metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.10Q8, C20.2Dic6, C12.1Dic10, C30.14SD16, C15⋊3C8⋊11C4, C15⋊5(C4.Q8), C4.7(C15⋊Q8), C20.42(C4×S3), (C2×C30).30D4, (C2×C20).59D6, C12.10(C4×D5), C6.6(Q8⋊D5), C4⋊Dic3.9D5, C4⋊Dic5.9S3, C30.26(C4⋊C4), C60.126(C2×C4), (C2×C12).59D10, C6.6(D4.D5), C5⋊3(C12.Q8), C3⋊2(C20.Q8), C10.6(D4.S3), (C2×C60).185C22, C4.11(D30.C2), C10.6(Q8⋊2S3), C6.4(C10.D4), C2.3(C20.D6), C2.3(C30.D4), C2.3(Dic15⋊5C4), C10.10(Dic3⋊C4), C22.16(C15⋊D4), (C2×C4).190(S3×D5), (C5×C4⋊Dic3).8C2, (C3×C4⋊Dic5).8C2, (C2×C15⋊3C8).13C2, (C2×C6).48(C5⋊D4), (C2×C10).48(C3⋊D4), SmallGroup(480,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.SD16
G = < a,b,c | a30=b8=1, c2=a15, bab-1=a-1, cac-1=a11, cbc-1=b3 >
Subgroups: 268 in 72 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C30, C4.Q8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, C60, C2×C30, C2×C5⋊2C8, C4⋊Dic5, C5×C4⋊C4, C12.Q8, C15⋊3C8, C6×Dic5, C10×Dic3, C2×C60, C20.Q8, C3×C4⋊Dic5, C5×C4⋊Dic3, C2×C15⋊3C8, C30.SD16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, SD16, D10, Dic6, C4×S3, C3⋊D4, C4.Q8, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, D4.S3, Q8⋊2S3, S3×D5, C10.D4, D4.D5, Q8⋊D5, C12.Q8, D30.C2, C15⋊D4, C15⋊Q8, C20.Q8, C30.D4, C20.D6, Dic15⋊5C4, C30.SD16
►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 53 347 191 101 250 76 235)(2 52 348 190 102 249 77 234)(3 51 349 189 103 248 78 233)(4 50 350 188 104 247 79 232)(5 49 351 187 105 246 80 231)(6 48 352 186 106 245 81 230)(7 47 353 185 107 244 82 229)(8 46 354 184 108 243 83 228)(9 45 355 183 109 242 84 227)(10 44 356 182 110 241 85 226)(11 43 357 181 111 270 86 225)(12 42 358 210 112 269 87 224)(13 41 359 209 113 268 88 223)(14 40 360 208 114 267 89 222)(15 39 331 207 115 266 90 221)(16 38 332 206 116 265 61 220)(17 37 333 205 117 264 62 219)(18 36 334 204 118 263 63 218)(19 35 335 203 119 262 64 217)(20 34 336 202 120 261 65 216)(21 33 337 201 91 260 66 215)(22 32 338 200 92 259 67 214)(23 31 339 199 93 258 68 213)(24 60 340 198 94 257 69 212)(25 59 341 197 95 256 70 211)(26 58 342 196 96 255 71 240)(27 57 343 195 97 254 72 239)(28 56 344 194 98 253 73 238)(29 55 345 193 99 252 74 237)(30 54 346 192 100 251 75 236)(121 465 286 402 381 436 179 329)(122 464 287 401 382 435 180 328)(123 463 288 400 383 434 151 327)(124 462 289 399 384 433 152 326)(125 461 290 398 385 432 153 325)(126 460 291 397 386 431 154 324)(127 459 292 396 387 430 155 323)(128 458 293 395 388 429 156 322)(129 457 294 394 389 428 157 321)(130 456 295 393 390 427 158 320)(131 455 296 392 361 426 159 319)(132 454 297 391 362 425 160 318)(133 453 298 420 363 424 161 317)(134 452 299 419 364 423 162 316)(135 451 300 418 365 422 163 315)(136 480 271 417 366 421 164 314)(137 479 272 416 367 450 165 313)(138 478 273 415 368 449 166 312)(139 477 274 414 369 448 167 311)(140 476 275 413 370 447 168 310)(141 475 276 412 371 446 169 309)(142 474 277 411 372 445 170 308)(143 473 278 410 373 444 171 307)(144 472 279 409 374 443 172 306)(145 471 280 408 375 442 173 305)(146 470 281 407 376 441 174 304)(147 469 282 406 377 440 175 303)(148 468 283 405 378 439 176 302)(149 467 284 404 379 438 177 301)(150 466 285 403 380 437 178 330)
(1 272 16 287)(2 283 17 298)(3 294 18 279)(4 275 19 290)(5 286 20 271)(6 297 21 282)(7 278 22 293)(8 289 23 274)(9 300 24 285)(10 281 25 296)(11 292 26 277)(12 273 27 288)(13 284 28 299)(14 295 29 280)(15 276 30 291)(31 448 46 433)(32 429 47 444)(33 440 48 425)(34 421 49 436)(35 432 50 447)(36 443 51 428)(37 424 52 439)(38 435 53 450)(39 446 54 431)(40 427 55 442)(41 438 56 423)(42 449 57 434)(43 430 58 445)(44 441 59 426)(45 422 60 437)(61 382 76 367)(62 363 77 378)(63 374 78 389)(64 385 79 370)(65 366 80 381)(66 377 81 362)(67 388 82 373)(68 369 83 384)(69 380 84 365)(70 361 85 376)(71 372 86 387)(72 383 87 368)(73 364 88 379)(74 375 89 390)(75 386 90 371)(91 175 106 160)(92 156 107 171)(93 167 108 152)(94 178 109 163)(95 159 110 174)(96 170 111 155)(97 151 112 166)(98 162 113 177)(99 173 114 158)(100 154 115 169)(101 165 116 180)(102 176 117 161)(103 157 118 172)(104 168 119 153)(105 179 120 164)(121 336 136 351)(122 347 137 332)(123 358 138 343)(124 339 139 354)(125 350 140 335)(126 331 141 346)(127 342 142 357)(128 353 143 338)(129 334 144 349)(130 345 145 360)(131 356 146 341)(132 337 147 352)(133 348 148 333)(134 359 149 344)(135 340 150 355)(181 396 196 411)(182 407 197 392)(183 418 198 403)(184 399 199 414)(185 410 200 395)(186 391 201 406)(187 402 202 417)(188 413 203 398)(189 394 204 409)(190 405 205 420)(191 416 206 401)(192 397 207 412)(193 408 208 393)(194 419 209 404)(195 400 210 415)(211 319 226 304)(212 330 227 315)(213 311 228 326)(214 322 229 307)(215 303 230 318)(216 314 231 329)(217 325 232 310)(218 306 233 321)(219 317 234 302)(220 328 235 313)(221 309 236 324)(222 320 237 305)(223 301 238 316)(224 312 239 327)(225 323 240 308)(241 470 256 455)(242 451 257 466)(243 462 258 477)(244 473 259 458)(245 454 260 469)(246 465 261 480)(247 476 262 461)(248 457 263 472)(249 468 264 453)(250 479 265 464)(251 460 266 475)(252 471 267 456)(253 452 268 467)(254 463 269 478)(255 474 270 459)
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), (1,53,347,191,101,250,76,235)(2,52,348,190,102,249,77,234)(3,51,349,189,103,248,78,233)(4,50,350,188,104,247,79,232)(5,49,351,187,105,246,80,231)(6,48,352,186,106,245,81,230)(7,47,353,185,107,244,82,229)(8,46,354,184,108,243,83,228)(9,45,355,183,109,242,84,227)(10,44,356,182,110,241,85,226)(11,43,357,181,111,270,86,225)(12,42,358,210,112,269,87,224)(13,41,359,209,113,268,88,223)(14,40,360,208,114,267,89,222)(15,39,331,207,115,266,90,221)(16,38,332,206,116,265,61,220)(17,37,333,205,117,264,62,219)(18,36,334,204,118,263,63,218)(19,35,335,203,119,262,64,217)(20,34,336,202,120,261,65,216)(21,33,337,201,91,260,66,215)(22,32,338,200,92,259,67,214)(23,31,339,199,93,258,68,213)(24,60,340,198,94,257,69,212)(25,59,341,197,95,256,70,211)(26,58,342,196,96,255,71,240)(27,57,343,195,97,254,72,239)(28,56,344,194,98,253,73,238)(29,55,345,193,99,252,74,237)(30,54,346,192,100,251,75,236)(121,465,286,402,381,436,179,329)(122,464,287,401,382,435,180,328)(123,463,288,400,383,434,151,327)(124,462,289,399,384,433,152,326)(125,461,290,398,385,432,153,325)(126,460,291,397,386,431,154,324)(127,459,292,396,387,430,155,323)(128,458,293,395,388,429,156,322)(129,457,294,394,389,428,157,321)(130,456,295,393,390,427,158,320)(131,455,296,392,361,426,159,319)(132,454,297,391,362,425,160,318)(133,453,298,420,363,424,161,317)(134,452,299,419,364,423,162,316)(135,451,300,418,365,422,163,315)(136,480,271,417,366,421,164,314)(137,479,272,416,367,450,165,313)(138,478,273,415,368,449,166,312)(139,477,274,414,369,448,167,311)(140,476,275,413,370,447,168,310)(141,475,276,412,371,446,169,309)(142,474,277,411,372,445,170,308)(143,473,278,410,373,444,171,307)(144,472,279,409,374,443,172,306)(145,471,280,408,375,442,173,305)(146,470,281,407,376,441,174,304)(147,469,282,406,377,440,175,303)(148,468,283,405,378,439,176,302)(149,467,284,404,379,438,177,301)(150,466,285,403,380,437,178,330), (1,272,16,287)(2,283,17,298)(3,294,18,279)(4,275,19,290)(5,286,20,271)(6,297,21,282)(7,278,22,293)(8,289,23,274)(9,300,24,285)(10,281,25,296)(11,292,26,277)(12,273,27,288)(13,284,28,299)(14,295,29,280)(15,276,30,291)(31,448,46,433)(32,429,47,444)(33,440,48,425)(34,421,49,436)(35,432,50,447)(36,443,51,428)(37,424,52,439)(38,435,53,450)(39,446,54,431)(40,427,55,442)(41,438,56,423)(42,449,57,434)(43,430,58,445)(44,441,59,426)(45,422,60,437)(61,382,76,367)(62,363,77,378)(63,374,78,389)(64,385,79,370)(65,366,80,381)(66,377,81,362)(67,388,82,373)(68,369,83,384)(69,380,84,365)(70,361,85,376)(71,372,86,387)(72,383,87,368)(73,364,88,379)(74,375,89,390)(75,386,90,371)(91,175,106,160)(92,156,107,171)(93,167,108,152)(94,178,109,163)(95,159,110,174)(96,170,111,155)(97,151,112,166)(98,162,113,177)(99,173,114,158)(100,154,115,169)(101,165,116,180)(102,176,117,161)(103,157,118,172)(104,168,119,153)(105,179,120,164)(121,336,136,351)(122,347,137,332)(123,358,138,343)(124,339,139,354)(125,350,140,335)(126,331,141,346)(127,342,142,357)(128,353,143,338)(129,334,144,349)(130,345,145,360)(131,356,146,341)(132,337,147,352)(133,348,148,333)(134,359,149,344)(135,340,150,355)(181,396,196,411)(182,407,197,392)(183,418,198,403)(184,399,199,414)(185,410,200,395)(186,391,201,406)(187,402,202,417)(188,413,203,398)(189,394,204,409)(190,405,205,420)(191,416,206,401)(192,397,207,412)(193,408,208,393)(194,419,209,404)(195,400,210,415)(211,319,226,304)(212,330,227,315)(213,311,228,326)(214,322,229,307)(215,303,230,318)(216,314,231,329)(217,325,232,310)(218,306,233,321)(219,317,234,302)(220,328,235,313)(221,309,236,324)(222,320,237,305)(223,301,238,316)(224,312,239,327)(225,323,240,308)(241,470,256,455)(242,451,257,466)(243,462,258,477)(244,473,259,458)(245,454,260,469)(246,465,261,480)(247,476,262,461)(248,457,263,472)(249,468,264,453)(250,479,265,464)(251,460,266,475)(252,471,267,456)(253,452,268,467)(254,463,269,478)(255,474,270,459)>;
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), (1,53,347,191,101,250,76,235)(2,52,348,190,102,249,77,234)(3,51,349,189,103,248,78,233)(4,50,350,188,104,247,79,232)(5,49,351,187,105,246,80,231)(6,48,352,186,106,245,81,230)(7,47,353,185,107,244,82,229)(8,46,354,184,108,243,83,228)(9,45,355,183,109,242,84,227)(10,44,356,182,110,241,85,226)(11,43,357,181,111,270,86,225)(12,42,358,210,112,269,87,224)(13,41,359,209,113,268,88,223)(14,40,360,208,114,267,89,222)(15,39,331,207,115,266,90,221)(16,38,332,206,116,265,61,220)(17,37,333,205,117,264,62,219)(18,36,334,204,118,263,63,218)(19,35,335,203,119,262,64,217)(20,34,336,202,120,261,65,216)(21,33,337,201,91,260,66,215)(22,32,338,200,92,259,67,214)(23,31,339,199,93,258,68,213)(24,60,340,198,94,257,69,212)(25,59,341,197,95,256,70,211)(26,58,342,196,96,255,71,240)(27,57,343,195,97,254,72,239)(28,56,344,194,98,253,73,238)(29,55,345,193,99,252,74,237)(30,54,346,192,100,251,75,236)(121,465,286,402,381,436,179,329)(122,464,287,401,382,435,180,328)(123,463,288,400,383,434,151,327)(124,462,289,399,384,433,152,326)(125,461,290,398,385,432,153,325)(126,460,291,397,386,431,154,324)(127,459,292,396,387,430,155,323)(128,458,293,395,388,429,156,322)(129,457,294,394,389,428,157,321)(130,456,295,393,390,427,158,320)(131,455,296,392,361,426,159,319)(132,454,297,391,362,425,160,318)(133,453,298,420,363,424,161,317)(134,452,299,419,364,423,162,316)(135,451,300,418,365,422,163,315)(136,480,271,417,366,421,164,314)(137,479,272,416,367,450,165,313)(138,478,273,415,368,449,166,312)(139,477,274,414,369,448,167,311)(140,476,275,413,370,447,168,310)(141,475,276,412,371,446,169,309)(142,474,277,411,372,445,170,308)(143,473,278,410,373,444,171,307)(144,472,279,409,374,443,172,306)(145,471,280,408,375,442,173,305)(146,470,281,407,376,441,174,304)(147,469,282,406,377,440,175,303)(148,468,283,405,378,439,176,302)(149,467,284,404,379,438,177,301)(150,466,285,403,380,437,178,330), (1,272,16,287)(2,283,17,298)(3,294,18,279)(4,275,19,290)(5,286,20,271)(6,297,21,282)(7,278,22,293)(8,289,23,274)(9,300,24,285)(10,281,25,296)(11,292,26,277)(12,273,27,288)(13,284,28,299)(14,295,29,280)(15,276,30,291)(31,448,46,433)(32,429,47,444)(33,440,48,425)(34,421,49,436)(35,432,50,447)(36,443,51,428)(37,424,52,439)(38,435,53,450)(39,446,54,431)(40,427,55,442)(41,438,56,423)(42,449,57,434)(43,430,58,445)(44,441,59,426)(45,422,60,437)(61,382,76,367)(62,363,77,378)(63,374,78,389)(64,385,79,370)(65,366,80,381)(66,377,81,362)(67,388,82,373)(68,369,83,384)(69,380,84,365)(70,361,85,376)(71,372,86,387)(72,383,87,368)(73,364,88,379)(74,375,89,390)(75,386,90,371)(91,175,106,160)(92,156,107,171)(93,167,108,152)(94,178,109,163)(95,159,110,174)(96,170,111,155)(97,151,112,166)(98,162,113,177)(99,173,114,158)(100,154,115,169)(101,165,116,180)(102,176,117,161)(103,157,118,172)(104,168,119,153)(105,179,120,164)(121,336,136,351)(122,347,137,332)(123,358,138,343)(124,339,139,354)(125,350,140,335)(126,331,141,346)(127,342,142,357)(128,353,143,338)(129,334,144,349)(130,345,145,360)(131,356,146,341)(132,337,147,352)(133,348,148,333)(134,359,149,344)(135,340,150,355)(181,396,196,411)(182,407,197,392)(183,418,198,403)(184,399,199,414)(185,410,200,395)(186,391,201,406)(187,402,202,417)(188,413,203,398)(189,394,204,409)(190,405,205,420)(191,416,206,401)(192,397,207,412)(193,408,208,393)(194,419,209,404)(195,400,210,415)(211,319,226,304)(212,330,227,315)(213,311,228,326)(214,322,229,307)(215,303,230,318)(216,314,231,329)(217,325,232,310)(218,306,233,321)(219,317,234,302)(220,328,235,313)(221,309,236,324)(222,320,237,305)(223,301,238,316)(224,312,239,327)(225,323,240,308)(241,470,256,455)(242,451,257,466)(243,462,258,477)(244,473,259,458)(245,454,260,469)(246,465,261,480)(247,476,262,461)(248,457,263,472)(249,468,264,453)(250,479,265,464)(251,460,266,475)(252,471,267,456)(253,452,268,467)(254,463,269,478)(255,474,270,459) );
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)], [(1,53,347,191,101,250,76,235),(2,52,348,190,102,249,77,234),(3,51,349,189,103,248,78,233),(4,50,350,188,104,247,79,232),(5,49,351,187,105,246,80,231),(6,48,352,186,106,245,81,230),(7,47,353,185,107,244,82,229),(8,46,354,184,108,243,83,228),(9,45,355,183,109,242,84,227),(10,44,356,182,110,241,85,226),(11,43,357,181,111,270,86,225),(12,42,358,210,112,269,87,224),(13,41,359,209,113,268,88,223),(14,40,360,208,114,267,89,222),(15,39,331,207,115,266,90,221),(16,38,332,206,116,265,61,220),(17,37,333,205,117,264,62,219),(18,36,334,204,118,263,63,218),(19,35,335,203,119,262,64,217),(20,34,336,202,120,261,65,216),(21,33,337,201,91,260,66,215),(22,32,338,200,92,259,67,214),(23,31,339,199,93,258,68,213),(24,60,340,198,94,257,69,212),(25,59,341,197,95,256,70,211),(26,58,342,196,96,255,71,240),(27,57,343,195,97,254,72,239),(28,56,344,194,98,253,73,238),(29,55,345,193,99,252,74,237),(30,54,346,192,100,251,75,236),(121,465,286,402,381,436,179,329),(122,464,287,401,382,435,180,328),(123,463,288,400,383,434,151,327),(124,462,289,399,384,433,152,326),(125,461,290,398,385,432,153,325),(126,460,291,397,386,431,154,324),(127,459,292,396,387,430,155,323),(128,458,293,395,388,429,156,322),(129,457,294,394,389,428,157,321),(130,456,295,393,390,427,158,320),(131,455,296,392,361,426,159,319),(132,454,297,391,362,425,160,318),(133,453,298,420,363,424,161,317),(134,452,299,419,364,423,162,316),(135,451,300,418,365,422,163,315),(136,480,271,417,366,421,164,314),(137,479,272,416,367,450,165,313),(138,478,273,415,368,449,166,312),(139,477,274,414,369,448,167,311),(140,476,275,413,370,447,168,310),(141,475,276,412,371,446,169,309),(142,474,277,411,372,445,170,308),(143,473,278,410,373,444,171,307),(144,472,279,409,374,443,172,306),(145,471,280,408,375,442,173,305),(146,470,281,407,376,441,174,304),(147,469,282,406,377,440,175,303),(148,468,283,405,378,439,176,302),(149,467,284,404,379,438,177,301),(150,466,285,403,380,437,178,330)], [(1,272,16,287),(2,283,17,298),(3,294,18,279),(4,275,19,290),(5,286,20,271),(6,297,21,282),(7,278,22,293),(8,289,23,274),(9,300,24,285),(10,281,25,296),(11,292,26,277),(12,273,27,288),(13,284,28,299),(14,295,29,280),(15,276,30,291),(31,448,46,433),(32,429,47,444),(33,440,48,425),(34,421,49,436),(35,432,50,447),(36,443,51,428),(37,424,52,439),(38,435,53,450),(39,446,54,431),(40,427,55,442),(41,438,56,423),(42,449,57,434),(43,430,58,445),(44,441,59,426),(45,422,60,437),(61,382,76,367),(62,363,77,378),(63,374,78,389),(64,385,79,370),(65,366,80,381),(66,377,81,362),(67,388,82,373),(68,369,83,384),(69,380,84,365),(70,361,85,376),(71,372,86,387),(72,383,87,368),(73,364,88,379),(74,375,89,390),(75,386,90,371),(91,175,106,160),(92,156,107,171),(93,167,108,152),(94,178,109,163),(95,159,110,174),(96,170,111,155),(97,151,112,166),(98,162,113,177),(99,173,114,158),(100,154,115,169),(101,165,116,180),(102,176,117,161),(103,157,118,172),(104,168,119,153),(105,179,120,164),(121,336,136,351),(122,347,137,332),(123,358,138,343),(124,339,139,354),(125,350,140,335),(126,331,141,346),(127,342,142,357),(128,353,143,338),(129,334,144,349),(130,345,145,360),(131,356,146,341),(132,337,147,352),(133,348,148,333),(134,359,149,344),(135,340,150,355),(181,396,196,411),(182,407,197,392),(183,418,198,403),(184,399,199,414),(185,410,200,395),(186,391,201,406),(187,402,202,417),(188,413,203,398),(189,394,204,409),(190,405,205,420),(191,416,206,401),(192,397,207,412),(193,408,208,393),(194,419,209,404),(195,400,210,415),(211,319,226,304),(212,330,227,315),(213,311,228,326),(214,322,229,307),(215,303,230,318),(216,314,231,329),(217,325,232,310),(218,306,233,321),(219,317,234,302),(220,328,235,313),(221,309,236,324),(222,320,237,305),(223,301,238,316),(224,312,239,327),(225,323,240,308),(241,470,256,455),(242,451,257,466),(243,462,258,477),(244,473,259,458),(245,454,260,469),(246,465,261,480),(247,476,262,461),(248,457,263,472),(249,468,264,453),(250,479,265,464),(251,460,266,475),(252,471,267,456),(253,452,268,467),(254,463,269,478),(255,474,270,459)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | - | + | + | - | + | + | - | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D5 | D6 | SD16 | D10 | Dic6 | C4×S3 | C3⋊D4 | Dic10 | C4×D5 | C5⋊D4 | D4.S3 | Q8⋊2S3 | S3×D5 | D4.D5 | Q8⋊D5 | D30.C2 | C15⋊Q8 | C15⋊D4 | C30.D4 | C20.D6 |
| kernel | C30.SD16 | C3×C4⋊Dic5 | C5×C4⋊Dic3 | C2×C15⋊3C8 | C15⋊3C8 | C4⋊Dic5 | C60 | C2×C30 | C4⋊Dic3 | C2×C20 | C30 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C30.SD16 ►in GL6(𝔽241)
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 42 | 108 | 0 | 0 | 0 | 0 |
| 93 | 199 | 0 | 0 | 0 | 0 |
| 0 | 0 | 131 | 111 | 0 | 0 |
| 0 | 0 | 221 | 110 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 0 | 11 | 38 |
| 41 | 85 | 0 | 0 | 0 | 0 |
| 156 | 200 | 0 | 0 | 0 | 0 |
| 0 | 0 | 92 | 234 | 0 | 0 |
| 0 | 0 | 142 | 149 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 237 |
| 0 | 0 | 0 | 0 | 210 | 184 |
G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,52,0,0,0,0,0,0,1,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[42,93,0,0,0,0,108,199,0,0,0,0,0,0,131,221,0,0,0,0,111,110,0,0,0,0,0,0,0,11,0,0,0,0,22,38],[41,156,0,0,0,0,85,200,0,0,0,0,0,0,92,142,0,0,0,0,234,149,0,0,0,0,0,0,57,210,0,0,0,0,237,184] >;
C30.SD16 in GAP, Magma, Sage, TeX
C_{30}.{\rm SD}_{16} % in TeX
G:=Group("C30.SD16"); // GroupNames label
G:=SmallGroup(480,62);
// by ID
G=gap.SmallGroup(480,62);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,148,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^30=b^8=1,c^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=b^3>;
// generators/relations