metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.23C42, C30.13M4(2), C3⋊C8⋊4Dic5, C5⋊4(C24⋊C4), C3⋊1(C40⋊8C4), C12.77(C4×D5), C5⋊2C8⋊4Dic3, C15⋊10(C8⋊C4), C6.6(C4×Dic5), C20.109(C4×S3), C60.146(C2×C4), (C2×C20).320D6, C6.1(C8⋊D5), C4.21(D5×Dic3), C4.21(S3×Dic5), C10.6(C8⋊S3), (C2×C12).324D10, C20.47(C2×Dic3), C10.18(C4×Dic3), C2.6(Dic3×Dic5), C12.26(C2×Dic5), (C2×C60).222C22, (C4×Dic15).18C2, (C2×Dic15).18C4, C2.1(D30.5C4), C22.9(D30.C2), (C5×C3⋊C8)⋊12C4, (C2×C3⋊C8).8D5, (C3×C5⋊2C8)⋊6C4, (C10×C3⋊C8).10C2, (C2×C5⋊2C8).8S3, (C2×C6).17(C4×D5), (C6×C5⋊2C8).10C2, (C2×C10).40(C4×S3), (C2×C30).76(C2×C4), (C2×C4).225(S3×D5), SmallGroup(480,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.23C42
G = < a,b,c | a30=1, b4=c4=a15, bab-1=a19, cac-1=a11, cbc-1=a15b >
Subgroups: 284 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, Dic3, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C30, C8⋊C4, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, Dic15, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, C24⋊C4, C5×C3⋊C8, C3×C5⋊2C8, C2×Dic15, C2×C60, C40⋊8C4, C6×C5⋊2C8, C10×C3⋊C8, C4×Dic15, C30.23C42
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, C8⋊S3, C4×Dic3, S3×D5, C8⋊D5, C4×Dic5, C24⋊C4, D5×Dic3, S3×Dic5, D30.C2, C40⋊8C4, D30.5C4, Dic3×Dic5, C30.23C42
►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 67 261 16 326 82 246)(2 330 68 250 17 315 83 265)(3 319 69 269 18 304 84 254)(4 308 70 258 19 323 85 243)(5 327 71 247 20 312 86 262)(6 316 72 266 21 301 87 251)(7 305 73 255 22 320 88 270)(8 324 74 244 23 309 89 259)(9 313 75 263 24 328 90 248)(10 302 76 252 25 317 61 267)(11 321 77 241 26 306 62 256)(12 310 78 260 27 325 63 245)(13 329 79 249 28 314 64 264)(14 318 80 268 29 303 65 253)(15 307 81 257 30 322 66 242)(31 335 120 285 46 350 105 300)(32 354 91 274 47 339 106 289)(33 343 92 293 48 358 107 278)(34 332 93 282 49 347 108 297)(35 351 94 271 50 336 109 286)(36 340 95 290 51 355 110 275)(37 359 96 279 52 344 111 294)(38 348 97 298 53 333 112 283)(39 337 98 287 54 352 113 272)(40 356 99 276 55 341 114 291)(41 345 100 295 56 360 115 280)(42 334 101 284 57 349 116 299)(43 353 102 273 58 338 117 288)(44 342 103 292 59 357 118 277)(45 331 104 281 60 346 119 296)(121 424 197 364 136 439 182 379)(122 443 198 383 137 428 183 368)(123 432 199 372 138 447 184 387)(124 421 200 361 139 436 185 376)(125 440 201 380 140 425 186 365)(126 429 202 369 141 444 187 384)(127 448 203 388 142 433 188 373)(128 437 204 377 143 422 189 362)(129 426 205 366 144 441 190 381)(130 445 206 385 145 430 191 370)(131 434 207 374 146 449 192 389)(132 423 208 363 147 438 193 378)(133 442 209 382 148 427 194 367)(134 431 210 371 149 446 195 386)(135 450 181 390 150 435 196 375)(151 459 227 399 166 474 212 414)(152 478 228 418 167 463 213 403)(153 467 229 407 168 452 214 392)(154 456 230 396 169 471 215 411)(155 475 231 415 170 460 216 400)(156 464 232 404 171 479 217 419)(157 453 233 393 172 468 218 408)(158 472 234 412 173 457 219 397)(159 461 235 401 174 476 220 416)(160 480 236 420 175 465 221 405)(161 469 237 409 176 454 222 394)(162 458 238 398 177 473 223 413)(163 477 239 417 178 462 224 402)(164 466 240 406 179 451 225 391)(165 455 211 395 180 470 226 410)
(1 240 104 149 16 225 119 134)(2 221 105 130 17 236 120 145)(3 232 106 141 18 217 91 126)(4 213 107 122 19 228 92 137)(5 224 108 133 20 239 93 148)(6 235 109 144 21 220 94 129)(7 216 110 125 22 231 95 140)(8 227 111 136 23 212 96 121)(9 238 112 147 24 223 97 132)(10 219 113 128 25 234 98 143)(11 230 114 139 26 215 99 124)(12 211 115 150 27 226 100 135)(13 222 116 131 28 237 101 146)(14 233 117 142 29 218 102 127)(15 214 118 123 30 229 103 138)(31 206 83 175 46 191 68 160)(32 187 84 156 47 202 69 171)(33 198 85 167 48 183 70 152)(34 209 86 178 49 194 71 163)(35 190 87 159 50 205 72 174)(36 201 88 170 51 186 73 155)(37 182 89 151 52 197 74 166)(38 193 90 162 53 208 75 177)(39 204 61 173 54 189 76 158)(40 185 62 154 55 200 77 169)(41 196 63 165 56 181 78 180)(42 207 64 176 57 192 79 161)(43 188 65 157 58 203 80 172)(44 199 66 168 59 184 81 153)(45 210 67 179 60 195 82 164)(241 456 356 361 256 471 341 376)(242 467 357 372 257 452 342 387)(243 478 358 383 258 463 343 368)(244 459 359 364 259 474 344 379)(245 470 360 375 260 455 345 390)(246 451 331 386 261 466 346 371)(247 462 332 367 262 477 347 382)(248 473 333 378 263 458 348 363)(249 454 334 389 264 469 349 374)(250 465 335 370 265 480 350 385)(251 476 336 381 266 461 351 366)(252 457 337 362 267 472 352 377)(253 468 338 373 268 453 353 388)(254 479 339 384 269 464 354 369)(255 460 340 365 270 475 355 380)(271 441 316 416 286 426 301 401)(272 422 317 397 287 437 302 412)(273 433 318 408 288 448 303 393)(274 444 319 419 289 429 304 404)(275 425 320 400 290 440 305 415)(276 436 321 411 291 421 306 396)(277 447 322 392 292 432 307 407)(278 428 323 403 293 443 308 418)(279 439 324 414 294 424 309 399)(280 450 325 395 295 435 310 410)(281 431 326 406 296 446 311 391)(282 442 327 417 297 427 312 402)(283 423 328 398 298 438 313 413)(284 434 329 409 299 449 314 394)(285 445 330 420 300 430 315 405)
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), (1,311,67,261,16,326,82,246)(2,330,68,250,17,315,83,265)(3,319,69,269,18,304,84,254)(4,308,70,258,19,323,85,243)(5,327,71,247,20,312,86,262)(6,316,72,266,21,301,87,251)(7,305,73,255,22,320,88,270)(8,324,74,244,23,309,89,259)(9,313,75,263,24,328,90,248)(10,302,76,252,25,317,61,267)(11,321,77,241,26,306,62,256)(12,310,78,260,27,325,63,245)(13,329,79,249,28,314,64,264)(14,318,80,268,29,303,65,253)(15,307,81,257,30,322,66,242)(31,335,120,285,46,350,105,300)(32,354,91,274,47,339,106,289)(33,343,92,293,48,358,107,278)(34,332,93,282,49,347,108,297)(35,351,94,271,50,336,109,286)(36,340,95,290,51,355,110,275)(37,359,96,279,52,344,111,294)(38,348,97,298,53,333,112,283)(39,337,98,287,54,352,113,272)(40,356,99,276,55,341,114,291)(41,345,100,295,56,360,115,280)(42,334,101,284,57,349,116,299)(43,353,102,273,58,338,117,288)(44,342,103,292,59,357,118,277)(45,331,104,281,60,346,119,296)(121,424,197,364,136,439,182,379)(122,443,198,383,137,428,183,368)(123,432,199,372,138,447,184,387)(124,421,200,361,139,436,185,376)(125,440,201,380,140,425,186,365)(126,429,202,369,141,444,187,384)(127,448,203,388,142,433,188,373)(128,437,204,377,143,422,189,362)(129,426,205,366,144,441,190,381)(130,445,206,385,145,430,191,370)(131,434,207,374,146,449,192,389)(132,423,208,363,147,438,193,378)(133,442,209,382,148,427,194,367)(134,431,210,371,149,446,195,386)(135,450,181,390,150,435,196,375)(151,459,227,399,166,474,212,414)(152,478,228,418,167,463,213,403)(153,467,229,407,168,452,214,392)(154,456,230,396,169,471,215,411)(155,475,231,415,170,460,216,400)(156,464,232,404,171,479,217,419)(157,453,233,393,172,468,218,408)(158,472,234,412,173,457,219,397)(159,461,235,401,174,476,220,416)(160,480,236,420,175,465,221,405)(161,469,237,409,176,454,222,394)(162,458,238,398,177,473,223,413)(163,477,239,417,178,462,224,402)(164,466,240,406,179,451,225,391)(165,455,211,395,180,470,226,410), (1,240,104,149,16,225,119,134)(2,221,105,130,17,236,120,145)(3,232,106,141,18,217,91,126)(4,213,107,122,19,228,92,137)(5,224,108,133,20,239,93,148)(6,235,109,144,21,220,94,129)(7,216,110,125,22,231,95,140)(8,227,111,136,23,212,96,121)(9,238,112,147,24,223,97,132)(10,219,113,128,25,234,98,143)(11,230,114,139,26,215,99,124)(12,211,115,150,27,226,100,135)(13,222,116,131,28,237,101,146)(14,233,117,142,29,218,102,127)(15,214,118,123,30,229,103,138)(31,206,83,175,46,191,68,160)(32,187,84,156,47,202,69,171)(33,198,85,167,48,183,70,152)(34,209,86,178,49,194,71,163)(35,190,87,159,50,205,72,174)(36,201,88,170,51,186,73,155)(37,182,89,151,52,197,74,166)(38,193,90,162,53,208,75,177)(39,204,61,173,54,189,76,158)(40,185,62,154,55,200,77,169)(41,196,63,165,56,181,78,180)(42,207,64,176,57,192,79,161)(43,188,65,157,58,203,80,172)(44,199,66,168,59,184,81,153)(45,210,67,179,60,195,82,164)(241,456,356,361,256,471,341,376)(242,467,357,372,257,452,342,387)(243,478,358,383,258,463,343,368)(244,459,359,364,259,474,344,379)(245,470,360,375,260,455,345,390)(246,451,331,386,261,466,346,371)(247,462,332,367,262,477,347,382)(248,473,333,378,263,458,348,363)(249,454,334,389,264,469,349,374)(250,465,335,370,265,480,350,385)(251,476,336,381,266,461,351,366)(252,457,337,362,267,472,352,377)(253,468,338,373,268,453,353,388)(254,479,339,384,269,464,354,369)(255,460,340,365,270,475,355,380)(271,441,316,416,286,426,301,401)(272,422,317,397,287,437,302,412)(273,433,318,408,288,448,303,393)(274,444,319,419,289,429,304,404)(275,425,320,400,290,440,305,415)(276,436,321,411,291,421,306,396)(277,447,322,392,292,432,307,407)(278,428,323,403,293,443,308,418)(279,439,324,414,294,424,309,399)(280,450,325,395,295,435,310,410)(281,431,326,406,296,446,311,391)(282,442,327,417,297,427,312,402)(283,423,328,398,298,438,313,413)(284,434,329,409,299,449,314,394)(285,445,330,420,300,430,315,405)>;
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), (1,311,67,261,16,326,82,246)(2,330,68,250,17,315,83,265)(3,319,69,269,18,304,84,254)(4,308,70,258,19,323,85,243)(5,327,71,247,20,312,86,262)(6,316,72,266,21,301,87,251)(7,305,73,255,22,320,88,270)(8,324,74,244,23,309,89,259)(9,313,75,263,24,328,90,248)(10,302,76,252,25,317,61,267)(11,321,77,241,26,306,62,256)(12,310,78,260,27,325,63,245)(13,329,79,249,28,314,64,264)(14,318,80,268,29,303,65,253)(15,307,81,257,30,322,66,242)(31,335,120,285,46,350,105,300)(32,354,91,274,47,339,106,289)(33,343,92,293,48,358,107,278)(34,332,93,282,49,347,108,297)(35,351,94,271,50,336,109,286)(36,340,95,290,51,355,110,275)(37,359,96,279,52,344,111,294)(38,348,97,298,53,333,112,283)(39,337,98,287,54,352,113,272)(40,356,99,276,55,341,114,291)(41,345,100,295,56,360,115,280)(42,334,101,284,57,349,116,299)(43,353,102,273,58,338,117,288)(44,342,103,292,59,357,118,277)(45,331,104,281,60,346,119,296)(121,424,197,364,136,439,182,379)(122,443,198,383,137,428,183,368)(123,432,199,372,138,447,184,387)(124,421,200,361,139,436,185,376)(125,440,201,380,140,425,186,365)(126,429,202,369,141,444,187,384)(127,448,203,388,142,433,188,373)(128,437,204,377,143,422,189,362)(129,426,205,366,144,441,190,381)(130,445,206,385,145,430,191,370)(131,434,207,374,146,449,192,389)(132,423,208,363,147,438,193,378)(133,442,209,382,148,427,194,367)(134,431,210,371,149,446,195,386)(135,450,181,390,150,435,196,375)(151,459,227,399,166,474,212,414)(152,478,228,418,167,463,213,403)(153,467,229,407,168,452,214,392)(154,456,230,396,169,471,215,411)(155,475,231,415,170,460,216,400)(156,464,232,404,171,479,217,419)(157,453,233,393,172,468,218,408)(158,472,234,412,173,457,219,397)(159,461,235,401,174,476,220,416)(160,480,236,420,175,465,221,405)(161,469,237,409,176,454,222,394)(162,458,238,398,177,473,223,413)(163,477,239,417,178,462,224,402)(164,466,240,406,179,451,225,391)(165,455,211,395,180,470,226,410), (1,240,104,149,16,225,119,134)(2,221,105,130,17,236,120,145)(3,232,106,141,18,217,91,126)(4,213,107,122,19,228,92,137)(5,224,108,133,20,239,93,148)(6,235,109,144,21,220,94,129)(7,216,110,125,22,231,95,140)(8,227,111,136,23,212,96,121)(9,238,112,147,24,223,97,132)(10,219,113,128,25,234,98,143)(11,230,114,139,26,215,99,124)(12,211,115,150,27,226,100,135)(13,222,116,131,28,237,101,146)(14,233,117,142,29,218,102,127)(15,214,118,123,30,229,103,138)(31,206,83,175,46,191,68,160)(32,187,84,156,47,202,69,171)(33,198,85,167,48,183,70,152)(34,209,86,178,49,194,71,163)(35,190,87,159,50,205,72,174)(36,201,88,170,51,186,73,155)(37,182,89,151,52,197,74,166)(38,193,90,162,53,208,75,177)(39,204,61,173,54,189,76,158)(40,185,62,154,55,200,77,169)(41,196,63,165,56,181,78,180)(42,207,64,176,57,192,79,161)(43,188,65,157,58,203,80,172)(44,199,66,168,59,184,81,153)(45,210,67,179,60,195,82,164)(241,456,356,361,256,471,341,376)(242,467,357,372,257,452,342,387)(243,478,358,383,258,463,343,368)(244,459,359,364,259,474,344,379)(245,470,360,375,260,455,345,390)(246,451,331,386,261,466,346,371)(247,462,332,367,262,477,347,382)(248,473,333,378,263,458,348,363)(249,454,334,389,264,469,349,374)(250,465,335,370,265,480,350,385)(251,476,336,381,266,461,351,366)(252,457,337,362,267,472,352,377)(253,468,338,373,268,453,353,388)(254,479,339,384,269,464,354,369)(255,460,340,365,270,475,355,380)(271,441,316,416,286,426,301,401)(272,422,317,397,287,437,302,412)(273,433,318,408,288,448,303,393)(274,444,319,419,289,429,304,404)(275,425,320,400,290,440,305,415)(276,436,321,411,291,421,306,396)(277,447,322,392,292,432,307,407)(278,428,323,403,293,443,308,418)(279,439,324,414,294,424,309,399)(280,450,325,395,295,435,310,410)(281,431,326,406,296,446,311,391)(282,442,327,417,297,427,312,402)(283,423,328,398,298,438,313,413)(284,434,329,409,299,449,314,394)(285,445,330,420,300,430,315,405) );
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)], [(1,311,67,261,16,326,82,246),(2,330,68,250,17,315,83,265),(3,319,69,269,18,304,84,254),(4,308,70,258,19,323,85,243),(5,327,71,247,20,312,86,262),(6,316,72,266,21,301,87,251),(7,305,73,255,22,320,88,270),(8,324,74,244,23,309,89,259),(9,313,75,263,24,328,90,248),(10,302,76,252,25,317,61,267),(11,321,77,241,26,306,62,256),(12,310,78,260,27,325,63,245),(13,329,79,249,28,314,64,264),(14,318,80,268,29,303,65,253),(15,307,81,257,30,322,66,242),(31,335,120,285,46,350,105,300),(32,354,91,274,47,339,106,289),(33,343,92,293,48,358,107,278),(34,332,93,282,49,347,108,297),(35,351,94,271,50,336,109,286),(36,340,95,290,51,355,110,275),(37,359,96,279,52,344,111,294),(38,348,97,298,53,333,112,283),(39,337,98,287,54,352,113,272),(40,356,99,276,55,341,114,291),(41,345,100,295,56,360,115,280),(42,334,101,284,57,349,116,299),(43,353,102,273,58,338,117,288),(44,342,103,292,59,357,118,277),(45,331,104,281,60,346,119,296),(121,424,197,364,136,439,182,379),(122,443,198,383,137,428,183,368),(123,432,199,372,138,447,184,387),(124,421,200,361,139,436,185,376),(125,440,201,380,140,425,186,365),(126,429,202,369,141,444,187,384),(127,448,203,388,142,433,188,373),(128,437,204,377,143,422,189,362),(129,426,205,366,144,441,190,381),(130,445,206,385,145,430,191,370),(131,434,207,374,146,449,192,389),(132,423,208,363,147,438,193,378),(133,442,209,382,148,427,194,367),(134,431,210,371,149,446,195,386),(135,450,181,390,150,435,196,375),(151,459,227,399,166,474,212,414),(152,478,228,418,167,463,213,403),(153,467,229,407,168,452,214,392),(154,456,230,396,169,471,215,411),(155,475,231,415,170,460,216,400),(156,464,232,404,171,479,217,419),(157,453,233,393,172,468,218,408),(158,472,234,412,173,457,219,397),(159,461,235,401,174,476,220,416),(160,480,236,420,175,465,221,405),(161,469,237,409,176,454,222,394),(162,458,238,398,177,473,223,413),(163,477,239,417,178,462,224,402),(164,466,240,406,179,451,225,391),(165,455,211,395,180,470,226,410)], [(1,240,104,149,16,225,119,134),(2,221,105,130,17,236,120,145),(3,232,106,141,18,217,91,126),(4,213,107,122,19,228,92,137),(5,224,108,133,20,239,93,148),(6,235,109,144,21,220,94,129),(7,216,110,125,22,231,95,140),(8,227,111,136,23,212,96,121),(9,238,112,147,24,223,97,132),(10,219,113,128,25,234,98,143),(11,230,114,139,26,215,99,124),(12,211,115,150,27,226,100,135),(13,222,116,131,28,237,101,146),(14,233,117,142,29,218,102,127),(15,214,118,123,30,229,103,138),(31,206,83,175,46,191,68,160),(32,187,84,156,47,202,69,171),(33,198,85,167,48,183,70,152),(34,209,86,178,49,194,71,163),(35,190,87,159,50,205,72,174),(36,201,88,170,51,186,73,155),(37,182,89,151,52,197,74,166),(38,193,90,162,53,208,75,177),(39,204,61,173,54,189,76,158),(40,185,62,154,55,200,77,169),(41,196,63,165,56,181,78,180),(42,207,64,176,57,192,79,161),(43,188,65,157,58,203,80,172),(44,199,66,168,59,184,81,153),(45,210,67,179,60,195,82,164),(241,456,356,361,256,471,341,376),(242,467,357,372,257,452,342,387),(243,478,358,383,258,463,343,368),(244,459,359,364,259,474,344,379),(245,470,360,375,260,455,345,390),(246,451,331,386,261,466,346,371),(247,462,332,367,262,477,347,382),(248,473,333,378,263,458,348,363),(249,454,334,389,264,469,349,374),(250,465,335,370,265,480,350,385),(251,476,336,381,266,461,351,366),(252,457,337,362,267,472,352,377),(253,468,338,373,268,453,353,388),(254,479,339,384,269,464,354,369),(255,460,340,365,270,475,355,380),(271,441,316,416,286,426,301,401),(272,422,317,397,287,437,302,412),(273,433,318,408,288,448,303,393),(274,444,319,419,289,429,304,404),(275,425,320,400,290,440,305,415),(276,436,321,411,291,421,306,396),(277,447,322,392,292,432,307,407),(278,428,323,403,293,443,308,418),(279,439,324,414,294,424,309,399),(280,450,325,395,295,435,310,410),(281,431,326,406,296,446,311,391),(282,442,327,417,297,427,312,402),(283,423,328,398,298,438,313,413),(284,434,329,409,299,449,314,394),(285,445,330,420,300,430,315,405)]])
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 | 15A | 15B | 20A | ··· | 20H | 24A | ··· | 24H | 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 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 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×S3 | C4×D5 | C4×D5 | C8⋊S3 | C8⋊D5 | S3×D5 | D5×Dic3 | S3×Dic5 | D30.C2 | D30.5C4 |
| kernel | C30.23C42 | C6×C5⋊2C8 | C10×C3⋊C8 | C4×Dic15 | C5×C3⋊C8 | C3×C5⋊2C8 | C2×Dic15 | C2×C5⋊2C8 | C2×C3⋊C8 | C5⋊2C8 | C2×C20 | C30 | C3⋊C8 | C2×C12 | C20 | C2×C10 | 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 | 2 | 2 | 4 | 4 | 8 | 16 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C30.23C42 ►in GL4(𝔽241) generated by
| 0 | 240 | 0 | 0 |
| 1 | 190 | 0 | 0 |
| 0 | 0 | 1 | 187 |
| 0 | 0 | 67 | 239 |
| 162 | 13 | 0 | 0 |
| 81 | 79 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 130 | 217 | 0 | 0 |
| 24 | 111 | 0 | 0 |
| 0 | 0 | 128 | 155 |
| 0 | 0 | 70 | 113 |
G:=sub<GL(4,GF(241))| [0,1,0,0,240,190,0,0,0,0,1,67,0,0,187,239],[162,81,0,0,13,79,0,0,0,0,177,0,0,0,0,177],[130,24,0,0,217,111,0,0,0,0,128,70,0,0,155,113] >;
C30.23C42 in GAP, Magma, Sage, TeX
C_{30}._{23}C_4^2 % in TeX
G:=Group("C30.23C4^2"); // GroupNames label
G:=SmallGroup(480,30);
// by ID
G=gap.SmallGroup(480,30);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,92,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^30=1,b^4=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