metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.21C42, C30.11M4(2), C3⋊C8⋊3Dic5, C15⋊3C8⋊15C4, C15⋊8(C8⋊C4), C3⋊2(C40⋊8C4), C12.39(C4×D5), C6.4(C4×Dic5), C20.108(C4×S3), C60.144(C2×C4), (C2×C20).318D6, C6.7(C8⋊D5), (C4×Dic5).6S3, (C6×Dic5).6C4, C4.20(S3×Dic5), (C2×C12).322D10, C5⋊3(C42.S3), (C12×Dic5).7C2, C10.16(C4×Dic3), C12.25(C2×Dic5), C2.4(Dic3×Dic5), C22.9(D5×Dic3), (C2×C60).220C22, C4.20(D30.C2), (C2×Dic5).3Dic3, C10.6(C4.Dic3), C2.1(C20.32D6), (C5×C3⋊C8)⋊11C4, (C2×C3⋊C8).7D5, (C10×C3⋊C8).9C2, (C2×C6).43(C4×D5), (C2×C30).74(C2×C4), (C2×C4).223(S3×D5), (C2×C15⋊3C8).19C2, (C2×C10).30(C2×Dic3), SmallGroup(480,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.21C42
G = < a,b,c | a30=b4=1, c4=a15, bab-1=a19, cac-1=a11, cbc-1=a15b >
Subgroups: 236 in 80 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C30, C30, C8⋊C4, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C3×Dic5, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, C42.S3, C5×C3⋊C8, C15⋊3C8, C6×Dic5, C2×C60, C40⋊8C4, C12×Dic5, C10×C3⋊C8, C2×C15⋊3C8, C30.21C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, C42, M4(2), Dic5, D10, C4×S3, C2×Dic3, C8⋊C4, C4×D5, C2×Dic5, C4.Dic3, C4×Dic3, S3×D5, C8⋊D5, C4×Dic5, C42.S3, D5×Dic3, S3×Dic5, D30.C2, C40⋊8C4, C20.32D6, Dic3×Dic5, C30.21C42
►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)
(1 321 81 256)(2 310 82 245)(3 329 83 264)(4 318 84 253)(5 307 85 242)(6 326 86 261)(7 315 87 250)(8 304 88 269)(9 323 89 258)(10 312 90 247)(11 301 61 266)(12 320 62 255)(13 309 63 244)(14 328 64 263)(15 317 65 252)(16 306 66 241)(17 325 67 260)(18 314 68 249)(19 303 69 268)(20 322 70 257)(21 311 71 246)(22 330 72 265)(23 319 73 254)(24 308 74 243)(25 327 75 262)(26 316 76 251)(27 305 77 270)(28 324 78 259)(29 313 79 248)(30 302 80 267)(31 337 100 282)(32 356 101 271)(33 345 102 290)(34 334 103 279)(35 353 104 298)(36 342 105 287)(37 331 106 276)(38 350 107 295)(39 339 108 284)(40 358 109 273)(41 347 110 292)(42 336 111 281)(43 355 112 300)(44 344 113 289)(45 333 114 278)(46 352 115 297)(47 341 116 286)(48 360 117 275)(49 349 118 294)(50 338 119 283)(51 357 120 272)(52 346 91 291)(53 335 92 280)(54 354 93 299)(55 343 94 288)(56 332 95 277)(57 351 96 296)(58 340 97 285)(59 359 98 274)(60 348 99 293)(121 430 204 385)(122 449 205 374)(123 438 206 363)(124 427 207 382)(125 446 208 371)(126 435 209 390)(127 424 210 379)(128 443 181 368)(129 432 182 387)(130 421 183 376)(131 440 184 365)(132 429 185 384)(133 448 186 373)(134 437 187 362)(135 426 188 381)(136 445 189 370)(137 434 190 389)(138 423 191 378)(139 442 192 367)(140 431 193 386)(141 450 194 375)(142 439 195 364)(143 428 196 383)(144 447 197 372)(145 436 198 361)(146 425 199 380)(147 444 200 369)(148 433 201 388)(149 422 202 377)(150 441 203 366)(151 470 212 395)(152 459 213 414)(153 478 214 403)(154 467 215 392)(155 456 216 411)(156 475 217 400)(157 464 218 419)(158 453 219 408)(159 472 220 397)(160 461 221 416)(161 480 222 405)(162 469 223 394)(163 458 224 413)(164 477 225 402)(165 466 226 391)(166 455 227 410)(167 474 228 399)(168 463 229 418)(169 452 230 407)(170 471 231 396)(171 460 232 415)(172 479 233 404)(173 468 234 393)(174 457 235 412)(175 476 236 401)(176 465 237 420)(177 454 238 409)(178 473 239 398)(179 462 240 417)(180 451 211 406)
(1 180 32 150 16 165 47 135)(2 161 33 131 17 176 48 146)(3 172 34 142 18 157 49 127)(4 153 35 123 19 168 50 138)(5 164 36 134 20 179 51 149)(6 175 37 145 21 160 52 130)(7 156 38 126 22 171 53 141)(8 167 39 137 23 152 54 122)(9 178 40 148 24 163 55 133)(10 159 41 129 25 174 56 144)(11 170 42 140 26 155 57 125)(12 151 43 121 27 166 58 136)(13 162 44 132 28 177 59 147)(14 173 45 143 29 158 60 128)(15 154 46 124 30 169 31 139)(61 231 111 193 76 216 96 208)(62 212 112 204 77 227 97 189)(63 223 113 185 78 238 98 200)(64 234 114 196 79 219 99 181)(65 215 115 207 80 230 100 192)(66 226 116 188 81 211 101 203)(67 237 117 199 82 222 102 184)(68 218 118 210 83 233 103 195)(69 229 119 191 84 214 104 206)(70 240 120 202 85 225 105 187)(71 221 91 183 86 236 106 198)(72 232 92 194 87 217 107 209)(73 213 93 205 88 228 108 190)(74 224 94 186 89 239 109 201)(75 235 95 197 90 220 110 182)(241 406 286 366 256 391 271 381)(242 417 287 377 257 402 272 362)(243 398 288 388 258 413 273 373)(244 409 289 369 259 394 274 384)(245 420 290 380 260 405 275 365)(246 401 291 361 261 416 276 376)(247 412 292 372 262 397 277 387)(248 393 293 383 263 408 278 368)(249 404 294 364 264 419 279 379)(250 415 295 375 265 400 280 390)(251 396 296 386 266 411 281 371)(252 407 297 367 267 392 282 382)(253 418 298 378 268 403 283 363)(254 399 299 389 269 414 284 374)(255 410 300 370 270 395 285 385)(301 456 336 446 316 471 351 431)(302 467 337 427 317 452 352 442)(303 478 338 438 318 463 353 423)(304 459 339 449 319 474 354 434)(305 470 340 430 320 455 355 445)(306 451 341 441 321 466 356 426)(307 462 342 422 322 477 357 437)(308 473 343 433 323 458 358 448)(309 454 344 444 324 469 359 429)(310 465 345 425 325 480 360 440)(311 476 346 436 326 461 331 421)(312 457 347 447 327 472 332 432)(313 468 348 428 328 453 333 443)(314 479 349 439 329 464 334 424)(315 460 350 450 330 475 335 435)
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), (1,321,81,256)(2,310,82,245)(3,329,83,264)(4,318,84,253)(5,307,85,242)(6,326,86,261)(7,315,87,250)(8,304,88,269)(9,323,89,258)(10,312,90,247)(11,301,61,266)(12,320,62,255)(13,309,63,244)(14,328,64,263)(15,317,65,252)(16,306,66,241)(17,325,67,260)(18,314,68,249)(19,303,69,268)(20,322,70,257)(21,311,71,246)(22,330,72,265)(23,319,73,254)(24,308,74,243)(25,327,75,262)(26,316,76,251)(27,305,77,270)(28,324,78,259)(29,313,79,248)(30,302,80,267)(31,337,100,282)(32,356,101,271)(33,345,102,290)(34,334,103,279)(35,353,104,298)(36,342,105,287)(37,331,106,276)(38,350,107,295)(39,339,108,284)(40,358,109,273)(41,347,110,292)(42,336,111,281)(43,355,112,300)(44,344,113,289)(45,333,114,278)(46,352,115,297)(47,341,116,286)(48,360,117,275)(49,349,118,294)(50,338,119,283)(51,357,120,272)(52,346,91,291)(53,335,92,280)(54,354,93,299)(55,343,94,288)(56,332,95,277)(57,351,96,296)(58,340,97,285)(59,359,98,274)(60,348,99,293)(121,430,204,385)(122,449,205,374)(123,438,206,363)(124,427,207,382)(125,446,208,371)(126,435,209,390)(127,424,210,379)(128,443,181,368)(129,432,182,387)(130,421,183,376)(131,440,184,365)(132,429,185,384)(133,448,186,373)(134,437,187,362)(135,426,188,381)(136,445,189,370)(137,434,190,389)(138,423,191,378)(139,442,192,367)(140,431,193,386)(141,450,194,375)(142,439,195,364)(143,428,196,383)(144,447,197,372)(145,436,198,361)(146,425,199,380)(147,444,200,369)(148,433,201,388)(149,422,202,377)(150,441,203,366)(151,470,212,395)(152,459,213,414)(153,478,214,403)(154,467,215,392)(155,456,216,411)(156,475,217,400)(157,464,218,419)(158,453,219,408)(159,472,220,397)(160,461,221,416)(161,480,222,405)(162,469,223,394)(163,458,224,413)(164,477,225,402)(165,466,226,391)(166,455,227,410)(167,474,228,399)(168,463,229,418)(169,452,230,407)(170,471,231,396)(171,460,232,415)(172,479,233,404)(173,468,234,393)(174,457,235,412)(175,476,236,401)(176,465,237,420)(177,454,238,409)(178,473,239,398)(179,462,240,417)(180,451,211,406), (1,180,32,150,16,165,47,135)(2,161,33,131,17,176,48,146)(3,172,34,142,18,157,49,127)(4,153,35,123,19,168,50,138)(5,164,36,134,20,179,51,149)(6,175,37,145,21,160,52,130)(7,156,38,126,22,171,53,141)(8,167,39,137,23,152,54,122)(9,178,40,148,24,163,55,133)(10,159,41,129,25,174,56,144)(11,170,42,140,26,155,57,125)(12,151,43,121,27,166,58,136)(13,162,44,132,28,177,59,147)(14,173,45,143,29,158,60,128)(15,154,46,124,30,169,31,139)(61,231,111,193,76,216,96,208)(62,212,112,204,77,227,97,189)(63,223,113,185,78,238,98,200)(64,234,114,196,79,219,99,181)(65,215,115,207,80,230,100,192)(66,226,116,188,81,211,101,203)(67,237,117,199,82,222,102,184)(68,218,118,210,83,233,103,195)(69,229,119,191,84,214,104,206)(70,240,120,202,85,225,105,187)(71,221,91,183,86,236,106,198)(72,232,92,194,87,217,107,209)(73,213,93,205,88,228,108,190)(74,224,94,186,89,239,109,201)(75,235,95,197,90,220,110,182)(241,406,286,366,256,391,271,381)(242,417,287,377,257,402,272,362)(243,398,288,388,258,413,273,373)(244,409,289,369,259,394,274,384)(245,420,290,380,260,405,275,365)(246,401,291,361,261,416,276,376)(247,412,292,372,262,397,277,387)(248,393,293,383,263,408,278,368)(249,404,294,364,264,419,279,379)(250,415,295,375,265,400,280,390)(251,396,296,386,266,411,281,371)(252,407,297,367,267,392,282,382)(253,418,298,378,268,403,283,363)(254,399,299,389,269,414,284,374)(255,410,300,370,270,395,285,385)(301,456,336,446,316,471,351,431)(302,467,337,427,317,452,352,442)(303,478,338,438,318,463,353,423)(304,459,339,449,319,474,354,434)(305,470,340,430,320,455,355,445)(306,451,341,441,321,466,356,426)(307,462,342,422,322,477,357,437)(308,473,343,433,323,458,358,448)(309,454,344,444,324,469,359,429)(310,465,345,425,325,480,360,440)(311,476,346,436,326,461,331,421)(312,457,347,447,327,472,332,432)(313,468,348,428,328,453,333,443)(314,479,349,439,329,464,334,424)(315,460,350,450,330,475,335,435)>;
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), (1,321,81,256)(2,310,82,245)(3,329,83,264)(4,318,84,253)(5,307,85,242)(6,326,86,261)(7,315,87,250)(8,304,88,269)(9,323,89,258)(10,312,90,247)(11,301,61,266)(12,320,62,255)(13,309,63,244)(14,328,64,263)(15,317,65,252)(16,306,66,241)(17,325,67,260)(18,314,68,249)(19,303,69,268)(20,322,70,257)(21,311,71,246)(22,330,72,265)(23,319,73,254)(24,308,74,243)(25,327,75,262)(26,316,76,251)(27,305,77,270)(28,324,78,259)(29,313,79,248)(30,302,80,267)(31,337,100,282)(32,356,101,271)(33,345,102,290)(34,334,103,279)(35,353,104,298)(36,342,105,287)(37,331,106,276)(38,350,107,295)(39,339,108,284)(40,358,109,273)(41,347,110,292)(42,336,111,281)(43,355,112,300)(44,344,113,289)(45,333,114,278)(46,352,115,297)(47,341,116,286)(48,360,117,275)(49,349,118,294)(50,338,119,283)(51,357,120,272)(52,346,91,291)(53,335,92,280)(54,354,93,299)(55,343,94,288)(56,332,95,277)(57,351,96,296)(58,340,97,285)(59,359,98,274)(60,348,99,293)(121,430,204,385)(122,449,205,374)(123,438,206,363)(124,427,207,382)(125,446,208,371)(126,435,209,390)(127,424,210,379)(128,443,181,368)(129,432,182,387)(130,421,183,376)(131,440,184,365)(132,429,185,384)(133,448,186,373)(134,437,187,362)(135,426,188,381)(136,445,189,370)(137,434,190,389)(138,423,191,378)(139,442,192,367)(140,431,193,386)(141,450,194,375)(142,439,195,364)(143,428,196,383)(144,447,197,372)(145,436,198,361)(146,425,199,380)(147,444,200,369)(148,433,201,388)(149,422,202,377)(150,441,203,366)(151,470,212,395)(152,459,213,414)(153,478,214,403)(154,467,215,392)(155,456,216,411)(156,475,217,400)(157,464,218,419)(158,453,219,408)(159,472,220,397)(160,461,221,416)(161,480,222,405)(162,469,223,394)(163,458,224,413)(164,477,225,402)(165,466,226,391)(166,455,227,410)(167,474,228,399)(168,463,229,418)(169,452,230,407)(170,471,231,396)(171,460,232,415)(172,479,233,404)(173,468,234,393)(174,457,235,412)(175,476,236,401)(176,465,237,420)(177,454,238,409)(178,473,239,398)(179,462,240,417)(180,451,211,406), (1,180,32,150,16,165,47,135)(2,161,33,131,17,176,48,146)(3,172,34,142,18,157,49,127)(4,153,35,123,19,168,50,138)(5,164,36,134,20,179,51,149)(6,175,37,145,21,160,52,130)(7,156,38,126,22,171,53,141)(8,167,39,137,23,152,54,122)(9,178,40,148,24,163,55,133)(10,159,41,129,25,174,56,144)(11,170,42,140,26,155,57,125)(12,151,43,121,27,166,58,136)(13,162,44,132,28,177,59,147)(14,173,45,143,29,158,60,128)(15,154,46,124,30,169,31,139)(61,231,111,193,76,216,96,208)(62,212,112,204,77,227,97,189)(63,223,113,185,78,238,98,200)(64,234,114,196,79,219,99,181)(65,215,115,207,80,230,100,192)(66,226,116,188,81,211,101,203)(67,237,117,199,82,222,102,184)(68,218,118,210,83,233,103,195)(69,229,119,191,84,214,104,206)(70,240,120,202,85,225,105,187)(71,221,91,183,86,236,106,198)(72,232,92,194,87,217,107,209)(73,213,93,205,88,228,108,190)(74,224,94,186,89,239,109,201)(75,235,95,197,90,220,110,182)(241,406,286,366,256,391,271,381)(242,417,287,377,257,402,272,362)(243,398,288,388,258,413,273,373)(244,409,289,369,259,394,274,384)(245,420,290,380,260,405,275,365)(246,401,291,361,261,416,276,376)(247,412,292,372,262,397,277,387)(248,393,293,383,263,408,278,368)(249,404,294,364,264,419,279,379)(250,415,295,375,265,400,280,390)(251,396,296,386,266,411,281,371)(252,407,297,367,267,392,282,382)(253,418,298,378,268,403,283,363)(254,399,299,389,269,414,284,374)(255,410,300,370,270,395,285,385)(301,456,336,446,316,471,351,431)(302,467,337,427,317,452,352,442)(303,478,338,438,318,463,353,423)(304,459,339,449,319,474,354,434)(305,470,340,430,320,455,355,445)(306,451,341,441,321,466,356,426)(307,462,342,422,322,477,357,437)(308,473,343,433,323,458,358,448)(309,454,344,444,324,469,359,429)(310,465,345,425,325,480,360,440)(311,476,346,436,326,461,331,421)(312,457,347,447,327,472,332,432)(313,468,348,428,328,453,333,443)(314,479,349,439,329,464,334,424)(315,460,350,450,330,475,335,435) );
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)], [(1,321,81,256),(2,310,82,245),(3,329,83,264),(4,318,84,253),(5,307,85,242),(6,326,86,261),(7,315,87,250),(8,304,88,269),(9,323,89,258),(10,312,90,247),(11,301,61,266),(12,320,62,255),(13,309,63,244),(14,328,64,263),(15,317,65,252),(16,306,66,241),(17,325,67,260),(18,314,68,249),(19,303,69,268),(20,322,70,257),(21,311,71,246),(22,330,72,265),(23,319,73,254),(24,308,74,243),(25,327,75,262),(26,316,76,251),(27,305,77,270),(28,324,78,259),(29,313,79,248),(30,302,80,267),(31,337,100,282),(32,356,101,271),(33,345,102,290),(34,334,103,279),(35,353,104,298),(36,342,105,287),(37,331,106,276),(38,350,107,295),(39,339,108,284),(40,358,109,273),(41,347,110,292),(42,336,111,281),(43,355,112,300),(44,344,113,289),(45,333,114,278),(46,352,115,297),(47,341,116,286),(48,360,117,275),(49,349,118,294),(50,338,119,283),(51,357,120,272),(52,346,91,291),(53,335,92,280),(54,354,93,299),(55,343,94,288),(56,332,95,277),(57,351,96,296),(58,340,97,285),(59,359,98,274),(60,348,99,293),(121,430,204,385),(122,449,205,374),(123,438,206,363),(124,427,207,382),(125,446,208,371),(126,435,209,390),(127,424,210,379),(128,443,181,368),(129,432,182,387),(130,421,183,376),(131,440,184,365),(132,429,185,384),(133,448,186,373),(134,437,187,362),(135,426,188,381),(136,445,189,370),(137,434,190,389),(138,423,191,378),(139,442,192,367),(140,431,193,386),(141,450,194,375),(142,439,195,364),(143,428,196,383),(144,447,197,372),(145,436,198,361),(146,425,199,380),(147,444,200,369),(148,433,201,388),(149,422,202,377),(150,441,203,366),(151,470,212,395),(152,459,213,414),(153,478,214,403),(154,467,215,392),(155,456,216,411),(156,475,217,400),(157,464,218,419),(158,453,219,408),(159,472,220,397),(160,461,221,416),(161,480,222,405),(162,469,223,394),(163,458,224,413),(164,477,225,402),(165,466,226,391),(166,455,227,410),(167,474,228,399),(168,463,229,418),(169,452,230,407),(170,471,231,396),(171,460,232,415),(172,479,233,404),(173,468,234,393),(174,457,235,412),(175,476,236,401),(176,465,237,420),(177,454,238,409),(178,473,239,398),(179,462,240,417),(180,451,211,406)], [(1,180,32,150,16,165,47,135),(2,161,33,131,17,176,48,146),(3,172,34,142,18,157,49,127),(4,153,35,123,19,168,50,138),(5,164,36,134,20,179,51,149),(6,175,37,145,21,160,52,130),(7,156,38,126,22,171,53,141),(8,167,39,137,23,152,54,122),(9,178,40,148,24,163,55,133),(10,159,41,129,25,174,56,144),(11,170,42,140,26,155,57,125),(12,151,43,121,27,166,58,136),(13,162,44,132,28,177,59,147),(14,173,45,143,29,158,60,128),(15,154,46,124,30,169,31,139),(61,231,111,193,76,216,96,208),(62,212,112,204,77,227,97,189),(63,223,113,185,78,238,98,200),(64,234,114,196,79,219,99,181),(65,215,115,207,80,230,100,192),(66,226,116,188,81,211,101,203),(67,237,117,199,82,222,102,184),(68,218,118,210,83,233,103,195),(69,229,119,191,84,214,104,206),(70,240,120,202,85,225,105,187),(71,221,91,183,86,236,106,198),(72,232,92,194,87,217,107,209),(73,213,93,205,88,228,108,190),(74,224,94,186,89,239,109,201),(75,235,95,197,90,220,110,182),(241,406,286,366,256,391,271,381),(242,417,287,377,257,402,272,362),(243,398,288,388,258,413,273,373),(244,409,289,369,259,394,274,384),(245,420,290,380,260,405,275,365),(246,401,291,361,261,416,276,376),(247,412,292,372,262,397,277,387),(248,393,293,383,263,408,278,368),(249,404,294,364,264,419,279,379),(250,415,295,375,265,400,280,390),(251,396,296,386,266,411,281,371),(252,407,297,367,267,392,282,382),(253,418,298,378,268,403,283,363),(254,399,299,389,269,414,284,374),(255,410,300,370,270,395,285,385),(301,456,336,446,316,471,351,431),(302,467,337,427,317,452,352,442),(303,478,338,438,318,463,353,423),(304,459,339,449,319,474,354,434),(305,470,340,430,320,455,355,445),(306,451,341,441,321,466,356,426),(307,462,342,422,322,477,357,437),(308,473,343,433,323,458,358,448),(309,454,344,444,324,469,359,429),(310,465,345,425,325,480,360,440),(311,476,346,436,326,461,331,421),(312,457,347,447,327,472,332,432),(313,468,348,428,328,453,333,443),(314,479,349,439,329,464,334,424),(315,460,350,450,330,475,335,435)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 2 | ··· | 2 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | Dic3 | D6 | M4(2) | Dic5 | D10 | C4×S3 | C4×D5 | C4×D5 | C4.Dic3 | C8⋊D5 | S3×D5 | S3×Dic5 | D30.C2 | D5×Dic3 | C20.32D6 |
| kernel | C30.21C42 | C12×Dic5 | C10×C3⋊C8 | C2×C15⋊3C8 | C5×C3⋊C8 | C15⋊3C8 | C6×Dic5 | C4×Dic5 | C2×C3⋊C8 | C2×Dic5 | C2×C20 | C30 | C3⋊C8 | C2×C12 | C20 | C12 | C2×C6 | C10 | C6 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 4 | 4 | 8 | 16 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C30.21C42 ►in GL6(𝔽241)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 188 | 52 |
| 171 | 101 | 0 | 0 | 0 | 0 |
| 140 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 142 | 43 | 0 | 0 |
| 0 | 0 | 198 | 99 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 54 |
| 0 | 0 | 0 | 0 | 138 | 206 |
| 16 | 71 | 0 | 0 | 0 | 0 |
| 55 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 176 | 240 | 0 | 0 |
| 0 | 0 | 64 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
G:=sub<GL(6,GF(241))| [1,240,0,0,0,0,1,0,0,0,0,0,0,0,1,240,0,0,0,0,1,0,0,0,0,0,0,0,240,188,0,0,0,0,1,52],[171,140,0,0,0,0,101,70,0,0,0,0,0,0,142,198,0,0,0,0,43,99,0,0,0,0,0,0,35,138,0,0,0,0,54,206],[16,55,0,0,0,0,71,225,0,0,0,0,0,0,176,64,0,0,0,0,240,65,0,0,0,0,0,0,64,0,0,0,0,0,0,64] >;
C30.21C42 in GAP, Magma, Sage, TeX
C_{30}._{21}C_4^2 % in TeX
G:=Group("C30.21C4^2"); // GroupNames label
G:=SmallGroup(480,28);
// by ID
G=gap.SmallGroup(480,28);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=1,c^4=a^15,b*a*b^-1=a^19,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations