direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Dic30, C60⋊9Q8, C20⋊8Dic6, C12⋊8Dic10, C42.3D15, C15⋊15(C4×Q8), C5⋊5(C4×Dic6), (C4×C20).5S3, (C4×C60).5C2, C4.9(C4×D15), (C4×C12).5D5, C3⋊4(C4×Dic10), C20.84(C4×S3), (C2×C4).75D30, C12.52(C4×D5), C30.65(C2×Q8), C60.189(C2×C4), (C2×C20).410D6, C60⋊5C4.27C2, C2.1(C2×Dic30), C6.89(C4○D20), (C2×C12).390D10, (C4×Dic15).9C2, C6.32(C2×Dic10), C10.32(C2×Dic6), C30.163(C4○D4), C10.89(C4○D12), C30.152(C22×C4), (C2×C30).265C23, (C2×C60).458C22, (C2×Dic30).20C2, Dic15.29(C2×C4), C30.4Q8.17C2, C2.1(D60⋊11C2), C22.8(C22×D15), (C2×Dic15).155C22, C2.4(C2×C4×D15), C6.57(C2×C4×D5), C10.89(S3×C2×C4), (C2×C6).261(C22×D5), (C2×C10).260(C22×S3), SmallGroup(480,833)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Dic30
G = < a,b,c | a4=b60=1, c2=b30, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 612 in 140 conjugacy classes, 71 normal (39 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C42, C42, C4⋊C4, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C30, C4×Q8, Dic10, C2×Dic5, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, Dic15, Dic15, C60, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, C4×Dic6, Dic30, C2×Dic15, C2×C60, C4×Dic10, C4×Dic15, C30.4Q8, C60⋊5C4, C4×C60, C2×Dic30, C4×Dic30
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, D6, C22×C4, C2×Q8, C4○D4, D10, Dic6, C4×S3, C22×S3, D15, C4×Q8, Dic10, C4×D5, C22×D5, C2×Dic6, S3×C2×C4, C4○D12, D30, C2×Dic10, C2×C4×D5, C4○D20, C4×Dic6, Dic30, C4×D15, C22×D15, C4×Dic10, C2×Dic30, C2×C4×D15, D60⋊11C2, C4×Dic30
►Regular action on 480 points
Generators in S480
(1 124 300 216)(2 125 241 217)(3 126 242 218)(4 127 243 219)(5 128 244 220)(6 129 245 221)(7 130 246 222)(8 131 247 223)(9 132 248 224)(10 133 249 225)(11 134 250 226)(12 135 251 227)(13 136 252 228)(14 137 253 229)(15 138 254 230)(16 139 255 231)(17 140 256 232)(18 141 257 233)(19 142 258 234)(20 143 259 235)(21 144 260 236)(22 145 261 237)(23 146 262 238)(24 147 263 239)(25 148 264 240)(26 149 265 181)(27 150 266 182)(28 151 267 183)(29 152 268 184)(30 153 269 185)(31 154 270 186)(32 155 271 187)(33 156 272 188)(34 157 273 189)(35 158 274 190)(36 159 275 191)(37 160 276 192)(38 161 277 193)(39 162 278 194)(40 163 279 195)(41 164 280 196)(42 165 281 197)(43 166 282 198)(44 167 283 199)(45 168 284 200)(46 169 285 201)(47 170 286 202)(48 171 287 203)(49 172 288 204)(50 173 289 205)(51 174 290 206)(52 175 291 207)(53 176 292 208)(54 177 293 209)(55 178 294 210)(56 179 295 211)(57 180 296 212)(58 121 297 213)(59 122 298 214)(60 123 299 215)(61 369 308 448)(62 370 309 449)(63 371 310 450)(64 372 311 451)(65 373 312 452)(66 374 313 453)(67 375 314 454)(68 376 315 455)(69 377 316 456)(70 378 317 457)(71 379 318 458)(72 380 319 459)(73 381 320 460)(74 382 321 461)(75 383 322 462)(76 384 323 463)(77 385 324 464)(78 386 325 465)(79 387 326 466)(80 388 327 467)(81 389 328 468)(82 390 329 469)(83 391 330 470)(84 392 331 471)(85 393 332 472)(86 394 333 473)(87 395 334 474)(88 396 335 475)(89 397 336 476)(90 398 337 477)(91 399 338 478)(92 400 339 479)(93 401 340 480)(94 402 341 421)(95 403 342 422)(96 404 343 423)(97 405 344 424)(98 406 345 425)(99 407 346 426)(100 408 347 427)(101 409 348 428)(102 410 349 429)(103 411 350 430)(104 412 351 431)(105 413 352 432)(106 414 353 433)(107 415 354 434)(108 416 355 435)(109 417 356 436)(110 418 357 437)(111 419 358 438)(112 420 359 439)(113 361 360 440)(114 362 301 441)(115 363 302 442)(116 364 303 443)(117 365 304 444)(118 366 305 445)(119 367 306 446)(120 368 307 447)
