metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.24C42, (C2×C30).5Q8, (C6×Dic5)⋊5C4, (C2×C6).36D20, (C2×C30).35D4, C30.34(C4⋊C4), C6.7(C4×Dic5), (C10×Dic3)⋊8C4, (C2×C10).36D12, C6.5(C4⋊Dic5), (C2×C6).6Dic10, (C2×C10).6Dic6, C23.59(S3×D5), C10.25(D6⋊C4), C22.4(C15⋊Q8), (C2×Dic5)⋊3Dic3, (C2×Dic3)⋊2Dic5, (C2×Dic15)⋊11C4, C2.2(D6⋊Dic5), C5⋊3(C6.C42), C10.19(C4×Dic3), C2.7(Dic3×Dic5), (C22×C10).96D6, (C22×C6).79D10, C2.2(D30⋊4C4), C30.65(C22⋊C4), C10.12(C4⋊Dic3), C15⋊4(C2.C42), C6.9(C10.D4), C6.10(C23.D5), C2.2(C30.Q8), C2.2(Dic15⋊5C4), C2.2(C6.Dic10), C6.10(D10⋊C4), (C22×Dic3).1D5, (C22×Dic5).3S3, C22.12(D5×Dic3), C22.12(S3×Dic5), C10.16(Dic3⋊C4), C3⋊1(C10.10C42), C2.2(D10⋊Dic3), C22.18(C15⋊D4), C22.18(C3⋊D20), C22.18(C5⋊D12), (C22×C30).25C22, (C22×Dic15).7C2, C10.21(C6.D4), C22.12(D30.C2), (C2×C6).48(C4×D5), (C2×C6×Dic5).1C2, (C2×C30).89(C2×C4), (C2×C10).43(C4×S3), (Dic3×C2×C10).1C2, (C2×C6).50(C5⋊D4), (C2×C6).13(C2×Dic5), (C2×C10).29(C3⋊D4), (C2×C10).33(C2×Dic3), SmallGroup(480,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.24C42
G = < a,b,c | a30=b4=c4=1, bab-1=a19, cac-1=a11, cbc-1=a15b >
Subgroups: 572 in 152 conjugacy classes, 80 normal (70 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C23, C10, Dic3, C12, C2×C6, C15, C22×C4, Dic5, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C2.C42, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C22×Dic3, C22×Dic3, C22×C12, C5×Dic3, C3×Dic5, Dic15, C2×C30, C22×Dic5, C22×Dic5, C22×C20, C6.C42, C6×Dic5, C6×Dic5, C10×Dic3, C10×Dic3, C2×Dic15, C2×Dic15, C22×C30, C10.10C42, C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C30.24C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic5, D10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, S3×D5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C6.C42, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C10.10C42, Dic3×Dic5, D10⋊Dic3, D6⋊Dic5, D30⋊4C4, C30.Q8, Dic15⋊5C4, C6.Dic10, C30.24C42
►Regular action on 480 points
Generators in S480
(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)
(1 311 82 251)(2 330 83 270)(3 319 84 259)(4 308 85 248)(5 327 86 267)(6 316 87 256)(7 305 88 245)(8 324 89 264)(9 313 90 253)(10 302 61 242)(11 321 62 261)(12 310 63 250)(13 329 64 269)(14 318 65 258)(15 307 66 247)(16 326 67 266)(17 315 68 255)(18 304 69 244)(19 323 70 263)(20 312 71 252)(21 301 72 241)(22 320 73 260)(23 309 74 249)(24 328 75 268)(25 317 76 257)(26 306 77 246)(27 325 78 265)(28 314 79 254)(29 303 80 243)(30 322 81 262)(31 351 101 291)(32 340 102 280)(33 359 103 299)(34 348 104 288)(35 337 105 277)(36 356 106 296)(37 345 107 285)(38 334 108 274)(39 353 109 293)(40 342 110 282)(41 331 111 271)(42 350 112 290)(43 339 113 279)(44 358 114 298)(45 347 115 287)(46 336 116 276)(47 355 117 295)(48 344 118 284)(49 333 119 273)(50 352 120 292)(51 341 91 281)(52 360 92 300)(53 349 93 289)(54 338 94 278)(55 357 95 297)(56 346 96 286)(57 335 97 275)(58 354 98 294)(59 343 99 283)(60 332 100 272)(121 446 202 361)(122 435 203 380)(123 424 204 369)(124 443 205 388)(125 432 206 377)(126 421 207 366)(127 440 208 385)(128 429 209 374)(129 448 210 363)(130 437 181 382)(131 426 182 371)(132 445 183 390)(133 434 184 379)(134 423 185 368)(135 442 186 387)(136 431 187 376)(137 450 188 365)(138 