metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.29C42, C23.33D30, C22.11D60, C22.3Dic30, (C2×C60)⋊17C4, (C2×C12)⋊4Dic5, (C2×C20)⋊9Dic3, (C2×C4)⋊2Dic15, (C2×C6).17D20, C30.43(C4⋊C4), (C2×C30).13Q8, (C2×Dic15)⋊8C4, (C2×C10).17D12, (C2×C30).139D4, (C22×C20).7S3, (C22×C60).3C2, C2.5(C4×Dic15), C6.12(C4×Dic5), C2.2(C60⋊5C4), (C22×C4).4D15, (C22×C12).3D5, C10.37(D6⋊C4), C6.10(C4⋊Dic5), C5⋊4(C6.C42), C10.24(C4×Dic3), (C2×C6).12Dic10, (C2×C10).12Dic6, C22.12(C4×D15), C2.2(D30⋊3C4), C30.79(C22⋊C4), C10.17(C4⋊Dic3), C15⋊5(C2.C42), (C22×C10).130D6, (C22×C6).112D10, C6.13(C23.D5), C2.2(C30.4Q8), C6.22(D10⋊C4), C10.21(Dic3⋊C4), C3⋊2(C10.10C42), C2.2(C30.38D4), C6.14(C10.D4), C22.16(C15⋊7D4), (C22×Dic15).1C2, C22.10(C2×Dic15), C10.24(C6.D4), (C22×C30).135C22, (C2×C6).30(C4×D5), (C2×C10).55(C4×S3), (C2×C30).137(C2×C4), (C2×C6).71(C5⋊D4), (C2×C6).29(C2×Dic5), (C2×C10).71(C3⋊D4), (C2×C10).49(C2×Dic3), SmallGroup(480,191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.29C42
G = < a,b,c | a30=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a15b >
Subgroups: 660 in 152 conjugacy classes, 83 normal (37 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C30, C30, C2.C42, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×Dic3, C22×C12, Dic15, C60, C2×C30, C2×C30, C22×Dic5, C22×C20, C6.C42, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×C30, C10.10C42, C22×Dic15, C22×C60, C30.29C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic5, D10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, D15, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, Dic15, D30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C6.C42, Dic30, C4×D15, D60, C2×Dic15, C15⋊7D4, C10.10C42, C4×Dic15, C30.4Q8, C60⋊5C4, D30⋊3C4, C30.38D4, C30.29C42
►Regular action on 480 points
Generators in S480
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390)(391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450)(451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)
(1 311 82 251)(2 310 83 250)(3 309 84 249)(4 308 85 248)(5 307 86 247)(6 306 87 246)(7 305 88 245)(8 304 89 244)(9 303 90 243)(10 302 61 242)(11 301 62 241)(12 330 63 270)(13 329 64 269)(14 328 65 268)(15 327 66 267)(16 326 67 266)(17 325 68 265)(18 324 69 264)(19 323 70 263)(20 322 71 262)(21 321 72 261)(22 320 73 260)(23 319 74 259)(24 318 75 258)(25 317 76 257)(26 316 77 256)(27 315 78 255)(28 314 79 254)(29 313 80 253)(30 312 81 252)(31 351 101 281)(32 350 102 280)(33 349 103 279)(34 348 104 278)(35 347 105 277)(36 346 106 276)(37 345 107 275)(38 344 108 274)(39 343 109 273)(40 342 110 272)(41 341 111 271)(42 340 112 300)(43 339 113 299)(44 338 114 298)(45 337 115 297)(46 336 116 296)(47 335 117 295)(48 334 118 294)(49 333 119 293)(50 332 120 292)(51 331 91 291)(52 360 92 290)(53 359 93 289)(54 358 94 288)(55 357 95 287)(56 356 96 286)(57 355 97 285)(58 354 98 284)(59 353 99 283)(60 352 100 282)(121 448 204 363)(122 447 205 362)(123 446 206 361)(124 445 207 390)(125 444 208 389)(126 443 209 388)(127 442 210 387)(128 441 181 386)(129 440 182 385)(130 439 183 384)(131 438 184 383)(132 437 185 382)(133 436 186 381)(134 435 187 380)(135 434 188 379)(136 433 189 378)(137 432 190 377)(138 431 191 376)(139 430 192 375)(140 429 193 374)(141 428 194 373)(142 427 195 372)(143 426 196 371)(144 425 197 370)(145 424 198 369)(146 423 199 368)(147 422 200 367)(148 421 201 366)(149 450 202 365)(150 449 203 364)(151 477 223 402)(152 476 224 401)(153 475 225 400)(154 474 226 399)(155 473 227 398)(156 472 228 397)(157 471 229 396)(158 470 230 395)(159 469 231 394)(160 468 232 393)(161 467 233 392)(162 466 234 391)(163 465 235 420)(164 464 236 419)(165 463 237 418)(166 462 238 417)(167 461 239 416)(168 460 240 415)(169 459 211 414)(170 458 212 413)(171 457 213 412)(172 456 214 411)(173 455 215 410)(174 454 216 409)(175 453 217 408)(176 452 218 407)(177 451 219 406)(178 480 220 405)(179 479 221 404)(180 478 222 403)
(1 167 51 123)(2 168 52 124)(3 169 53 125)(4 170 54 126)(5 171 55 127)(6 172 56 128)(7 173 57 129)(8 174 58 130)(9 175 59 131)(10 176 60 132)(11 177 31 133)(12 178 32 134)(13 179 33 135)(14 180 34 136)(15 151 35 137)(16 152 36 138)(17 153 37 139)(18 154 38 140)(19 155 39 141)(20 156 40 142)(21 157 41 143)(22 158 42 144)(23 159 43 145)(24 160 44 146)(25 161 45 147)(26 162 46 148)(27 163 47 149)(28 164 48 150)(29 165 49 121)(30 166 50 122)(61 218 100 185)(62 219 101 186)(63 220 102 187)(64 221 103 188)(65 222 104 189)(66 223 105 190)(67 224 106 191)(68 225 107 192)(69 226 108 193)(70 227 109 194)(71 228 110 195)(72 229 111 196)(73 230 112 197)(74 231 113 198)(75 232 114 199)(76 233 115 200)(77 234 116 201)(78 235 117 202)(79 236 118 203)(80 237 119 204)(81 238 120 205)(82 239 91 206)(83 240 92 207)(84 211 93 208)(85 212 94 209)(86 213 95 210)(87 214 96 181)(88 215 97 182)(89 216 98 183)(90 217 99 184)(241 391 281 366)(242 392 282 367)(243 393 283 368)(244 394 284 369)(245 395 285 370)(246 396 286 371)(247 397 287 372)(248 398 288 373)(249 399 289 374)(250 400 290 375)(251 401 291 376)(252 402 292 377)(253 403 293 378)(254 404 294 379)(255 405 295 380)(256 406 296 381)(257 407 297 382)(258 408 298 383)(259 409 299 384)(260 410 300 385)(261 411 271 386)(262 412 272 387)(263 413 273 388)(264 414 274 389)(265 415 275 390)(266 416 276 361)(267 417 277 362)(268 418 278 363)(269 419 279 364)(270 420 280 365)(301 466 351 421)(302 467 352 422)(303 468 353 423)(304 469 354 424)(305 470 355 425)(306 471 356 426)(307 472 357 427)(308 473 358 428)(309 474 359 429)(310 475 360 430)(311 476 331 431)(312 477 332 432)(313 478 333 433)(314 479 334 434)(315 480 335 435)(316 451 336 436)(317 452 337 437)(318 453 338 438)(319 454 339 439)(320 455 340 440)(321 456 341 441)(322 457 342 442)(323 458 343 443)(324 459 344 444)(325 460 345 445)(326 461 346 446)(327 462 347 447)(328 463 348 448)(329 464 349 449)(330 465 350 450)
G:=sub<Sym(480)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450)(451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480), (1,311,82,251)(2,310,83,250)(3,309,84,249)(4,308,85,248)(5,307,86,247)(6,306,87,246)(7,305,88,245)(8,304,89,244)(9,303,90,243)(10,302,61,242)(11,301,62,241)(12,330,63,270)(13,329,64,269)(14,328,65,268)(15,327,66,267)(16,326,67,266)(17,325,68,265)(18,324,69,264)(19,323,70,263)(20,322,71,262)(21,321,72,261)(22,320,73,260)(23,319,74,259)(24,318,75,258)(25,317,76,257)(26,316,77,256)(27,315,78,255)(28,314,79,254)(29,313,80,253)(30,312,81,252)(31,351,101,281)(32,350,102,280)(33,349,103,279)(34,348,104,278)(35,347,105,277)(36,346,106,276)(37,345,107,275)(38,344,108,274)(39,343,109,273)(40,342,110,272)(41,341,111,271)(42,340,112,300)(43,339,113,299)(44,338,114,298)(45,337,115,297)(46,336,116,296)(47,335,117,295)(48,334,118,294)(49,333,119,293)(50,332,120,292)(51,331,91,291)(52,360,92,290)(53,359,93,289)(54,358,94,288)(55,357,95,287)(56,356,96,286)(57,355,97,285)(58,354,98,284)(59,353,99,283)(60,352,100,282)(121,448,204,363)(122,447,205,362)(123,446,206,361)(124,445,207,390)(125,444,208,389)(126,443,209,388)(127,442,210,387)(128,441,181,386)(129,440,182,385)(130,439,183,384)(131,438,184,383)(132,437,185,382)(133,436,186,381)(134,435,187,380)(135,434,188,379)(136,433,189,378)(137,432,190,377)(138,431,191,376)(139,430,192,375)(140,429,193,374)(141,428,194,373)(142,427,195,372)(143,426,196,371)(144,425,197,370)(145,424,198,369)(146,423,199,368)(147,422,200,367)(148,421,201,366)(149,450,202,365)(150,449,203,364)(151,477,223,402)(152,476,224,401)(153,475,225,400)(154,474,226,399)(155,473,227,398)(156,472,228,397)(157,471,229,396)(158,470,230,395)(159,469,231,394)(160,468,232,393)(161,467,233,392)(162,466,234,391)(163,465,235,420)(164,464,236,419)(165,463,237,418)(166,462,238,417)(167,461,239,416)(168,460,240,415)(169,459,211,414)(170,458,212,413)(171,457,213,412)(172,456,214,411)(173,455,215,410)(174,454,216,409)(175,453,217,408)(176,452,218,407)(177,451,219,406)(178,480,220,405)(179,479,221,404)(180,478,222,403), (1,167,51,123)(2,168,52,124)(3,169,53,125)(4,170,54,126)(5,171,55,127)(6,172,56,128)(7,173,57,129)(8,174,58,130)(9,175,59,131)(10,176,60,132)(11,177,31,133)(12,178,32,134)(13,179,33,135)(14,180,34,136)(15,151,35,137)(16,152,36,138)(17,153,37,139)(18,154,38,140)(19,155,39,141)(20,156,40,142)(21,157,41,143)(22,158,42,144)(23,159,43,145)(24,160,44,146)(25,161,45,147)(26,162,46,148)(27,163,47,149)(28,164,48,150)(29,165,49,121)(30,166,50,122)(61,218,100,185)(62,219,101,186)(63,220,102,187)(64,221,103,188)(65,222,104,189)(66,223,105,190)(67,224,106,191)(68,225,107,192)(69,226,108,193)(70,227,109,194)(71,228,110,195)(72,229,111,196)(73,230,112,197)(74,231,113,198)(75,232,114,199)(76,233,115,200)(77,234,116,201)(78,235,117,202)(79,236,118,203)(80,237,119,204)(81,238,120,205)(82,239,91,206)(83,240,92,207)(84,211,93,208)(85,212,94,209)(86,213,95,210)(87,214,96,181)(88,215,97,182)(89,216,98,183)(90,217,99,184)(241,391,281,366)(242,392,282,367)(243,393,283,368)(244,394,284,369)(245,395,285,370)(246,396,286,371)(247,397,287,372)(248,398,288,373)(249,399,289,374)(250,400,290,375)(251,401,291,376)(252,402,292,377)(253,403,293,378)(254,404,294,379)(255,405,295,380)(256,406,296,381)(257,407,297,382)(258,408,298,383)(259,409,299,384)(260,410,300,385)(261,411,271,386)(262,412,272,387)(263,413,273,388)(264,414,274,389)(265,415,275,390)(266,416,276,361)(267,417,277,362)(268,418,278,363)(269,419,279,364)(270,420,280,365)(301,466,351,421)(302,467,352,422)(303,468,353,423)(304,469,354,424)(305,470,355,425)(306,471,356,426)(307,472,357,427)(308,473,358,428)(309,474,359,429)(310,475,360,430)(311,476,331,431)(312,477,332,432)(313,478,333,433)(314,479,334,434)(315,480,335,435)(316,451,336,436)(317,452,337,437)(318,453,338,438)(319,454,339,439)(320,455,340,440)(321,456,341,441)(322,457,342,442)(323,458,343,443)(324,459,344,444)(325,460,345,445)(326,461,346,446)(327,462,347,447)(328,463,348,448)(329,464,349,449)(330,465,350,450)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450)(451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480), (1,311,82,251)(2,310,83,250)(3,309,84,249)(4,308,85,248)(5,307,86,247)(6,306,87,246)(7,305,88,245)(8,304,89,244)(9,303,90,243)(10,302,61,242)(11,301,62,241)(12,330,63,270)(13,329,64,269)(14,328,65,268)(15,327,66,267)(16,326,67,266)(17,325,68,265)(18,324,69,264)(19,323,70,263)(20,322,71,262)(21,321,72,261)(22,320,73,260)(23,319,74,259)(24,318,75,258)(25,317,76,257)(26,316,77,256)(27,315,78,255)(28,314,79,254)(29,313,80,253)(30,312,81,252)(31,351,101,281)(32,350,102,280)(33,349,103,279)(34,348,104,278)(35,347,105,277)(36,346,106,276)(37,345,107,275)(38,344,108,274)(39,343,109,273)(40,342,110,272)(41,341,111,271)(42,340,112,300)(43,339,113,299)(44,338,114,298)(45,337,115,297)(46,336,116,296)(47,335,117,295)(48,334,118,294)(49,333,119,293)(50,332,120,292)(51,331,91,291)(52,360,92,290)(53,359,93,289)(54,358,94,288)(55,357,95,287)(56,356,96,286)(57,355,97,285)(58,354,98,284)(59,353,99,283)(60,352,100,282)(121,448,204,363)(122,447,205,362)(123,446,206,361)(124,445,207,390)(125,444,208,389)(126,443,209,388)(127,442,210,387)(128,441,181,386)(129,440,182,385)(130,439,183,384)(131,438,184,383)(132,437,185,382)(133,436,186,381)(134,435,187,380)(135,434,188,379)(136,433,189,378)(137,432,190,377)(138,431,191,376)(139,430,192,375)(140,429,193,374)(141,428,194,373)(142,427,195,372)(143,426,196,371)(144,425,197,370)(145,424,198,369)(146,423,199,368)(147,422,200,367)(148,421,201,366)(149,450,202,365)(150,449,203,364)(151,477,223,402)(152,476,224,401)(153,475,225,400)(154,474,226,399)(155,473,227,398)(156,472,228,397)(157,471,229,396)(158,470,230,395)(159,469,231,394)(160,468,232,393)(161,467,233,392)(162,466,234,391)(163,465,235,420)(164,464,236,419)(165,463,237,418)(166,462,238,417)(167,461,239,416)(168,460,240,415)(169,459,211,414)(170,458,212,413)(171,457,213,412)(172,456,214,411)(173,455,215,410)(174,454,216,409)(175,453,217,408)(176,452,218,407)(177,451,219,406)(178,480,220,405)(179,479,221,404)(180,478,222,403), (1,167,51,123)(2,168,52,124)(3,169,53,125)(4,170,54,126)(5,171,55,127)(6,172,56,128)(7,173,57,129)(8,174,58,130)(9,175,59,131)(10,176,60,132)(11,177,31,133)(12,178,32,134)(13,179,33,135)(14,180,34,136)(15,151,35,137)(16,152,36,138)(17,153,37,139)(18,154,38,140)(19,155,39,141)(20,156,40,142)(21,157,41,143)(22,158,42,144)(23,159,43,145)(24,160,44,146)(25,161,45,147)(26,162,46,148)(27,163,47,149)(28,164,48,150)(29,165,49,121)(30,166,50,122)(61,218,100,185)(62,219,101,186)(63,220,102,187)(64,221,103,188)(65,222,104,189)(66,223,105,190)(67,224,106,191)(68,225,107,192)(69,226,108,193)(70,227,109,194)(71,228,110,195)(72,229,111,196)(73,230,112,197)(74,231,113,198)(75,232,114,199)(76,233,115,200)(77,234,116,201)(78,235,117,202)(79,236,118,203)(80,237,119,204)(81,238,120,205)(82,239,91,206)(83,240,92,207)(84,211,93,208)(85,212,94,209)(86,213,95,210)(87,214,96,181)(88,215,97,182)(89,216,98,183)(90,217,99,184)(241,391,281,366)(242,392,282,367)(243,393,283,368)(244,394,284,369)(245,395,285,370)(246,396,286,371)(247,397,287,372)(248,398,288,373)(249,399,289,374)(250,400,290,375)(251,401,291,376)(252,402,292,377)(253,403,293,378)(254,404,294,379)(255,405,295,380)(256,406,296,381)(257,407,297,382)(258,408,298,383)(259,409,299,384)(260,410,300,385)(261,411,271,386)(262,412,272,387)(263,413,273,388)(264,414,274,389)(265,415,275,390)(266,416,276,361)(267,417,277,362)(268,418,278,363)(269,419,279,364)(270,420,280,365)(301,466,351,421)(302,467,352,422)(303,468,353,423)(304,469,354,424)(305,470,355,425)(306,471,356,426)(307,472,357,427)(308,473,358,428)(309,474,359,429)(310,475,360,430)(311,476,331,431)(312,477,332,432)(313,478,333,433)(314,479,334,434)(315,480,335,435)(316,451,336,436)(317,452,337,437)(318,453,338,438)(319,454,339,439)(320,455,340,440)(321,456,341,441)(322,457,342,442)(323,458,343,443)(324,459,344,444)(325,460,345,445)(326,461,346,446)(327,462,347,447)(328,463,348,448)(329,464,349,449)(330,465,350,450) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270),(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300),(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330),(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390),(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420),(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450),(451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)], [(1,311,82,251),(2,310,83,250),(3,309,84,249),(4,308,85,248),(5,307,86,247),(6,306,87,246),(7,305,88,245),(8,304,89,244),(9,303,90,243),(10,302,61,242),(11,301,62,241),(12,330,63,270),(13,329,64,269),(14,328,65,268),(15,327,66,267),(16,326,67,266),(17,325,68,265),(18,324,69,264),(19,323,70,263),(20,322,71,262),(21,321,72,261),(22,320,73,260),(23,319,74,259),(24,318,75,258),(25,317,76,257),(26,316,77,256),(27,315,78,255),(28,314,79,254),(29,313,80,253),(30,312,81,252),(31,351,101,281),(32,350,102,280),(33,349,103,279),(34,348,104,278),(35,347,105,277),(36,346,106,276),(37,345,107,275),(38,344,108,274),(39,343,109,273),(40,342,110,272),(41,341,111,271),(42,340,112,300),(43,339,113,299),(44,338,114,298),(45,337,115,297),(46,336,116,296),(47,335,117,295),(48,334,118,294),(49,333,119,293),(50,332,120,292),(51,331,91,291),(52,360,92,290),(53,359,93,289),(54,358,94,288),(55,357,95,287),(56,356,96,286),(57,355,97,285),(58,354,98,284),(59,353,99,283),(60,352,100,282),(121,448,204,363),(122,447,205,362),(123,446,206,361),(124,445,207,390),(125,444,208,389),(126,443,209,388),(127,442,210,387),(128,441,181,386),(129,440,182,385),(130,439,183,384),(131,438,184,383),(132,437,185,382),(133,436,186,381),(134,435,187,380),(135,434,188,379),(136,433,189,378),(137,432,190,377),(138,431,191,376),(139,430,192,375),(140,429,193,374),(141,428,194,373),(142,427,195,372),(143,426,196,371),(144,425,197,370),(145,424,198,369),(146,423,199,368),(147,422,200,367),(148,421,201,366),(149,450,202,365),(150,449,203,364),(151,477,223,402),(152,476,224,401),(153,475,225,400),(154,474,226,399),(155,473,227,398),(156,472,228,397),(157,471,229,396),(158,470,230,395),(159,469,231,394),(160,468,232,393),(161,467,233,392),(162,466,234,391),(163,465,235,420),(164,464,236,419),(165,463,237,418),(166,462,238,417),(167,461,239,416),(168,460,240,415),(169,459,211,414),(170,458,212,413),(171,457,213,412),(172,456,214,411),(173,455,215,410),(174,454,216,409),(175,453,217,408),(176,452,218,407),(177,451,219,406),(178,480,220,405),(179,479,221,404),(180,478,222,403)], [(1,167,51,123),(2,168,52,124),(3,169,53,125),(4,170,54,126),(5,171,55,127),(6,172,56,128),(7,173,57,129),(8,174,58,130),(9,175,59,131),(10,176,60,132),(11,177,31,133),(12,178,32,134),(13,179,33,135),(14,180,34,136),(15,151,35,137),(16,152,36,138),(17,153,37,139),(18,154,38,140),(19,155,39,141),(20,156,40,142),(21,157,41,143),(22,158,42,144),(23,159,43,145),(24,160,44,146),(25,161,45,147),(26,162,46,148),(27,163,47,149),(28,164,48,150),(29,165,49,121),(30,166,50,122),(61,218,100,185),(62,219,101,186),(63,220,102,187),(64,221,103,188),(65,222,104,189),(66,223,105,190),(67,224,106,191),(68,225,107,192),(69,226,108,193),(70,227,109,194),(71,228,110,195),(72,229,111,196),(73,230,112,197),(74,231,113,198),(75,232,114,199),(76,233,115,200),(77,234,116,201),(78,235,117,202),(79,236,118,203),(80,237,119,204),(81,238,120,205),(82,239,91,206),(83,240,92,207),(84,211,93,208),(85,212,94,209),(86,213,95,210),(87,214,96,181),(88,215,97,182),(89,216,98,183),(90,217,99,184),(241,391,281,366),(242,392,282,367),(243,393,283,368),(244,394,284,369),(245,395,285,370),(246,396,286,371),(247,397,287,372),(248,398,288,373),(249,399,289,374),(250,400,290,375),(251,401,291,376),(252,402,292,377),(253,403,293,378),(254,404,294,379),(255,405,295,380),(256,406,296,381),(257,407,297,382),(258,408,298,383),(259,409,299,384),(260,410,300,385),(261,411,271,386),(262,412,272,387),(263,413,273,388),(264,414,274,389),(265,415,275,390),(266,416,276,361),(267,417,277,362),(268,418,278,363),(269,419,279,364),(270,420,280,365),(301,466,351,421),(302,467,352,422),(303,468,353,423),(304,469,354,424),(305,470,355,425),(306,471,356,426),(307,472,357,427),(308,473,358,428),(309,474,359,429),(310,475,360,430),(311,476,331,431),(312,477,332,432),(313,478,333,433),(314,479,334,434),(315,480,335,435),(316,451,336,436),(317,452,337,437),(318,453,338,438),(319,454,339,439),(320,455,340,440),(321,456,341,441),(322,457,342,442),(323,458,343,443),(324,459,344,444),(325,460,345,445),(326,461,346,446),(327,462,347,447),(328,463,348,448),(329,464,349,449),(330,465,350,450)]])
132 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 30 | ··· | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | + | - | + | + | - | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | D5 | Dic3 | D6 | Dic5 | D10 | Dic6 | C4×S3 | D12 | C3⋊D4 | D15 | Dic10 | C4×D5 | D20 | C5⋊D4 | Dic15 | D30 | Dic30 | C4×D15 | D60 | C15⋊7D4 |
| kernel | C30.29C42 | C22×Dic15 | C22×C60 | C2×Dic15 | C2×C60 | C22×C20 | C2×C30 | C2×C30 | C22×C12 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C2×C10 | C2×C10 | C2×C10 | C2×C10 | C22×C4 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 3 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 4 | 8 | 16 | 8 | 16 |
Matrix representation of C30.29C42 ►in GL6(𝔽61)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 11 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 53 |
| 0 | 0 | 0 | 0 | 55 | 24 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 18 | 0 | 0 |
| 0 | 0 | 11 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 57 |
| 0 | 0 | 0 | 0 | 58 | 29 |
G:=sub<GL(6,GF(61))| [16,0,0,0,0,0,0,42,0,0,0,0,0,0,42,11,0,0,0,0,0,16,0,0,0,0,0,0,30,55,0,0,0,0,53,24],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,44,11,0,0,0,0,18,17,0,0,0,0,0,0,0,4,0,0,0,0,15,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,58,0,0,0,0,57,29] >;
C30.29C42 in GAP, Magma, Sage, TeX
C_{30}._{29}C_4^2 % in TeX
G:=Group("C30.29C4^2"); // GroupNames label
G:=SmallGroup(480,191);
// by ID
G=gap.SmallGroup(480,191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations