389,342,419),(314,387,344,417),(316,385,346,415),(318,383,348,413),(320,381,350,411),(322,379,352,409),(324,377,354,407),(326,375,356,405),(328,373,358,403),(330,371,360,401)]])
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