439 189 384)(139 428 190 373)(140 447 191 362)(141 436 192 381)(142 425 193 370)(143 444 194 389)(144 433 195 378)(145 422 196 367)(146 441 197 386)(147 430 198 375)(148 449 199 364)(149 438 200 383)(150 427 201 372)(151 474 235 404)(152 463 236 393)(153 452 237 412)(154 471 238 401)(155 460 239 420)(156 479 240 409)(157 468 211 398)(158 457 212 417)(159 476 213 406)(160 465 214 395)(161 454 215 414)(162 473 216 403)(163 462 217 392)(164 451 218 411)(165 470 219 400)(166 459 220 419)(167 478 221 408)(168 467 222 397)(169 456 223 416)(170 475 224 405)(171 464 225 394)(172 453 226 413)(173 472 227 402)(174 461 228 391)(175 480 229 410)(176 469 230 399)(177 458 231 418)(178 477 232 407)(179 466 233 396)(180 455 234 415)
(1 164 51 121)(2 175 52 132)(3 156 53 143)(4 167 54 124)(5 178 55 135)(6 159 56 146)(7 170 57 127)(8 151 58 138)(9 162 59 149)(10 173 60 130)(11 154 31 141)(12 165 32 122)(13 176 33 133)(14 157 34 144)(15 168 35 125)(16 179 36 136)(17 160 37 147)(18 171 38 128)(19 152 39 139)(20 163 40 150)(21 174 41 131)(22 155 42 142)(23 166 43 123)(24 177 44 134)(25 158 45 145)(26 169 46 126)(27 180 47 137)(28 161 48 148)(29 172 49 129)(30 153 50 140)(61 227 100 181)(62 238 101 192)(63 219 102 203)(64 230 103 184)(65 211 104 195)(66 222 105 206)(67 233 106 187)(68 214 107 198)(69 225 108 209)(70 236 109 190)(71 217 110 201)(72 228 111 182)(73 239 112 193)(74 220 113 204)(75 231 114 185)(76 212 115 196)(77 223 116 207)(78 234 117 188)(79 215 118 199)(80 226 119 210)(81 237 120 191)(82 218 91 202)(83 229 92 183)(84 240 93 194)(85 221 94 205)(86 232 95 186)(87 213 96 197)(88 224 97 208)(89 235 98 189)(90 216 99 200)(241 406 271 386)(242 417 272 367)(243 398 273 378)(244 409 274 389)(245 420 275 370)(246 401 276 381)(247 412 277 362)(248 393 278 373)(249 404 279 384)(250 415 280 365)(251 396 281 376)(252 407 282 387)(253 418 283 368)(254 399 284 379)(255 410 285 390)(256 391 286 371)(257 402 287 382)(258 413 288 363)(259 394 289 374)(260 405 290 385)(261 416 291 366)(262 397 292 377)(263 408 293 388)(264 419 294 369)(265 400 295 380)(266 411 296 361)(267 392 297 372)(268 403 298 383)(269 414 299 364)(270 395 300 375)(301 476 331 441)(302 457 332 422)(303 468 333 433)(304 479 334 444)(305 460 335 425)(306 471 336 436)(307 452 337 447)(308 463 338 428)(309 474 339 439)(310 455 340 450)(311 466 341 431)(312 477 342 442)(313 458 343 423)(314 469 344 434)(315 480 345 445)(316 461 346 426)(317 472 347 437)(318 453 348 448)(319 464 349 429)(320 475 350 440)(321 456 351 421)(322 467 352 432)(323 478 353 443)(324 459 354 424)(325 470 355 435)(326 451 356 446)(327 462 357 427)(328 473 358 438)(329 454 359 449)(330 465 360 430)
G:=sub<Sym(480)| (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), (1,311,82,251)(2,330,83,270)(3,319,84,259)(4,308,85,248)(5,327,86,267)(6,316,87,256)(7,305,88,245)(8,324,89,264)(9,313,90,253)(10,302,61,242)(11,321,62,261)(12,310,63,250)(13,329,64,269)(14,318,65,258)(15,307,66,247)(16,326,67,266)(17,315,68,255)(18,304,69,244)(19,323,70,263)(20,312,71,252)(21,301,72,241)(22,320,73,260)(23,309,74,249)(24,328,75,268)(25,317,76,257)(26,306,77,246)(27,325,78,265)(28,314,79,254)(29,303,80,243)(30,322,81,262)(31,351,101,291)(32,340,102,280)(33,359,103,299)(34,348,104,288)(35,337,105,277)(36,356,106,296)(37,345,107,285)(38,334,108,274)(39,353,109,293)(40,342,110,282)(41,331,111,271)(42,350,112,290)(43,339,113,279)(44,358,114,298)(45,347,115,287)(46,336,116,276)(47,355,117,295)(48,344,118,284)(49,333,119,273)(50,352,120,292)(51,341,91,281)(52,360,92,300)(53,349,93,289)(54,338,94,278)(55,357,95,297)(56,346,96,286)(57,335,97,275)(58,354,98,294)(59,343,99,283)(60,332,100,272)(121,446,202,361)(122,435,203,380)(123,424,204,369)(124,443,205,388)(125,432,206,377)(126,421,207,366)(127,440,208,385)(128,429,209,374)(129,448,210,363)(130,437,181,382)(131,426,182,371)(132,445,183,390)(133,434,184,379)(134,423,185,368)(135,442,186,387)(136,431,187,376)(137,450,188,365)(138,439,189,384)(139,428,190,373)(140,447,191,362)(141,436,192,381)(142,425,193,370)(143,444,194,389)(144,433,195,378)(145,422,196,367)(146,441,197,386)(147,430,198,375)(148,449,199,364)(149,438,200,383)(150,427,201,372)(151,474,235,404)(152,463,236,393)(153,452,237,412)(154,471,238,401)(155,460,239,420)(156,479,240,409)(157,468,211,398)(158,457,212,417)(159,476,213,406)(160,465,214,395)(161,454,215,414)(162,473,216,403)(163,462,217,392)(164,451,218,411)(165,470,219,400)(166,459,220,419)(167,478,221,408)(168,467,222,397)(169,456,223,416)(170,475,224,405)(171,464,225,394)(172,453,226,413)(173,472,227,402)(174,461,228,391)(175,480,229,410)(176,469,230,399)(177,458,231,418)(178,477,232,407)(179,466,233,396)(180,455,234,415), (1,164,51,121)(2,175,52,132)(3,156,53,143)(4,167,54,124)(5,178,55,135)(6,159,56,146)(7,170,57,127)(8,151,58,138)(9,162,59,149)(10,173,60,130)(11,154,31,141)(12,165,32,122)(13,176,33,133)(14,157,34,144)(15,168,35,125)(16,179,36,136)(17,160,37,147)(18,171,38,128)(19,152,39,139)(20,163,40,150)(21,174,41,131)(22,155,42,142)(23,166,43,123)(24,177,44,134)(25,158,45,145)(26,169,46,126)(27,180,47,137)(28,161,48,148)(29,172,49,129)(30,153,50,140)(61,227,100,181)(62,238,101,192)(63,219,102,203)(64,230,103,184)(65,211,104,195)(66,222,105,206)(67,233,106,187)(68,214,107,198)(69,225,108,209)(70,236,109,190)(71,217,110,201)(72,228,111,182)(73,239,112,193)(74,220,113,204)(75,231,114,185)(76,212,115,196)(77,223,116,207)(78,234,117,188)(79,215,118,199)(80,226,119,210)(81,237,120,191)(82,218,91,202)(83,229,92,183)(84,240,93,194)(85,221,94,205)(86,232,95,186)(87,213,96,197)(88,224,97,208)(89,235,98,189)(90,216,99,200)(241,406,271,386)(242,417,272,367)(243,398,273,378)(244,409,274,389)(245,420,275,370)(246,401,276,381)(247,412,277,362)(248,393,278,373)(249,404,279,384)(250,415,280,365)(251,396,281,376)(252,407,282,387)(253,418,283,368)(254,399,284,379)(255,410,285,390)(256,391,286,371)(257,402,287,382)(258,413,288,363)(259,394,289,374)(260,405,290,385)(261,416,291,366)(262,397,292,377)(263,408,293,388)(264,419,294,369)(265,400,295,380)(266,411,296,361)(267,392,297,372)(268,403,298,383)(269,414,299,364)(270,395,300,375)(301,476,331,441)(302,457,332,422)(303,468,333,433)(304,479,334,444)(305,460,335,425)(306,471,336,436)(307,452,337,447)(308,463,338,428)(309,474,339,439)(310,455,340,450)(311,466,341,431)(312,477,342,442)(313,458,343,423)(314,469,344,434)(315,480,345,445)(316,461,346,426)(317,472,347,437)(318,453,348,448)(319,464,349,429)(320,475,350,440)(321,456,351,421)(322,467,352,432)(323,478,353,443)(324,459,354,424)(325,470,355,435)(326,451,356,446)(327,462,357,427)(328,473,358,438)(329,454,359,449)(330,465,360,430)>;
G:=Group( (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), (1,311,82,251)(2,330,83,270)(3,319,84,259)(4,308,85,248)(5,327,86,267)(6,316,87,256)(7,305,88,245)(8,324,89,264)(9,313,90,253)(10,302,61,242)(11,321,62,261)(12,310,63,250)(13,329,64,269)(14,318,65,258)(15,307,66,247)(16,326,67,266)(17,315,68,255)(18,304,69,244)(19,323,70,263)(20,312,71,252)(21,301,72,241)(22,320,73,260)(23,309,74,249)(24,328,75,268)(25,317,76,257)(26,306,77,246)(27,325,78,265)(28,314,79,254)(29,303,80,243)(30,322,81,262)(31,351,101,291)(32,340,102,280)(33,359,103,299)(34,348,104,288)(35,337,105,277)(36,356,106,296)(37,345,107,285)(38,334,108,274)(39,353,109,293)(40,342,110,282)(41,331,111,271)(42,350,112,290)(43,339,113,279)(44,358,114,298)(45,347,115,287)(46,336,116,276)(47,355,117,295)(48,344,118,284)(49,333,119,273)(50,352,120,292)(51,341,91,281)(52,360,92,300)(53,349,93,289)(54,338,94,278)(55,357,95,297)(56,346,96,286)(57,335,97,275)(58,354,98,294)(59,343,99,283)(60,332,100,272)(121,446,202,361)(122,435,203,380)(123,424,204,369)(124,443,205,388)(125,432,206,377)(126,421,207,366)(127,440,208,385)(128,429,209,374)(129,448,210,363)(130,437,181,382)(131,426,182,371)(132,445,183,390)(133,434,184,379)(134,423,185,368)(135,442,186,387)(136,431,187,376)(137,450,188,365)(138,439,189,384)(139,428,190,373)(140,447,191,362)(141,436,192,381)(142,425,193,370)(143,444,194,389)(144,433,195,378)(145,422,196,367)(146,441,197,386)(147,430,198,375)(148,449,199,364)(149,438,200,383)(150,427,201,372)(151,474,235,404)(152,463,236,393)(153,452,237,412)(154,471,238,401)(155,460,239,420)(156,479,240,409)(157,468,211,398)(158,457,212,417)(159,476,213,406)(160,465,214,395)(161,454,215,414)(162,473,216,403)(163,462,217,392)(164,451,218,411)(165,470,219,400)(166,459,220,419)(167,478,221,408)(168,467,222,397)(169,456,223,416)(170,475,224,405)(171,464,225,394)(172,453,226,413)(173,472,227,402)(174,461,228,391)(175,480,229,410)(176,469,230,399)(177,458,231,418)(178,477,232,407)(179,466,233,396)(180,455,234,415), (1,164,51,121)(2,175,52,132)(3,156,53,143)(4,167,54,124)(5,178,55,135)(6,159,56,146)(7,170,57,127)(8,151,58,138)(9,162,59,149)(10,173,60,130)(11,154,31,141)(12,165,32,122)(13,176,33,133)(14,157,34,144)(15,168,35,125)(16,179,36,136)(17,160,37,147)(18,171,38,128)(19,152,39,139)(20,163,40,150)(21,174,41,131)(22,155,42,142)(23,166,43,123)(24,177,44,134)(25,158,45,145)(26,169,46,126)(27,180,47,137)(28,161,48,148)(29,172,49,129)(30,153,50,140)(61,227,100,181)(62,238,101,192)(63,219,102,203)(64,230,103,184)(65,211,104,195)(66,222,105,206)(67,233,106,187)(68,214,107,198)(69,225,108,209)(70,236,109,190)(71,217,110,201)(72,228,111,182)(73,239,112,193)(74,220,113,204)(75,231,114,185)(76,212,115,196)(77,223,116,207)(78,234,117,188)(79,215,118,199)(80,226,119,210)(81,237,120,191)(82,218,91,202)(83,229,92,183)(84,240,93,194)(85,221,94,205)(86,232,95,186)(87,213,96,197)(88,224,97,208)(89,235,98,189)(90,216,99,200)(241,406,271,386)(242,417,272,367)(243,398,273,378)(244,409,274,389)(245,420,275,370)(246,401,276,381)(247,412,277,362)(248,393,278,373)(249,404,279,384)(250,415,280,365)(251,396,281,376)(252,407,282,387)(253,418,283,368)(254,399,284,379)(255,410,285,390)(256,391,286,371)(257,402,287,382)(258,413,288,363)(259,394,289,374)(260,405,290,385)(261,416,291,366)(262,397,292,377)(263,408,293,388)(264,419,294,369)(265,400,295,380)(266,411,296,361)(267,392,297,372)(268,403,298,383)(269,414,299,364)(270,395,300,375)(301,476,331,441)(302,457,332,422)(303,468,333,433)(304,479,334,444)(305,460,335,425)(306,471,336,436)(307,452,337,447)(308,463,338,428)(309,474,339,439)(310,455,340,450)(311,466,341,431)(312,477,342,442)(313,458,343,423)(314,469,344,434)(315,480,345,445)(316,461,346,426)(317,472,347,437)(318,453,348,448)(319,464,349,429)(320,475,350,440)(321,456,351,421)(322,467,352,432)(323,478,353,443)(324,459,354,424)(325,470,355,435)(326,451,356,446)(327,462,357,427)(328,473,358,438)(329,454,359,449)(330,465,360,430) );
G=PermutationGroup([[(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)], [(1,311,82,251),(2,330,83,270),(3,319,84,259),(4,308,85,248),(5,327,86,267),(6,316,87,256),(7,305,88,245),(8,324,89,264),(9,313,90,253),(10,302,61,242),(11,321,62,261),(12,310,63,250),(13,329,64,269),(14,318,65,258),(15,307,66,247),(16,326,67,266),(17,315,68,255),(18,304,69,244),(19,323,70,263),(20,312,71,252),(21,301,72,241),(22,320,73,260),(23,309,74,249),(24,328,75,268),(25,317,76,257),(26,306,77,246),(27,325,78,265),(28,314,79,254),(29,303,80,243),(30,322,81,262),(31,351,101,291),(32,340,102,280),(33,359,103,299),(34,348,104,288),(35,337,105,277),(36,356,106,296),(37,345,107,285),(38,334,108,274),(39,353,109,293),(40,342,110,282),(41,331,111,271),(42,350,112,290),(43,339,113,279),(44,358,114,298),(45,347,115,287),(46,336,116,276),(47,355,117,295),(48,344,118,284),(49,333,119,273),(50,352,120,292),(51,341,91,281),(52,360,92,300),(53,349,93,289),(54,338,94,278),(55,357,95,297),(56,346,96,286),(57,335,97,275),(58,354,98,294),(59,343,99,283),(60,332,100,272),(121,446,202,361),(122,435,203,380),(123,424,204,369),(124,443,205,388),(125,432,206,377),(126,421,207,366),(127,440,208,385),(128,429,209,374),(129,448,210,363),(130,437,181,382),(131,426,182,371),(132,445,183,390),(133,434,184,379),(134,423,185,368),(135,442,186,387),(136,431,187,376),(137,450,188,365),(138,439,189,384),(139,428,190,373),(140,447,191,362),(141,436,192,381),(142,425,193,370),(143,444,194,389),(144,433,195,378),(145,422,196,367),(146,441,197,386),(147,430,198,375),(148,449,199,364),(149,438,200,383),(150,427,201,372),(151,474,235,404),(152,463,236,393),(153,452,237,412),(154,471,238,401),(155,460,239,420),(156,479,240,409),(157,468,211,398),(158,457,212,417),(159,476,213,406),(160,465,214,395),(161,454,215,414),(162,473,216,403),(163,462,217,392),(164,451,218,411),(165,470,219,400),(166,459,220,419),(167,478,221,408),(168,467,222,397),(169,456,223,416),(170,475,224,405),(171,464,225,394),(172,453,226,413),(173,472,227,402),(174,461,228,391),(175,480,229,410),(176,469,230,399),(177,458,231,418),(178,477,232,407),(179,466,233,396),(180,455,234,415)], [(1,164,51,121),(2,175,52,132),(3,156,53,143),(4,167,54,124),(5,178,55,135),(6,159,56,146),(7,170,57,127),(8,151,58,138),(9,162,59,149),(10,173,60,130),(11,154,31,141),(12,165,32,122),(13,176,33,133),(14,157,34,144),(15,168,35,125),(16,179,36,136),(17,160,37,147),(18,171,38,128),(19,152,39,139),(20,163,40,150),(21,174,41,131),(22,155,42,142),(23,166,43,123),(24,177,44,134),(25,158,45,145),(26,169,46,126),(27,180,47,137),(28,161,48,148),(29,172,49,129),(30,153,50,140),(61,227,100,181),(62,238,101,192),(63,219,102,203),(64,230,103,184),(65,211,104,195),(66,222,105,206),(67,233,106,187),(68,214,107,198),(69,225,108,209),(70,236,109,190),(71,217,110,201),(72,228,111,182),(73,239,112,193),(74,220,113,204),(75,231,114,185),(76,212,115,196),(77,223,116,207),(78,234,117,188),(79,215,118,199),(80,226,119,210),(81,237,120,191),(82,218,91,202),(83,229,92,183),(84,240,93,194),(85,221,94,205),(86,232,95,186),(87,213,96,197),(88,224,97,208),(89,235,98,189),(90,216,99,200),(241,406,271,386),(242,417,272,367),(243,398,273,378),(244,409,274,389),(245,420,275,370),(246,401,276,381),(247,412,277,362),(248,393,278,373),(249,404,279,384),(250,415,280,365),(251,396,281,376),(252,407,282,387),(253,418,283,368),(254,399,284,379),(255,410,285,390),(256,391,286,371),(257,402,287,382),(258,413,288,363),(259,394,289,374),(260,405,290,385),(261,416,291,366),(262,397,292,377),(263,408,293,388),(264,419,294,369),(265,400,295,380),(266,411,296,361),(267,392,297,372),(268,403,298,383),(269,414,299,364),(270,395,300,375),(301,476,331,441),(302,457,332,422),(303,468,333,433),(304,479,334,444),(305,460,335,425),(306,471,336,436),(307,452,337,447),(308,463,338,428),(309,474,339,439),(310,455,340,450),(311,466,341,431),(312,477,342,442),(313,458,343,423),(314,469,344,434),(315,480,345,445),(316,461,346,426),(317,472,347,437),(318,453,348,448),(319,464,349,429),(320,475,350,440),(321,456,351,421),(322,467,352,432),(323,478,353,443),(324,459,354,424),(325,470,355,435),(326,451,356,446),(327,462,357,427),(328,473,358,438),(329,454,359,449),(330,465,360,430)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 20A | ··· | 20P | 30A | ··· | 30N |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | - | + | - | + | + | - | - | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | D5 | Dic3 | D6 | Dic5 | D10 | Dic6 | C4×S3 | D12 | C3⋊D4 | Dic10 | C4×D5 | D20 | C5⋊D4 | S3×D5 | D5×Dic3 | S3×Dic5 | D30.C2 | C15⋊D4 | C3⋊D20 | C5⋊D12 | C15⋊Q8 |
| kernel | C30.24C42 | C2×C6×Dic5 | Dic3×C2×C10 | C22×Dic15 | C6×Dic5 | C10×Dic3 | C2×Dic15 | C22×Dic5 | C2×C30 | C2×C30 | C22×Dic3 | C2×Dic5 | C22×C10 | C2×Dic3 | C22×C6 | C2×C10 | C2×C10 | C2×C10 | C2×C10 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 3 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of C30.24C42 ►in GL6(𝔽61)
| 60 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 44 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 12 |
| 0 | 0 | 0 | 0 | 53 | 55 |
| 52 | 9 | 0 | 0 | 0 | 0 |
| 18 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 54 |
| 0 | 0 | 0 | 0 | 7 | 32 |
G:=sub<GL(6,GF(61))| [60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,60,44],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,6,53,0,0,0,0,12,55],[52,18,0,0,0,0,9,9,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,29,7,0,0,0,0,54,32] >;
C30.24C42 in GAP, Magma, Sage, TeX
C_{30}._{24}C_4^2 % in TeX
G:=Group("C30.24C4^2"); // GroupNames label
G:=SmallGroup(480,70);
// by ID
G=gap.SmallGroup(480,70);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=c^4=1,b*a*b^-1=a^19,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations