metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊1Q8, C4⋊1Dic30, C20⋊3Dic6, Dic15⋊3Q8, C12⋊3Dic10, Dic15.21D4, C3⋊4(C20⋊Q8), C5⋊4(C12⋊Q8), C4⋊C4.4D15, C15⋊13(C4⋊Q8), C2.4(Q8×D15), C6.39(Q8×D5), C2.11(D4×D15), (C2×C20).36D6, (C2×C4).10D30, C6.102(D4×D5), C30.92(C2×Q8), C10.39(S3×Q8), C10.104(S3×D4), C30.310(C2×D4), C60⋊5C4.13C2, C2.7(C2×Dic30), (C2×C12).138D10, (C2×C60).63C22, (C4×Dic15).1C2, (C2×Dic30).4C2, C6.36(C2×Dic10), C10.36(C2×Dic6), C30.4Q8.2C2, (C2×C30).285C23, (C2×Dic15).9C22, C22.46(C22×D15), (C5×C4⋊C4).5S3, (C3×C4⋊C4).5D5, (C15×C4⋊C4).5C2, (C2×C6).281(C22×D5), (C2×C10).280(C22×S3), SmallGroup(480,853)
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=b-1 >
Subgroups: 708 in 136 conjugacy classes, 59 normal (33 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C2×C12, C30, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, Dic15, Dic15, C60, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×Dic10, C12⋊Q8, Dic30, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C20⋊Q8, C4×Dic15, C30.4Q8, C60⋊5C4, C15×C4⋊C4, C2×Dic30, C4⋊Dic30
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, Dic6, C22×S3, D15, C4⋊Q8, Dic10, C22×D5, C2×Dic6, S3×D4, S3×Q8, D30, C2×Dic10, D4×D5, Q8×D5, C12⋊Q8, Dic30, C22×D15, C20⋊Q8, C2×Dic30, D4×D15, Q8×D15, C4⋊Dic30
►Regular action on 480 points
Generators in S480
(1 196 274 413)(2 414 275 197)(3 198 276 415)(4 416 277 199)(5 200 278 417)(6 418 279 201)(7 202 280 419)(8 420 281 203)(9 204 282 361)(10 362 283 205)(11 206 284 363)(12 364 285 207)(13 208 286 365)(14 366 287 209)(15 210 288 367)(16 368 289 211)(17 212 290 369)(18 370 291 213)(19 214 292 371)(20 372 293 215)(21 216 294 373)(22 374 295 217)(23 218 296 375)(24 376 297 219)(25 220 298 377)(26 378 299 221)(27 222 300 379)(28 380 241 223)(29 224 242 381)(30 382 243 225)(31 226 244 383)(32 384 245 227)(33 228 246 385)(34 386 247 229)(35 230 248 387)(36 388 249 231)(37 232 250 389)(38 390 251 233)(39 234 252 391)(40 392 253 235)(41 236 254 393)(42 394 255 237)(43 238 256 395)(44 396 257 239)(45 240 258 397)(46 398 259 181)(47 182 260 399)(48 400 261 183)(49 184 262 401)(50 402 263 185)(51 186 264 403)(52 404 265 187)(53 188 266 405)(54 406 267 189)(55 190 268 407)(56 408 269 191)(57 192 270 409)(58 410 271 193)(59 194 272 411)(60 412 273 195)(61 327 122 463)(62 464 123 328)(63 329 124 465)(64 466 125 330)(65 331 126 467)(66 468 127 332)(67 333 128 469)(68 470 129 334)(69 335 130 471)(70 472 131 336)(71 337 132 473)(72 474 133 338)(73 339 134 475)(74 476 135 340)(75 341 136 477)(76 478 137 342)(77 343 138 479)(78 480 139 344)(79 345 140 421)(80 422 141 346)(81 347 142 423)(82 424 143 348)(83 349 144 425)(84 426 145 350)(85 351 146 427)(86 428 147 352)(87 353 148 429)(88 430 149 354)(89 355 150 431)(90 432 151 356)(91 357 152 433)(92 434 153 358)(93 359 154 435)(94 436 155 360)(95 301 156 437)(96 438 157 302)(97 303 158 439)(98 440 159 304)(99 305 160 441)(100 442 161 306)(101 307 162 443)(102 444 163 308)(103 309 164 445)(104 446 165 310)(105 311 166 447)(106 448 167 312)(107 313 168 449)(108 450 169 314)(109 315 170 451)(110 452 171 316)(111 317 172 453)(112 454 173 318)(113 319 174 455)(114 456 175 320)(115 321 176 457)(116 458 177 322)(117 323 178 459)(118 460 179 324)(119 325 180 461)(120 462 121 326)
(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)
(1 316 31 346)(2 315 32 345)(3 314 33 344)(4 313 34 343)(5 312 35 342)(6 311 36 341)(7 310 37 340)(8 309 38 339)(9 308 39 338)(10 307 40 337)(11 306 41 336)(12 305 42 335)(13 304 43 334)(14 303 44 333)(15 302 45 332)(16 301 46 331)(17 360 47 330)(18 359 48 329)(19 358 49 328)(20 357 50 327)(21 356 51 326)(22 355 52 325)(23 354 53 324)(24 353 54 323)(25 352 55 322)(26 351 56 321)(27 350 57 320)(28 349 58 319)(29 348 59 318)(30 347 60 317)(61 215 91 185)(62 214 92 184)(63 213 93 183)(64 212 94 182)(65 211 95 181)(66 210 96 240)(67 209 97 239)(68 208 98 238)(69 207 99 237)(70 206 100 236)(71 205 101 235)(72 204 102 234)(73 203 103 233)(74 202 104 232)(75 201 105 231)(76 200 106 230)(77 199 107 229)(78 198 108 228)(79 197 109 227)(80 196 110 226)(81 195 111 225)(82 194 112 224)(83 193 113 223)(84 192 114 222)(85 191 115 221)(86 190 116 220)(87 189 117 219)(88 188 118 218)(89 187 119 217)(90 186 120 216)(121 373 151 403)(122 372 152 402)(123 371 153 401)(124 370 154 400)(125 369 155 399)(126 368 156 398)(127 367 157 397)(128 366 158 396)(129 365 159 395)(130 364 160 394)(131 363 161 393)(132 362 162 392)(133 361 163 391)(134 420 164 390)(135 419 165 389)(136 418 166 388)(137 417 167 387)(138 416 168 386)(139 415 169 385)(140 414 170 384)(141 413 171 383)(142 412 172 382)(143 411 173 381)(144 410 174 380)(145 409 175 379)(146 408 176 378)(147 407 177 377)(148 406 178 376)(149 405 179 375)(150 404 180 374)(241 425 271 455)(242 424 272 454)(243 423 273 453)(244 422 274 452)(245 421 275 451)(246 480 276 450)(247 479 277 449)(248 478 278 448)(249 477 279 447)(250 476 280 446)(251 475 281 445)(252 474 282 444)(253 473 283 443)(254 472 284 442)(255 471 285 441)(256 470 286 440)(257 469 287 439)(258 468 288 438)(259 467 289 437)(260 466 290 436)(261 465 291 435)(262 464 292 434)(263 463 293 433)(264 462 294 432)(265 461 295 431)(266 460 296 430)(267 459 297 429)(268 458 298 428)(269 457 299 427)(270 456 300 426)
G:=sub<Sym(480)| (1,196,274,413)(2,414,275,197)(3,198,276,415)(4,416,277,199)(5,200,278,417)(6,418,279,201)(7,202,280,419)(8,420,281,203)(9,204,282,361)(10,362,283,205)(11,206,284,363)(12,364,285,207)(13,208,286,365)(14,366,287,209)(15,210,288,367)(16,368,289,211)(17,212,290,369)(18,370,291,213)(19,214,292,371)(20,372,293,215)(21,216,294,373)(22,374,295,217)(23,218,296,375)(24,376,297,219)(25,220,298,377)(26,378,299,221)(27,222,300,379)(28,380,241,223)(29,224,242,381)(30,382,243,225)(31,226,244,383)(32,384,245,227)(33,228,246,385)(34,386,247,229)(35,230,248,387)(36,388,249,231)(37,232,250,389)(38,390,251,233)(39,234,252,391)(40,392,253,235)(41,236,254,393)(42,394,255,237)(43,238,256,395)(44,396,257,239)(45,240,258,397)(46,398,259,181)(47,182,260,399)(48,400,261,183)(49,184,262,401)(50,402,263,185)(51,186,264,403)(52,404,265,187)(53,188,266,405)(54,406,267,189)(55,190,268,407)(56,408,269,191)(57,192,270,409)(58,410,271,193)(59,194,272,411)(60,412,273,195)(61,327,122,463)(62,464,123,328)(63,329,124,465)(64,466,125,330)(65,331,126,467)(66,468,127,332)(67,333,128,469)(68,470,129,334)(69,335,130,471)(70,472,131,336)(71,337,132,473)(72,474,133,338)(73,339,134,475)(74,476,135,340)(75,341,136,477)(76,478,137,342)(77,343,138,479)(78,480,139,344)(79,345,140,421)(80,422,141,346)(81,347,142,423)(82,424,143,348)(83,349,144,425)(84,426,145,350)(85,351,146,427)(86,428,147,352)(87,353,148,429)(88,430,149,354)(89,355,150,431)(90,432,151,356)(91,357,152,433)(92,434,153,358)(93,359,154,435)(94,436,155,360)(95,301,156,437)(96,438,157,302)(97,303,158,439)(98,440,159,304)(99,305,160,441)(100,442,161,306)(101,307,162,443)(102,444,163,308)(103,309,164,445)(104,446,165,310)(105,311,166,447)(106,448,167,312)(107,313,168,449)(108,450,169,314)(109,315,170,451)(110,452,171,316)(111,317,172,453)(112,454,173,318)(113,319,174,455)(114,456,175,320)(115,321,176,457)(116,458,177,322)(117,323,178,459)(118,460,179,324)(119,325,180,461)(120,462,121,326), (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), (1,316,31,346)(2,315,32,345)(3,314,33,344)(4,313,34,343)(5,312,35,342)(6,311,36,341)(7,310,37,340)(8,309,38,339)(9,308,39,338)(10,307,40,337)(11,306,41,336)(12,305,42,335)(13,304,43,334)(14,303,44,333)(15,302,45,332)(16,301,46,331)(17,360,47,330)(18,359,48,329)(19,358,49,328)(20,357,50,327)(21,356,51,326)(22,355,52,325)(23,354,53,324)(24,353,54,323)(25,352,55,322)(26,351,56,321)(27,350,57,320)(28,349,58,319)(29,348,59,318)(30,347,60,317)(61,215,91,185)(62,214,92,184)(63,213,93,183)(64,212,94,182)(65,211,95,181)(66,210,96,240)(67,209,97,239)(68,208,98,238)(69,207,99,237)(70,206,100,236)(71,205,101,235)(72,204,102,234)(73,203,103,233)(74,202,104,232)(75,201,105,231)(76,200,106,230)(77,199,107,229)(78,198,108,228)(79,197,109,227)(80,196,110,226)(81,195,111,225)(82,194,112,224)(83,193,113,223)(84,192,114,222)(85,191,115,221)(86,190,116,220)(87,189,117,219)(88,188,118,218)(89,187,119,217)(90,186,120,216)(121,373,151,403)(122,372,152,402)(123,371,153,401)(124,370,154,400)(125,369,155,399)(126,368,156,398)(127,367,157,397)(128,366,158,396)(129,365,159,395)(130,364,160,394)(131,363,161,393)(132,362,162,392)(133,361,163,391)(134,420,164,390)(135,419,165,389)(136,418,166,388)(137,417,167,387)(138,416,168,386)(139,415,169,385)(140,414,170,384)(141,413,171,383)(142,412,172,382)(143,411,173,381)(144,410,174,380)(145,409,175,379)(146,408,176,378)(147,407,177,377)(148,406,178,376)(149,405,179,375)(150,404,180,374)(241,425,271,455)(242,424,272,454)(243,423,273,453)(244,422,274,452)(245,421,275,451)(246,480,276,450)(247,479,277,449)(248,478,278,448)(249,477,279,447)(250,476,280,446)(251,475,281,445)(252,474,282,444)(253,473,283,443)(254,472,284,442)(255,471,285,441)(256,470,286,440)(257,469,287,439)(258,468,288,438)(259,467,289,437)(260,466,290,436)(261,465,291,435)(262,464,292,434)(263,463,293,433)(264,462,294,432)(265,461,295,431)(266,460,296,430)(267,459,297,429)(268,458,298,428)(269,457,299,427)(270,456,300,426)>;
G:=Group( (1,196,274,413)(2,414,275,197)(3,198,276,415)(4,416,277,199)(5,200,278,417)(6,418,279,201)(7,202,280,419)(8,420,281,203)(9,204,282,361)(10,362,283,205)(11,206,284,363)(12,364,285,207)(13,208,286,365)(14,366,287,209)(15,210,288,367)(16,368,289,211)(17,212,290,369)(18,370,291,213)(19,214,292,371)(20,372,293,215)(21,216,294,373)(22,374,295,217)(23,218,296,375)(24,376,297,219)(25,220,298,377)(26,378,299,221)(27,222,300,379)(28,380,241,223)(29,224,242,381)(30,382,243,225)(31,226,244,383)(32,384,245,227)(33,228,246,385)(34,386,247,229)(35,230,248,387)(36,388,249,231)(37,232,250,389)(38,390,251,233)(39,234,252,391)(40,392,253,235)(41,236,254,393)(42,394,255,237)(43,238,256,395)(44,396,257,239)(45,240,258,397)(46,398,259,181)(47,182,260,399)(48,400,261,183)(49,184,262,401)(50,402,263,185)(51,186,264,403)(52,404,265,187)(53,188,266,405)(54,406,267,189)(55,190,268,407)(56,408,269,191)(57,192,270,409)(58,410,271,193)(59,194,272,411)(60,412,273,195)(61,327,122,463)(62,464,123,328)(63,329,124,465)(64,466,125,330)(65,331,126,467)(66,468,127,332)(67,333,128,469)(68,470,129,334)(69,335,130,471)(70,472,131,336)(71,337,132,473)(72,474,133,338)(73,339,134,475)(74,476,135,340)(75,341,136,477)(76,478,137,342)(77,343,138,479)(78,480,139,344)(79,345,140,421)(80,422,141,346)(81,347,142,423)(82,424,143,348)(83,349,144,425)(84,426,145,350)(85,351,146,427)(86,428,147,352)(87,353,148,429)(88,430,149,354)(89,355,150,431)(90,432,151,356)(91,357,152,433)(92,434,153,358)(93,359,154,435)(94,436,155,360)(95,301,156,437)(96,438,157,302)(97,303,158,439)(98,440,159,304)(99,305,160,441)(100,442,161,306)(101,307,162,443)(102,444,163,308)(103,309,164,445)(104,446,165,310)(105,311,166,447)(106,448,167,312)(107,313,168,449)(108,450,169,314)(109,315,170,451)(110,452,171,316)(111,317,172,453)(112,454,173,318)(113,319,174,455)(114,456,175,320)(115,321,176,457)(116,458,177,322)(117,323,178,459)(118,460,179,324)(119,325,180,461)(120,462,121,326), (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), (1,316,31,346)(2,315,32,345)(3,314,33,344)(4,313,34,343)(5,312,35,342)(6,311,36,341)(7,310,37,340)(8,309,38,339)(9,308,39,338)(10,307,40,337)(11,306,41,336)(12,305,42,335)(13,304,43,334)(14,303,44,333)(15,302,45,332)(16,301,46,331)(17,360,47,330)(18,359,48,329)(19,358,49,328)(20,357,50,327)(21,356,51,326)(22,355,52,325)(23,354,53,324)(24,353,54,323)(25,352,55,322)(26,351,56,321)(27,350,57,320)(28,349,58,319)(29,348,59,318)(30,347,60,317)(61,215,91,185)(62,214,92,184)(63,213,93,183)(64,212,94,182)(65,211,95,181)(66,210,96,240)(67,209,97,239)(68,208,98,238)(69,207,99,237)(70,206,100,236)(71,205,101,235)(72,204,102,234)(73,203,103,233)(74,202,104,232)(75,201,105,231)(76,200,106,230)(77,199,107,229)(78,198,108,228)(79,197,109,227)(80,196,110,226)(81,195,111,225)(82,194,112,224)(83,193,113,223)(84,192,114,222)(85,191,115,221)(86,190,116,220)(87,189,117,219)(88,188,118,218)(89,187,119,217)(90,186,120,216)(121,373,151,403)(122,372,152,402)(123,371,153,401)(124,370,154,400)(125,369,155,399)(126,368,156,398)(127,367,157,397)(128,366,158,396)(129,365,159,395)(130,364,160,394)(131,363,161,393)(132,362,162,392)(133,361,163,391)(134,420,164,390)(135,419,165,389)(136,418,166,388)(137,417,167,387)(138,416,168,386)(139,415,169,385)(140,414,170,384)(141,413,171,383)(142,412,172,382)(143,411,173,381)(144,410,174,380)(145,409,175,379)(146,408,176,378)(147,407,177,377)(148,406,178,376)(149,405,179,375)(150,404,180,374)(241,425,271,455)(242,424,272,454)(243,423,273,453)(244,422,274,452)(245,421,275,451)(246,480,276,450)(247,479,277,449)(248,478,278,448)(249,477,279,447)(250,476,280,446)(251,475,281,445)(252,474,282,444)(253,473,283,443)(254,472,284,442)(255,471,285,441)(256,470,286,440)(257,469,287,439)(258,468,288,438)(259,467,289,437)(260,466,290,436)(261,465,291,435)(262,464,292,434)(263,463,293,433)(264,462,294,432)(265,461,295,431)(266,460,296,430)(267,459,297,429)(268,458,298,428)(269,457,299,427)(270,456,300,426) );
G=PermutationGroup([[(1,196,274,413),(2,414,275,197),(3,198,276,415),(4,416,277,199),(5,200,278,417),(6,418,279,201),(7,202,280,419),(8,420,281,203),(9,204,282,361),(10,362,283,205),(11,206,284,363),(12,364,285,207),(13,208,286,365),(14,366,287,209),(15,210,288,367),(16,368,289,211),(17,212,290,369),(18,370,291,213),(19,214,292,371),(20,372,293,215),(21,216,294,373),(22,374,295,217),(23,218,296,375),(24,376,297,219),(25,220,298,377),(26,378,299,221),(27,222,300,379),(28,380,241,223),(29,224,242,381),(30,382,243,225),(31,226,244,383),(32,384,245,227),(33,228,246,385),(34,386,247,229),(35,230,248,387),(36,388,249,231),(37,232,250,389),(38,390,251,233),(39,234,252,391),(40,392,253,235),(41,236,254,393),(42,394,255,237),(43,238,256,395),(44,396,257,239),(45,240,258,397),(46,398,259,181),(47,182,260,399),(48,400,261,183),(49,184,262,401),(50,402,263,185),(51,186,264,403),(52,404,265,187),(53,188,266,405),(54,406,267,189),(55,190,268,407),(56,408,269,191),(57,192,270,409),(58,410,271,193),(59,194,272,411),(60,412,273,195),(61,327,122,463),(62,464,123,328),(63,329,124,465),(64,466,125,330),(65,331,126,467),(66,468,127,332),(67,333,128,469),(68,470,129,334),(69,335,130,471),(70,472,131,336),(71,337,132,473),(72,474,133,338),(73,339,134,475),(74,476,135,340),(75,341,136,477),(76,478,137,342),(77,343,138,479),(78,480,139,344),(79,345,140,421),(80,422,141,346),(81,347,142,423),(82,424,143,348),(83,349,144,425),(84,426,145,350),(85,351,146,427),(86,428,147,352),(87,353,148,429),(88,430,149,354),(89,355,150,431),(90,432,151,356),(91,357,152,433),(92,434,153,358),(93,359,154,435),(94,436,155,360),(95,301,156,437),(96,438,157,302),(97,303,158,439),(98,440,159,304),(99,305,160,441),(100,442,161,306),(101,307,162,443),(102,444,163,308),(103,309,164,445),(104,446,165,310),(105,311,166,447),(106,448,167,312),(107,313,168,449),(108,450,169,314),(109,315,170,451),(110,452,171,316),(111,317,172,453),(112,454,173,318),(113,319,174,455),(114,456,175,320),(115,321,176,457),(116,458,177,322),(117,323,178,459),(118,460,179,324),(119,325,180,461),(120,462,121,326)], [(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)], [(1,316,31,346),(2,315,32,345),(3,314,33,344),(4,313,34,343),(5,312,35,342),(6,311,36,341),(7,310,37,340),(8,309,38,339),(9,308,39,338),(10,307,40,337),(11,306,41,336),(12,305,42,335),(13,304,43,334),(14,303,44,333),(15,302,45,332),(16,301,46,331),(17,360,47,330),(18,359,48,329),(19,358,49,328),(20,357,50,327),(21,356,51,326),(22,355,52,325),(23,354,53,324),(24,353,54,323),(25,352,55,322),(26,351,56,321),(27,350,57,320),(28,349,58,319),(29,348,59,318),(30,347,60,317),(61,215,91,185),(62,214,92,184),(63,213,93,183),(64,212,94,182),(65,211,95,181),(66,210,96,240),(67,209,97,239),(68,208,98,238),(69,207,99,237),(70,206,100,236),(71,205,101,235),(72,204,102,234),(73,203,103,233),(74,202,104,232),(75,201,105,231),(76,200,106,230),(77,199,107,229),(78,198,108,228),(79,197,109,227),(80,196,110,226),(81,195,111,225),(82,194,112,224),(83,193,113,223),(84,192,114,222),(85,191,115,221),(86,190,116,220),(87,189,117,219),(88,188,118,218),(89,187,119,217),(90,186,120,216),(121,373,151,403),(122,372,152,402),(123,371,153,401),(124,370,154,400),(125,369,155,399),(126,368,156,398),(127,367,157,397),(128,366,158,396),(129,365,159,395),(130,364,160,394),(131,363,161,393),(132,362,162,392),(133,361,163,391),(134,420,164,390),(135,419,165,389),(136,418,166,388),(137,417,167,387),(138,416,168,386),(139,415,169,385),(140,414,170,384),(141,413,171,383),(142,412,172,382),(143,411,173,381),(144,410,174,380),(145,409,175,379),(146,408,176,378),(147,407,177,377),(148,406,178,376),(149,405,179,375),(150,404,180,374),(241,425,271,455),(242,424,272,454),(243,423,273,453),(244,422,274,452),(245,421,275,451),(246,480,276,450),(247,479,277,449),(248,478,278,448),(249,477,279,447),(250,476,280,446),(251,475,281,445),(252,474,282,444),(253,473,283,443),(254,472,284,442),(255,471,285,441),(256,470,286,440),(257,469,287,439),(258,468,288,438),(259,467,289,437),(260,466,290,436),(261,465,291,435),(262,464,292,434),(263,463,293,433),(264,462,294,432),(265,461,295,431),(266,460,296,430),(267,459,297,429),(268,458,298,428),(269,457,299,427),(270,456,300,426)]])
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