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G = C30.29C42order 480 = 25·3·5

5th non-split extension by C30 of C42 acting via C42/C2×C4=C2

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(C605C4), (C22×C4).4D15, (C22×C12).3D5, C10.37(D6⋊C4), C6.10(C4⋊Dic5), C54(C6.C42), C10.24(C4×Dic3), (C2×C6).12Dic10, (C2×C10).12Dic6, C22.12(C4×D15), C2.2(D303C4), C30.79(C22⋊C4), C10.17(C4⋊Dic3), C155(C2.C42), (C22×C10).130D6, (C22×C6).112D10, C6.13(C23.D5), C2.2(C30.4Q8), C6.22(D10⋊C4), C10.21(Dic3⋊C4), C32(C10.10C42), C2.2(C30.38D4), C6.14(C10.D4), C22.16(C157D4), (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

C1C30 — C30.29C42
C1C5C15C30C2×C30C22×C30C22×Dic15 — C30.29C42
C15C30 — C30.29C42
C1C23C22×C4

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 [×3], C2 [×4], C3, C4 [×6], C22 [×3], C22 [×4], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×10], C23, C10 [×3], C10 [×4], Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C15, C22×C4, C22×C4 [×2], Dic5 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×4], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×4], C2.C42, C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×2], C22×C10, C22×Dic3 [×2], C22×C12, Dic15 [×4], C60 [×2], C2×C30 [×3], C2×C30 [×4], C22×Dic5 [×2], C22×C20, C6.C42, C2×Dic15 [×4], C2×Dic15 [×4], C2×C60 [×2], C2×C60 [×2], C22×C30, C10.10C42, C22×Dic15 [×2], C22×C60, C30.29C42
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D4 [×3], Q8, D5, Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], D15, C2.C42, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, Dic15 [×2], D30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C6.C42, Dic30, C4×D15 [×2], D60, C2×Dic15, C157D4 [×2], C10.10C42, C4×Dic15, C30.4Q8 [×2], C605C4, D303C4 [×2], C30.38D4, C30.29C42

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

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

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

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

132 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order12···2344444···4556···610···1012···121515151520···2030···3060···60
size11···12222230···30222···22···22···222222···22···22···2

132 irreducible representations

dim1111122222222222222222222222
type+++++-+-+-+-++-+-+-+
imageC1C2C2C4C4S3D4Q8D5Dic3D6Dic5D10Dic6C4×S3D12C3⋊D4D15Dic10C4×D5D20C5⋊D4Dic15D30Dic30C4×D15D60C157D4
kernelC30.29C42C22×Dic15C22×C60C2×Dic15C2×C60C22×C20C2×C30C2×C30C22×C12C2×C20C22×C10C2×C12C22×C6C2×C10C2×C10C2×C10C2×C10C22×C4C2×C6C2×C6C2×C6C2×C6C2×C4C23C22C22C22C22
# reps121841312214224244484884816816

Matrix representation of C30.29C42 in GL6(𝔽61)

1600000
0420000
0042000
00111600
00003053
00005524
,
010000
6000000
00441800
00111700
0000015
000040
,
5000000
0500000
001000
000100
00003257
00005829

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

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