(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)
(1 414 31 384)(2 413 32 383)(3 412 33 382)(4 411 34 381)(5 410 35 380)(6 409 36 379)(7 408 37 378)(8 407 38 377)(9 406 39 376)(10 405 40 375)(11 404 41 374)(12 403 42 373)(13 402 43 372)(14 401 44 371)(15 400 45 370)(16 399 46 369)(17 398 47 368)(18 397 48 367)(19 396 49 366)(20 395 50 365)(21 394 51 364)(22 393 52 363)(23 392 53 362)(24 391 54 361)(25 390 55 420)(26 389 56 419)(27 388 57 418)(28 387 58 417)(29 386 59 416)(30 385 60 415)(61 231 91 201)(62 230 92 200)(63 229 93 199)(64 228 94 198)(65 227 95 197)(66 226 96 196)(67 225 97 195)(68 224 98 194)(69 223 99 193)(70 222 100 192)(71 221 101 191)(72 220 102 190)(73 219 103 189)(74 218 104 188)(75 217 105 187)(76 216 106 186)(77 215 107 185)(78 214 108 184)(79 213 109 183)(80 212 110 182)(81 211 111 181)(82 210 112 240)(83 209 113 239)(84 208 114 238)(85 207 115 237)(86 206 116 236)(87 205 117 235)(88 204 118 234)(89 203 119 233)(90 202 120 232)(121 356 151 326)(122 355 152 325)(123 354 153 324)(124 353 154 323)(125 352 155 322)(126 351 156 321)(127 350 157 320)(128 349 158 319)(129 348 159 318)(130 347 160 317)(131 346 161 316)(132 345 162 315)(133 344 163 314)(134 343 164 313)(135 342 165 312)(136 341 166 311)(137 340 167 310)(138 339 168 309)(139 338 169 308)(140 337 170 307)(141 336 171 306)(142 335 172 305)(143 334 173 304)(144 333 174 303)(145 332 175 302)(146 331 176 301)(147 330 177 360)(148 329 178 359)(149 328 179 358)(150 327 180 357)(241 432 271 462)(242 431 272 461)(243 430 273 460)(244 429 274 459)(245 428 275 458)(246 427 276 457)(247 426 277 456)(248 425 278 455)(249 424 279 454)(250 423 280 453)(251 422 281 452)(252 421 282 451)(253 480 283 450)(254 479 284 449)(255 478 285 448)(256 477 286 447)(257 476 287 446)(258 475 288 445)(259 474 289 444)(260 473 290 443)(261 472 291 442)(262 471 292 441)(263 470 293 440)(264 469 294 439)(265 468 295 438)(266 467 296 437)(267 466 297 436)(268 465 298 435)(269 464 299 434)(270 463 300 433)
G:=sub<Sym(480)| (1,124,300,216)(2,125,241,217)(3,126,242,218)(4,127,243,219)(5,128,244,220)(6,129,245,221)(7,130,246,222)(8,131,247,223)(9,132,248,224)(10,133,249,225)(11,134,250,226)(12,135,251,227)(13,136,252,228)(14,137,253,229)(15,138,254,230)(16,139,255,231)(17,140,256,232)(18,141,257,233)(19,142,258,234)(20,143,259,235)(21,144,260,236)(22,145,261,237)(23,146,262,238)(24,147,263,239)(25,148,264,240)(26,149,265,181)(27,150,266,182)(28,151,267,183)(29,152,268,184)(30,153,269,185)(31,154,270,186)(32,155,271,187)(33,156,272,188)(34,157,273,189)(35,158,274,190)(36,159,275,191)(37,160,276,192)(38,161,277,193)(39,162,278,194)(40,163,279,195)(41,164,280,196)(42,165,281,197)(43,166,282,198)(44,167,283,199)(45,168,284,200)(46,169,285,201)(47,170,286,202)(48,171,287,203)(49,172,288,204)(50,173,289,205)(51,174,290,206)(52,175,291,207)(53,176,292,208)(54,177,293,209)(55,178,294,210)(56,179,295,211)(57,180,296,212)(58,121,297,213)(59,122,298,214)(60,123,299,215)(61,369,308,448)(62,370,309,449)(63,371,310,450)(64,372,311,451)(65,373,312,452)(66,374,313,453)(67,375,314,454)(68,376,315,455)(69,377,316,456)(70,378,317,457)(71,379,318,458)(72,380,319,459)(73,381,320,460)(74,382,321,461)(75,383,322,462)(76,384,323,463)(77,385,324,464)(78,386,325,465)(79,387,326,466)(80,388,327,467)(81,389,328,468)(82,390,329,469)(83,391,330,470)(84,392,331,471)(85,393,332,472)(86,394,333,473)(87,395,334,474)(88,396,335,475)(89,397,336,476)(90,398,337,477)(91,399,338,478)(92,400,339,479)(93,401,340,480)(94,402,341,421)(95,403,342,422)(96,404,343,423)(97,405,344,424)(98,406,345,425)(99,407,346,426)(100,408,347,427)(101,409,348,428)(102,410,349,429)(103,411,350,430)(104,412,351,431)(105,413,352,432)(106,414,353,433)(107,415,354,434)(108,416,355,435)(109,417,356,436)(110,418,357,437)(111,419,358,438)(112,420,359,439)(113,361,360,440)(114,362,301,441)(115,363,302,442)(116,364,303,443)(117,365,304,444)(118,366,305,445)(119,367,306,446)(120,368,307,447), (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), (1,414,31,384)(2,413,32,383)(3,412,33,382)(4,411,34,381)(5,410,35,380)(6,409,36,379)(7,408,37,378)(8,407,38,377)(9,406,39,376)(10,405,40,375)(11,404,41,374)(12,403,42,373)(13,402,43,372)(14,401,44,371)(15,400,45,370)(16,399,46,369)(17,398,47,368)(18,397,48,367)(19,396,49,366)(20,395,50,365)(21,394,51,364)(22,393,52,363)(23,392,53,362)(24,391,54,361)(25,390,55,420)(26,389,56,419)(27,388,57,418)(28,387,58,417)(29,386,59,416)(30,385,60,415)(61,231,91,201)(62,230,92,200)(63,229,93,199)(64,228,94,198)(65,227,95,197)(66,226,96,196)(67,225,97,195)(68,224,98,194)(69,223,99,193)(70,222,100,192)(71,221,101,191)(72,220,102,190)(73,219,103,189)(74,218,104,188)(75,217,105,187)(76,216,106,186)(77,215,107,185)(78,214,108,184)(79,213,109,183)(80,212,110,182)(81,211,111,181)(82,210,112,240)(83,209,113,239)(84,208,114,238)(85,207,115,237)(86,206,116,236)(87,205,117,235)(88,204,118,234)(89,203,119,233)(90,202,120,232)(121,356,151,326)(122,355,152,325)(123,354,153,324)(124,353,154,323)(125,352,155,322)(126,351,156,321)(127,350,157,320)(128,349,158,319)(129,348,159,318)(130,347,160,317)(131,346,161,316)(132,345,162,315)(133,344,163,314)(134,343,164,313)(135,342,165,312)(136,341,166,311)(137,340,167,310)(138,339,168,309)(139,338,169,308)(140,337,170,307)(141,336,171,306)(142,335,172,305)(143,334,173,304)(144,333,174,303)(145,332,175,302)(146,331,176,301)(147,330,177,360)(148,329,178,359)(149,328,179,358)(150,327,180,357)(241,432,271,462)(242,431,272,461)(243,430,273,460)(244,429,274,459)(245,428,275,458)(246,427,276,457)(247,426,277,456)(248,425,278,455)(249,424,279,454)(250,423,280,453)(251,422,281,452)(252,421,282,451)(253,480,283,450)(254,479,284,449)(255,478,285,448)(256,477,286,447)(257,476,287,446)(258,475,288,445)(259,474,289,444)(260,473,290,443)(261,472,291,442)(262,471,292,441)(263,470,293,440)(264,469,294,439)(265,468,295,438)(266,467,296,437)(267,466,297,436)(268,465,298,435)(269,464,299,434)(270,463,300,433)>;
G:=Group( (1,124,300,216)(2,125,241,217)(3,126,242,218)(4,127,243,219)(5,128,244,220)(6,129,245,221)(7,130,246,222)(8,131,247,223)(9,132,248,224)(10,133,249,225)(11,134,250,226)(12,135,251,227)(13,136,252,228)(14,137,253,229)(15,138,254,230)(16,139,255,231)(17,140,256,232)(18,141,257,233)(19,142,258,234)(20,143,259,235)(21,144,260,236)(22,145,261,237)(23,146,262,238)(24,147,263,239)(25,148,264,240)(26,149,265,181)(27,150,266,182)(28,151,267,183)(29,152,268,184)(30,153,269,185)(31,154,270,186)(32,155,271,187)(33,156,272,188)(34,157,273,189)(35,158,274,190)(36,159,275,191)(37,160,276,192)(38,161,277,193)(39,162,278,194)(40,163,279,195)(41,164,280,196)(42,165,281,197)(43,166,282,198)(44,167,283,199)(45,168,284,200)(46,169,285,201)(47,170,286,202)(48,171,287,203)(49,172,288,204)(50,173,289,205)(51,174,290,206)(52,175,291,207)(53,176,292,208)(54,177,293,209)(55,178,294,210)(56,179,295,211)(57,180,296,212)(58,121,297,213)(59,122,298,214)(60,123,299,215)(61,369,308,448)(62,370,309,449)(63,371,310,450)(64,372,311,451)(65,373,312,452)(66,374,313,453)(67,375,314,454)(68,376,315,455)(69,377,316,456)(70,378,317,457)(71,379,318,458)(72,380,319,459)(73,381,320,460)(74,382,321,461)(75,383,322,462)(76,384,323,463)(77,385,324,464)(78,386,325,465)(79,387,326,466)(80,388,327,467)(81,389,328,468)(82,390,329,469)(83,391,330,470)(84,392,331,471)(85,393,332,472)(86,394,333,473)(87,395,334,474)(88,396,335,475)(89,397,336,476)(90,398,337,477)(91,399,338,478)(92,400,339,479)(93,401,340,480)(94,402,341,421)(95,403,342,422)(96,404,343,423)(97,405,344,424)(98,406,345,425)(99,407,346,426)(100,408,347,427)(101,409,348,428)(102,410,349,429)(103,411,350,430)(104,412,351,431)(105,413,352,432)(106,414,353,433)(107,415,354,434)(108,416,355,435)(109,417,356,436)(110,418,357,437)(111,419,358,438)(112,420,359,439)(113,361,360,440)(114,362,301,441)(115,363,302,442)(116,364,303,443)(117,365,304,444)(118,366,305,445)(119,367,306,446)(120,368,307,447), (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), (1,414,31,384)(2,413,32,383)(3,412,33,382)(4,411,34,381)(5,410,35,380)(6,409,36,379)(7,408,37,378)(8,407,38,377)(9,406,39,376)(10,405,40,375)(11,404,41,374)(12,403,42,373)(13,402,43,372)(14,401,44,371)(15,400,45,370)(16,399,46,369)(17,398,47,368)(18,397,48,367)(19,396,49,366)(20,395,50,365)(21,394,51,364)(22,393,52,363)(23,392,53,362)(24,391,54,361)(25,390,55,420)(26,389,56,419)(27,388,57,418)(28,387,58,417)(29,386,59,416)(30,385,60,415)(61,231,91,201)(62,230,92,200)(63,229,93,199)(64,228,94,198)(65,227,95,197)(66,226,96,196)(67,225,97,195)(68,224,98,194)(69,223,99,193)(70,222,100,192)(71,221,101,191)(72,220,102,190)(73,219,103,189)(74,218,104,188)(75,217,105,187)(76,216,106,186)(77,215,107,185)(78,214,108,184)(79,213,109,183)(80,212,110,182)(81,211,111,181)(82,210,112,240)(83,209,113,239)(84,208,114,238)(85,207,115,237)(86,206,116,236)(87,205,117,235)(88,204,118,234)(89,203,119,233)(90,202,120,232)(121,356,151,326)(122,355,152,325)(123,354,153,324)(124,353,154,323)(125,352,155,322)(126,351,156,321)(127,350,157,320)(128,349,158,319)(129,348,159,318)(130,347,160,317)(131,346,161,316)(132,345,162,315)(133,344,163,314)(134,343,164,313)(135,342,165,312)(136,341,166,311)(137,340,167,310)(138,339,168,309)(139,338,169,308)(140,337,170,307)(141,336,171,306)(142,335,172,305)(143,334,173,304)(144,333,174,303)(145,332,175,302)(146,331,176,301)(147,330,177,360)(148,329,178,359)(149,328,179,358)(150,327,180,357)(241,432,271,462)(242,431,272,461)(243,430,273,460)(244,429,274,459)(245,428,275,458)(246,427,276,457)(247,426,277,456)(248,425,278,455)(249,424,279,454)(250,423,280,453)(251,422,281,452)(252,421,282,451)(253,480,283,450)(254,479,284,449)(255,478,285,448)(256,477,286,447)(257,476,287,446)(258,475,288,445)(259,474,289,444)(260,473,290,443)(261,472,291,442)(262,471,292,441)(263,470,293,440)(264,469,294,439)(265,468,295,438)(266,467,296,437)(267,466,297,436)(268,465,298,435)(269,464,299,434)(270,463,300,433) );
G=PermutationGroup([[(1,124,300,216),(2,125,241,217),(3,126,242,218),(4,127,243,219),(5,128,244,220),(6,129,245,221),(7,130,246,222),(8,131,247,223),(9,132,248,224),(10,133,249,225),(11,134,250,226),(12,135,251,227),(13,136,252,228),(14,137,253,229),(15,138,254,230),(16,139,255,231),(17,140,256,232),(18,141,257,233),(19,142,258,234),(20,143,259,235),(21,144,260,236),(22,145,261,237),(23,146,262,238),(24,147,263,239),(25,148,264,240),(26,149,265,181),(27,150,266,182),(28,151,267,183),(29,152,268,184),(30,153,269,185),(31,154,270,186),(32,155,271,187),(33,156,272,188),(34,157,273,189),(35,158,274,190),(36,159,275,191),(37,160,276,192),(38,161,277,193),(39,162,278,194),(40,163,279,195),(41,164,280,196),(42,165,281,197),(43,166,282,198),(44,167,283,199),(45,168,284,200),(46,169,285,201),(47,170,286,202),(48,171,287,203),(49,172,288,204),(50,173,289,205),(51,174,290,206),(52,175,291,207),(53,176,292,208),(54,177,293,209),(55,178,294,210),(56,179,295,211),(57,180,296,212),(58,121,297,213),(59,122,298,214),(60,123,299,215),(61,369,308,448),(62,370,309,449),(63,371,310,450),(64,372,311,451),(65,373,312,452),(66,374,313,453),(67,375,314,454),(68,376,315,455),(69,377,316,456),(70,378,317,457),(71,379,318,458),(72,380,319,459),(73,381,320,460),(74,382,321,461),(75,383,322,462),(76,384,323,463),(77,385,324,464),(78,386,325,465),(79,387,326,466),(80,388,327,467),(81,389,328,468),(82,390,329,469),(83,391,330,470),(84,392,331,471),(85,393,332,472),(86,394,333,473),(87,395,334,474),(88,396,335,475),(89,397,336,476),(90,398,337,477),(91,399,338,478),(92,400,339,479),(93,401,340,480),(94,402,341,421),(95,403,342,422),(96,404,343,423),(97,405,344,424),(98,406,345,425),(99,407,346,426),(100,408,347,427),(101,409,348,428),(102,410,349,429),(103,411,350,430),(104,412,351,431),(105,413,352,432),(106,414,353,433),(107,415,354,434),(108,416,355,435),(109,417,356,436),(110,418,357,437),(111,419,358,438),(112,420,359,439),(113,361,360,440),(114,362,301,441),(115,363,302,442),(116,364,303,443),(117,365,304,444),(118,366,305,445),(119,367,306,446),(120,368,307,447)], [(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)], [(1,414,31,384),(2,413,32,383),(3,412,33,382),(4,411,34,381),(5,410,35,380),(6,409,36,379),(7,408,37,378),(8,407,38,377),(9,406,39,376),(10,405,40,375),(11,404,41,374),(12,403,42,373),(13,402,43,372),(14,401,44,371),(15,400,45,370),(16,399,46,369),(17,398,47,368),(18,397,48,367),(19,396,49,366),(20,395,50,365),(21,394,51,364),(22,393,52,363),(23,392,53,362),(24,391,54,361),(25,390,55,420),(26,389,56,419),(27,388,57,418),(28,387,58,417),(29,386,59,416),(30,385,60,415),(61,231,91,201),(62,230,92,200),(63,229,93,199),(64,228,94,198),(65,227,95,197),(66,226,96,196),(67,225,97,195),(68,224,98,194),(69,223,99,193),(70,222,100,192),(71,221,101,191),(72,220,102,190),(73,219,103,189),(74,218,104,188),(75,217,105,187),(76,216,106,186),(77,215,107,185),(78,214,108,184),(79,213,109,183),(80,212,110,182),(81,211,111,181),(82,210,112,240),(83,209,113,239),(84,208,114,238),(85,207,115,237),(86,206,116,236),(87,205,117,235),(88,204,118,234),(89,203,119,233),(90,202,120,232),(121,356,151,326),(122,355,152,325),(123,354,153,324),(124,353,154,323),(125,352,155,322),(126,351,156,321),(127,350,157,320),(128,349,158,319),(129,348,159,318),(130,347,160,317),(131,346,161,316),(132,345,162,315),(133,344,163,314),(134,343,164,313),(135,342,165,312),(136,341,166,311),(137,340,167,310),(138,339,168,309),(139,338,169,308),(140,337,170,307),(141,336,171,306),(142,335,172,305),(143,334,173,304),(144,333,174,303),(145,332,175,302),(146,331,176,301),(147,330,177,360),(148,329,178,359),(149,328,179,358),(150,327,180,357),(241,432,271,462),(242,431,272,461),(243,430,273,460),(244,429,274,459),(245,428,275,458),(246,427,276,457),(247,426,277,456),(248,425,278,455),(249,424,279,454),(250,423,280,453),(251,422,281,452),(252,421,282,451),(253,480,283,450),(254,479,284,449),(255,478,285,448),(256,477,286,447),(257,476,287,446),(258,475,288,445),(259,474,289,444),(260,473,290,443),(261,472,291,442),(262,471,292,441),(263,470,293,440),(264,469,294,439),(265,468,295,438),(266,467,296,437),(267,466,297,436),(268,465,298,435),(269,464,299,434),(270,463,300,433)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 30 | ··· | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D5 | D6 | C4○D4 | D10 | Dic6 | C4×S3 | D15 | Dic10 | C4×D5 | C4○D12 | D30 | C4○D20 | Dic30 | C4×D15 | D60⋊11C2 |
| kernel | C4×Dic30 | C4×Dic15 | C30.4Q8 | C60⋊5C4 | C4×C60 | C2×Dic30 | Dic30 | C4×C20 | C60 | C4×C12 | C2×C20 | C30 | C2×C12 | C20 | C20 | C42 | C12 | C12 | C10 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 4 | 8 | 8 | 4 | 12 | 8 | 16 | 16 | 16 |
Matrix representation of C4×Dic30 ►in GL4(𝔽61) generated by
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 0 | 60 | 0 | 0 |
| 1 | 18 | 0 | 0 |
| 0 | 0 | 2 | 22 |
| 0 | 0 | 39 | 33 |
| 23 | 53 | 0 | 0 |
| 5 | 38 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 46 | 50 |
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,50,0,0,0,0,50],[0,1,0,0,60,18,0,0,0,0,2,39,0,0,22,33],[23,5,0,0,53,38,0,0,0,0,11,46,0,0,0,50] >;
C4×Dic30 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{30} % in TeX
G:=Group("C4xDic30"); // GroupNames label
G:=SmallGroup(480,833);
// by ID
G=gap.SmallGroup(480,833);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,120,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^60=1,c^2=b^30,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations