metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.11C42, C30.20M4(2), C15⋊C8⋊6C4, C6.4(C4×F5), C15⋊6(C8⋊C4), (C2×C12).6F5, (C2×C60).10C4, C6.5(C4.F5), (C4×Dic5).8S3, (C2×C20).2Dic3, C5⋊2(C42.S3), Dic5.20(C4×S3), (C6×Dic5).19C4, C10.11(C4×Dic3), C2.2(C12.F5), C6.4(C22.F5), C3⋊1(C10.C42), (C12×Dic5).20C2, (C2×Dic5).202D6, C10.3(C4.Dic3), (C2×Dic5).10Dic3, C2.1(C15⋊8M4(2)), (C6×Dic5).261C22, C2.4(C4×C3⋊F5), (C2×C4).2(C3⋊F5), (C2×C6).35(C2×F5), (C2×C15⋊C8).6C2, (C2×C30).29(C2×C4), C22.11(C2×C3⋊F5), (C2×C10).5(C2×Dic3), (C3×Dic5).50(C2×C4), SmallGroup(480,307)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.11C42
G = < a,b,c | a30=c4=1, b4=a15, bab-1=a17, ac=ca, cbc-1=a15b >
Subgroups: 284 in 80 conjugacy classes, 41 normal (31 characteristic)
C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C30, C8⋊C4, C5⋊C8, C2×Dic5, C2×C20, C2×C3⋊C8, C4×C12, C3×Dic5, C3×Dic5, C60, C2×C30, C4×Dic5, C2×C5⋊C8, C42.S3, C15⋊C8, C6×Dic5, C2×C60, C10.C42, C12×Dic5, C2×C15⋊C8, C30.11C42
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, M4(2), F5, C4×S3, C2×Dic3, C8⋊C4, C2×F5, C4.Dic3, C4×Dic3, C3⋊F5, C4.F5, C4×F5, C22.F5, C42.S3, C2×C3⋊F5, C10.C42, C12.F5, C4×C3⋊F5, C15⋊8M4(2), C30.11C42
►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)
(1 305 81 269 16 320 66 254)(2 328 70 256 17 313 85 241)(3 321 89 243 18 306 74 258)(4 314 78 260 19 329 63 245)(5 307 67 247 20 322 82 262)(6 330 86 264 21 315 71 249)(7 323 75 251 22 308 90 266)(8 316 64 268 23 301 79 253)(9 309 83 255 24 324 68 270)(10 302 72 242 25 317 87 257)(11 325 61 259 26 310 76 244)(12 318 80 246 27 303 65 261)(13 311 69 263 28 326 84 248)(14 304 88 250 29 319 73 265)(15 327 77 267 30 312 62 252)(31 359 110 278 46 344 95 293)(32 352 99 295 47 337 114 280)(33 345 118 282 48 360 103 297)(34 338 107 299 49 353 92 284)(35 331 96 286 50 346 111 271)(36 354 115 273 51 339 100 288)(37 347 104 290 52 332 119 275)(38 340 93 277 53 355 108 292)(39 333 112 294 54 348 97 279)(40 356 101 281 55 341 116 296)(41 349 120 298 56 334 105 283)(42 342 109 285 57 357 94 300)(43 335 98 272 58 350 113 287)(44 358 117 289 59 343 102 274)(45 351 106 276 60 336 91 291)(121 436 202 383 136 421 187 368)(122 429 191 370 137 444 206 385)(123 422 210 387 138 437 195 372)(124 445 199 374 139 430 184 389)(125 438 188 361 140 423 203 376)(126 431 207 378 141 446 192 363)(127 424 196 365 142 439 181 380)(128 447 185 382 143 432 200 367)(129 440 204 369 144 425 189 384)(130 433 193 386 145 448 208 371)(131 426 182 373 146 441 197 388)(132 449 201 390 147 434 186 375)(133 442 190 377 148 427 205 362)(134 435 209 364 149 450 194 379)(135 428 198 381 150 443 183 366)(151 454 234 411 166 469 219 396)(152 477 223 398 167 462 238 413)(153 470 212 415 168 455 227 400)(154 463 231 402 169 478 216 417)(155 456 220 419 170 471 235 404)(156 479 239 406 171 464 224 391)(157 472 228 393 172 457 213 408)(158 465 217 410 173 480 232 395)(159 458 236 397 174 473 221 412)(160 451 225 414 175 466 240 399)(161 474 214 401 176 459 229 416)(162 467 233 418 177 452 218 403)(163 460 222 405 178 475 237 420)(164 453 211 392 179 468 226 407)(165 476 230 409 180 461 215 394)
(1 169 35 137)(2 170 36 138)(3 171 37 139)(4 172 38 140)(5 173 39 141)(6 174 40 142)(7 175 41 143)(8 176 42 144)(9 177 43 145)(10 178 44 146)(11 179 45 147)(12 180 46 148)(13 151 47 149)(14 152 48 150)(15 153 49 121)(16 154 50 122)(17 155 51 123)(18 156 52 124)(19 157 53 125)(20 158 54 126)(21 159 55 127)(22 160 56 128)(23 161 57 129)(24 162 58 130)(25 163 59 131)(26 164 60 132)(27 165 31 133)(28 166 32 134)(29 167 33 135)(30 168 34 136)(61 226 106 186)(62 227 107 187)(63 228 108 188)(64 229 109 189)(65 230 110 190)(66 231 111 191)(67 232 112 192)(68 233 113 193)(69 234 114 194)(70 235 115 195)(71 236 116 196)(72 237 117 197)(73 238 118 198)(74 239 119 199)(75 240 120 200)(76 211 91 201)(77 212 92 202)(78 213 93 203)(79 214 94 204)(80 215 95 205)(81 216 96 206)(82 217 97 207)(83 218 98 208)(84 219 99 209)(85 220 100 210)(86 221 101 181)(87 222 102 182)(88 223 103 183)(89 224 104 184)(90 225 105 185)(241 404 288 372)(242 405 289 373)(243 406 290 374)(244 407 291 375)(245 408 292 376)(246 409 293 377)(247 410 294 378)(248 411 295 379)(249 412 296 380)(250 413 297 381)(251 414 298 382)(252 415 299 383)(253 416 300 384)(254 417 271 385)(255 418 272 386)(256 419 273 387)(257 420 274 388)(258 391 275 389)(259 392 276 390)(260 393 277 361)(261 394 278 362)(262 395 279 363)(263 396 280 364)(264 397 281 365)(265 398 282 366)(266 399 283 367)(267 400 284 368)(268 401 285 369)(269 402 286 370)(270 403 287 371)(301 459 357 425)(302 460 358 426)(303 461 359 427)(304 462 360 428)(305 463 331 429)(306 464 332 430)(307 465 333 431)(308 466 334 432)(309 467 335 433)(310 468 336 434)(311 469 337 435)(312 470 338 436)(313 471 339 437)(314 472 340 438)(315 473 341 439)(316 474 342 440)(317 475 343 441)(318 476 344 442)(319 477 345 443)(320 478 346 444)(321 479 347 445)(322 480 348 446)(323 451 349 447)(324 452 350 448)(325 453 351 449)(326 454 352 450)(327 455 353 421)(328 456 354 422)(329 457 355 423)(330 458 356 424)
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), (1,305,81,269,16,320,66,254)(2,328,70,256,17,313,85,241)(3,321,89,243,18,306,74,258)(4,314,78,260,19,329,63,245)(5,307,67,247,20,322,82,262)(6,330,86,264,21,315,71,249)(7,323,75,251,22,308,90,266)(8,316,64,268,23,301,79,253)(9,309,83,255,24,324,68,270)(10,302,72,242,25,317,87,257)(11,325,61,259,26,310,76,244)(12,318,80,246,27,303,65,261)(13,311,69,263,28,326,84,248)(14,304,88,250,29,319,73,265)(15,327,77,267,30,312,62,252)(31,359,110,278,46,344,95,293)(32,352,99,295,47,337,114,280)(33,345,118,282,48,360,103,297)(34,338,107,299,49,353,92,284)(35,331,96,286,50,346,111,271)(36,354,115,273,51,339,100,288)(37,347,104,290,52,332,119,275)(38,340,93,277,53,355,108,292)(39,333,112,294,54,348,97,279)(40,356,101,281,55,341,116,296)(41,349,120,298,56,334,105,283)(42,342,109,285,57,357,94,300)(43,335,98,272,58,350,113,287)(44,358,117,289,59,343,102,274)(45,351,106,276,60,336,91,291)(121,436,202,383,136,421,187,368)(122,429,191,370,137,444,206,385)(123,422,210,387,138,437,195,372)(124,445,199,374,139,430,184,389)(125,438,188,361,140,423,203,376)(126,431,207,378,141,446,192,363)(127,424,196,365,142,439,181,380)(128,447,185,382,143,432,200,367)(129,440,204,369,144,425,189,384)(130,433,193,386,145,448,208,371)(131,426,182,373,146,441,197,388)(132,449,201,390,147,434,186,375)(133,442,190,377,148,427,205,362)(134,435,209,364,149,450,194,379)(135,428,198,381,150,443,183,366)(151,454,234,411,166,469,219,396)(152,477,223,398,167,462,238,413)(153,470,212,415,168,455,227,400)(154,463,231,402,169,478,216,417)(155,456,220,419,170,471,235,404)(156,479,239,406,171,464,224,391)(157,472,228,393,172,457,213,408)(158,465,217,410,173,480,232,395)(159,458,236,397,174,473,221,412)(160,451,225,414,175,466,240,399)(161,474,214,401,176,459,229,416)(162,467,233,418,177,452,218,403)(163,460,222,405,178,475,237,420)(164,453,211,392,179,468,226,407)(165,476,230,409,180,461,215,394), (1,169,35,137)(2,170,36,138)(3,171,37,139)(4,172,38,140)(5,173,39,141)(6,174,40,142)(7,175,41,143)(8,176,42,144)(9,177,43,145)(10,178,44,146)(11,179,45,147)(12,180,46,148)(13,151,47,149)(14,152,48,150)(15,153,49,121)(16,154,50,122)(17,155,51,123)(18,156,52,124)(19,157,53,125)(20,158,54,126)(21,159,55,127)(22,160,56,128)(23,161,57,129)(24,162,58,130)(25,163,59,131)(26,164,60,132)(27,165,31,133)(28,166,32,134)(29,167,33,135)(30,168,34,136)(61,226,106,186)(62,227,107,187)(63,228,108,188)(64,229,109,189)(65,230,110,190)(66,231,111,191)(67,232,112,192)(68,233,113,193)(69,234,114,194)(70,235,115,195)(71,236,116,196)(72,237,117,197)(73,238,118,198)(74,239,119,199)(75,240,120,200)(76,211,91,201)(77,212,92,202)(78,213,93,203)(79,214,94,204)(80,215,95,205)(81,216,96,206)(82,217,97,207)(83,218,98,208)(84,219,99,209)(85,220,100,210)(86,221,101,181)(87,222,102,182)(88,223,103,183)(89,224,104,184)(90,225,105,185)(241,404,288,372)(242,405,289,373)(243,406,290,374)(244,407,291,375)(245,408,292,376)(246,409,293,377)(247,410,294,378)(248,411,295,379)(249,412,296,380)(250,413,297,381)(251,414,298,382)(252,415,299,383)(253,416,300,384)(254,417,271,385)(255,418,272,386)(256,419,273,387)(257,420,274,388)(258,391,275,389)(259,392,276,390)(260,393,277,361)(261,394,278,362)(262,395,279,363)(263,396,280,364)(264,397,281,365)(265,398,282,366)(266,399,283,367)(267,400,284,368)(268,401,285,369)(269,402,286,370)(270,403,287,371)(301,459,357,425)(302,460,358,426)(303,461,359,427)(304,462,360,428)(305,463,331,429)(306,464,332,430)(307,465,333,431)(308,466,334,432)(309,467,335,433)(310,468,336,434)(311,469,337,435)(312,470,338,436)(313,471,339,437)(314,472,340,438)(315,473,341,439)(316,474,342,440)(317,475,343,441)(318,476,344,442)(319,477,345,443)(320,478,346,444)(321,479,347,445)(322,480,348,446)(323,451,349,447)(324,452,350,448)(325,453,351,449)(326,454,352,450)(327,455,353,421)(328,456,354,422)(329,457,355,423)(330,458,356,424)>;
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), (1,305,81,269,16,320,66,254)(2,328,70,256,17,313,85,241)(3,321,89,243,18,306,74,258)(4,314,78,260,19,329,63,245)(5,307,67,247,20,322,82,262)(6,330,86,264,21,315,71,249)(7,323,75,251,22,308,90,266)(8,316,64,268,23,301,79,253)(9,309,83,255,24,324,68,270)(10,302,72,242,25,317,87,257)(11,325,61,259,26,310,76,244)(12,318,80,246,27,303,65,261)(13,311,69,263,28,326,84,248)(14,304,88,250,29,319,73,265)(15,327,77,267,30,312,62,252)(31,359,110,278,46,344,95,293)(32,352,99,295,47,337,114,280)(33,345,118,282,48,360,103,297)(34,338,107,299,49,353,92,284)(35,331,96,286,50,346,111,271)(36,354,115,273,51,339,100,288)(37,347,104,290,52,332,119,275)(38,340,93,277,53,355,108,292)(39,333,112,294,54,348,97,279)(40,356,101,281,55,341,116,296)(41,349,120,298,56,334,105,283)(42,342,109,285,57,357,94,300)(43,335,98,272,58,350,113,287)(44,358,117,289,59,343,102,274)(45,351,106,276,60,336,91,291)(121,436,202,383,136,421,187,368)(122,429,191,370,137,444,206,385)(123,422,210,387,138,437,195,372)(124,445,199,374,139,430,184,389)(125,438,188,361,140,423,203,376)(126,431,207,378,141,446,192,363)(127,424,196,365,142,439,181,380)(128,447,185,382,143,432,200,367)(129,440,204,369,144,425,189,384)(130,433,193,386,145,448,208,371)(131,426,182,373,146,441,197,388)(132,449,201,390,147,434,186,375)(133,442,190,377,148,427,205,362)(134,435,209,364,149,450,194,379)(135,428,198,381,150,443,183,366)(151,454,234,411,166,469,219,396)(152,477,223,398,167,462,238,413)(153,470,212,415,168,455,227,400)(154,463,231,402,169,478,216,417)(155,456,220,419,170,471,235,404)(156,479,239,406,171,464,224,391)(157,472,228,393,172,457,213,408)(158,465,217,410,173,480,232,395)(159,458,236,397,174,473,221,412)(160,451,225,414,175,466,240,399)(161,474,214,401,176,459,229,416)(162,467,233,418,177,452,218,403)(163,460,222,405,178,475,237,420)(164,453,211,392,179,468,226,407)(165,476,230,409,180,461,215,394), (1,169,35,137)(2,170,36,138)(3,171,37,139)(4,172,38,140)(5,173,39,141)(6,174,40,142)(7,175,41,143)(8,176,42,144)(9,177,43,145)(10,178,44,146)(11,179,45,147)(12,180,46,148)(13,151,47,149)(14,152,48,150)(15,153,49,121)(16,154,50,122)(17,155,51,123)(18,156,52,124)(19,157,53,125)(20,158,54,126)(21,159,55,127)(22,160,56,128)(23,161,57,129)(24,162,58,130)(25,163,59,131)(26,164,60,132)(27,165,31,133)(28,166,32,134)(29,167,33,135)(30,168,34,136)(61,226,106,186)(62,227,107,187)(63,228,108,188)(64,229,109,189)(65,230,110,190)(66,231,111,191)(67,232,112,192)(68,233,113,193)(69,234,114,194)(70,235,115,195)(71,236,116,196)(72,237,117,197)(73,238,118,198)(74,239,119,199)(75,240,120,200)(76,211,91,201)(77,212,92,202)(78,213,93,203)(79,214,94,204)(80,215,95,205)(81,216,96,206)(82,217,97,207)(83,218,98,208)(84,219,99,209)(85,220,100,210)(86,221,101,181)(87,222,102,182)(88,223,103,183)(89,224,104,184)(90,225,105,185)(241,404,288,372)(242,405,289,373)(243,406,290,374)(244,407,291,375)(245,408,292,376)(246,409,293,377)(247,410,294,378)(248,411,295,379)(249,412,296,380)(250,413,297,381)(251,414,298,382)(252,415,299,383)(253,416,300,384)(254,417,271,385)(255,418,272,386)(256,419,273,387)(257,420,274,388)(258,391,275,389)(259,392,276,390)(260,393,277,361)(261,394,278,362)(262,395,279,363)(263,396,280,364)(264,397,281,365)(265,398,282,366)(266,399,283,367)(267,400,284,368)(268,401,285,369)(269,402,286,370)(270,403,287,371)(301,459,357,425)(302,460,358,426)(303,461,359,427)(304,462,360,428)(305,463,331,429)(306,464,332,430)(307,465,333,431)(308,466,334,432)(309,467,335,433)(310,468,336,434)(311,469,337,435)(312,470,338,436)(313,471,339,437)(314,472,340,438)(315,473,341,439)(316,474,342,440)(317,475,343,441)(318,476,344,442)(319,477,345,443)(320,478,346,444)(321,479,347,445)(322,480,348,446)(323,451,349,447)(324,452,350,448)(325,453,351,449)(326,454,352,450)(327,455,353,421)(328,456,354,422)(329,457,355,423)(330,458,356,424) );
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)], [(1,305,81,269,16,320,66,254),(2,328,70,256,17,313,85,241),(3,321,89,243,18,306,74,258),(4,314,78,260,19,329,63,245),(5,307,67,247,20,322,82,262),(6,330,86,264,21,315,71,249),(7,323,75,251,22,308,90,266),(8,316,64,268,23,301,79,253),(9,309,83,255,24,324,68,270),(10,302,72,242,25,317,87,257),(11,325,61,259,26,310,76,244),(12,318,80,246,27,303,65,261),(13,311,69,263,28,326,84,248),(14,304,88,250,29,319,73,265),(15,327,77,267,30,312,62,252),(31,359,110,278,46,344,95,293),(32,352,99,295,47,337,114,280),(33,345,118,282,48,360,103,297),(34,338,107,299,49,353,92,284),(35,331,96,286,50,346,111,271),(36,354,115,273,51,339,100,288),(37,347,104,290,52,332,119,275),(38,340,93,277,53,355,108,292),(39,333,112,294,54,348,97,279),(40,356,101,281,55,341,116,296),(41,349,120,298,56,334,105,283),(42,342,109,285,57,357,94,300),(43,335,98,272,58,350,113,287),(44,358,117,289,59,343,102,274),(45,351,106,276,60,336,91,291),(121,436,202,383,136,421,187,368),(122,429,191,370,137,444,206,385),(123,422,210,387,138,437,195,372),(124,445,199,374,139,430,184,389),(125,438,188,361,140,423,203,376),(126,431,207,378,141,446,192,363),(127,424,196,365,142,439,181,380),(128,447,185,382,143,432,200,367),(129,440,204,369,144,425,189,384),(130,433,193,386,145,448,208,371),(131,426,182,373,146,441,197,388),(132,449,201,390,147,434,186,375),(133,442,190,377,148,427,205,362),(134,435,209,364,149,450,194,379),(135,428,198,381,150,443,183,366),(151,454,234,411,166,469,219,396),(152,477,223,398,167,462,238,413),(153,470,212,415,168,455,227,400),(154,463,231,402,169,478,216,417),(155,456,220,419,170,471,235,404),(156,479,239,406,171,464,224,391),(157,472,228,393,172,457,213,408),(158,465,217,410,173,480,232,395),(159,458,236,397,174,473,221,412),(160,451,225,414,175,466,240,399),(161,474,214,401,176,459,229,416),(162,467,233,418,177,452,218,403),(163,460,222,405,178,475,237,420),(164,453,211,392,179,468,226,407),(165,476,230,409,180,461,215,394)], [(1,169,35,137),(2,170,36,138),(3,171,37,139),(4,172,38,140),(5,173,39,141),(6,174,40,142),(7,175,41,143),(8,176,42,144),(9,177,43,145),(10,178,44,146),(11,179,45,147),(12,180,46,148),(13,151,47,149),(14,152,48,150),(15,153,49,121),(16,154,50,122),(17,155,51,123),(18,156,52,124),(19,157,53,125),(20,158,54,126),(21,159,55,127),(22,160,56,128),(23,161,57,129),(24,162,58,130),(25,163,59,131),(26,164,60,132),(27,165,31,133),(28,166,32,134),(29,167,33,135),(30,168,34,136),(61,226,106,186),(62,227,107,187),(63,228,108,188),(64,229,109,189),(65,230,110,190),(66,231,111,191),(67,232,112,192),(68,233,113,193),(69,234,114,194),(70,235,115,195),(71,236,116,196),(72,237,117,197),(73,238,118,198),(74,239,119,199),(75,240,120,200),(76,211,91,201),(77,212,92,202),(78,213,93,203),(79,214,94,204),(80,215,95,205),(81,216,96,206),(82,217,97,207),(83,218,98,208),(84,219,99,209),(85,220,100,210),(86,221,101,181),(87,222,102,182),(88,223,103,183),(89,224,104,184),(90,225,105,185),(241,404,288,372),(242,405,289,373),(243,406,290,374),(244,407,291,375),(245,408,292,376),(246,409,293,377),(247,410,294,378),(248,411,295,379),(249,412,296,380),(250,413,297,381),(251,414,298,382),(252,415,299,383),(253,416,300,384),(254,417,271,385),(255,418,272,386),(256,419,273,387),(257,420,274,388),(258,391,275,389),(259,392,276,390),(260,393,277,361),(261,394,278,362),(262,395,279,363),(263,396,280,364),(264,397,281,365),(265,398,282,366),(266,399,283,367),(267,400,284,368),(268,401,285,369),(269,402,286,370),(270,403,287,371),(301,459,357,425),(302,460,358,426),(303,461,359,427),(304,462,360,428),(305,463,331,429),(306,464,332,430),(307,465,333,431),(308,466,334,432),(309,467,335,433),(310,468,336,434),(311,469,337,435),(312,470,338,436),(313,471,339,437),(314,472,340,438),(315,473,341,439),(316,474,342,440),(317,475,343,441),(318,476,344,442),(319,477,345,443),(320,478,346,444),(321,479,347,445),(322,480,348,446),(323,451,349,447),(324,452,350,448),(325,453,351,449),(326,454,352,450),(327,455,353,421),(328,456,354,422),(329,457,355,423),(330,458,356,424)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 8A | ··· | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 2 | 2 | 2 | 30 | ··· | 30 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | C4×S3 | C4.Dic3 | F5 | C2×F5 | C3⋊F5 | C4.F5 | C4×F5 | C22.F5 | C2×C3⋊F5 | C12.F5 | C4×C3⋊F5 | C15⋊8M4(2) |
| kernel | C30.11C42 | C12×Dic5 | C2×C15⋊C8 | C15⋊C8 | C6×Dic5 | C2×C60 | C4×Dic5 | C2×Dic5 | C2×Dic5 | C2×C20 | C30 | Dic5 | C10 | C2×C12 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C30.11C42 ►in GL6(𝔽241)
| 240 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 122 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 0 | 0 | 141 |
| 38 | 91 | 0 | 0 | 0 | 0 |
| 53 | 203 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,10,0,0,0,0,0,0,122,0,0,0,0,0,0,58,0,0,0,0,0,0,141],[38,53,0,0,0,0,91,203,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[177,0,0,0,0,0,0,177,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C30.11C42 in GAP, Magma, Sage, TeX
C_{30}._{11}C_4^2 % in TeX
G:=Group("C30.11C4^2"); // GroupNames label
G:=SmallGroup(480,307);
// by ID
G=gap.SmallGroup(480,307);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,100,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c|a^30=c^4=1,b^4=a^15,b*a*b^-1=a^17,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations