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G = C30.24C42order 480 = 25·3·5

7th non-split extension by C30 of C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.24C42, (C2×C30).5Q8, (C6×Dic5)⋊5C4, (C2×C6).36D20, (C2×C30).35D4, C30.34(C4⋊C4), C6.7(C4×Dic5), (C10×Dic3)⋊8C4, (C2×C10).36D12, C6.5(C4⋊Dic5), (C2×C6).6Dic10, (C2×C10).6Dic6, C23.59(S3×D5), C10.25(D6⋊C4), C22.4(C15⋊Q8), (C2×Dic5)⋊3Dic3, (C2×Dic3)⋊2Dic5, (C2×Dic15)⋊11C4, C2.2(D6⋊Dic5), C53(C6.C42), C10.19(C4×Dic3), C2.7(Dic3×Dic5), (C22×C10).96D6, (C22×C6).79D10, C2.2(D304C4), C30.65(C22⋊C4), C10.12(C4⋊Dic3), C154(C2.C42), C6.9(C10.D4), C6.10(C23.D5), C2.2(C30.Q8), C2.2(Dic155C4), C2.2(C6.Dic10), C6.10(D10⋊C4), (C22×Dic3).1D5, (C22×Dic5).3S3, C22.12(D5×Dic3), C22.12(S3×Dic5), C10.16(Dic3⋊C4), C31(C10.10C42), C2.2(D10⋊Dic3), C22.18(C15⋊D4), C22.18(C3⋊D20), C22.18(C5⋊D12), (C22×C30).25C22, (C22×Dic15).7C2, C10.21(C6.D4), C22.12(D30.C2), (C2×C6).48(C4×D5), (C2×C6×Dic5).1C2, (C2×C30).89(C2×C4), (C2×C10).43(C4×S3), (Dic3×C2×C10).1C2, (C2×C6).50(C5⋊D4), (C2×C6).13(C2×Dic5), (C2×C10).29(C3⋊D4), (C2×C10).33(C2×Dic3), SmallGroup(480,70)

Series: Derived Chief Lower central Upper central

C1C30 — C30.24C42
C1C5C15C30C2×C30C22×C30C2×C6×Dic5 — C30.24C42
C15C30 — C30.24C42
C1C23

Generators and relations for C30.24C42
 G = < a,b,c | a30=b4=c4=1, bab-1=a19, cac-1=a11, cbc-1=a15b >

Subgroups: 572 in 152 conjugacy classes, 80 normal (70 characteristic)
C1, C2 [×7], C3, C4 [×6], C22 [×7], C5, C6 [×7], C2×C4 [×12], C23, C10 [×7], Dic3 [×4], C12 [×2], C2×C6 [×7], C15, C22×C4 [×3], Dic5 [×4], C20 [×2], C2×C10 [×7], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×4], C22×C6, C30 [×7], C2.C42, C2×Dic5 [×2], C2×Dic5 [×6], C2×C20 [×4], C22×C10, C22×Dic3, C22×Dic3, C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C2×C30 [×7], C22×Dic5, C22×Dic5, C22×C20, C6.C42, C6×Dic5 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×Dic15 [×2], C22×C30, C10.10C42, C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C30.24C42
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], C2.C42, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, S3×D5, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C6.C42, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C10.10C42, Dic3×Dic5, D10⋊Dic3, D6⋊Dic5, D304C4, C30.Q8, Dic155C4, C6.Dic10, C30.24C42

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

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

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

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

84 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A···6G10A···10N12A···12H15A15B20A···20P30A···30N
order12···23444444444444556···610···1012···12151520···2030···30
size11···1266661010101030303030222···22···210···10446···64···4

84 irreducible representations

dim1111111222222222222222244444444
type++++++-+-+-+-+-++--+-++-
imageC1C2C2C2C4C4C4S3D4Q8D5Dic3D6Dic5D10Dic6C4×S3D12C3⋊D4Dic10C4×D5D20C5⋊D4S3×D5D5×Dic3S3×Dic5D30.C2C15⋊D4C3⋊D20C5⋊D12C15⋊Q8
kernelC30.24C42C2×C6×Dic5Dic3×C2×C10C22×Dic15C6×Dic5C10×Dic3C2×Dic15C22×Dic5C2×C30C2×C30C22×Dic3C2×Dic5C22×C10C2×Dic3C22×C6C2×C10C2×C10C2×C10C2×C10C2×C6C2×C6C2×C6C2×C6C23C22C22C22C22C22C22C22
# reps1111444131221422424484822222222

Matrix representation of C30.24C42 in GL6(𝔽61)

60600000
100000
00606000
001000
0000060
0000144
,
5000000
0500000
0050000
0005000
0000612
00005355
,
5290000
1890000
00606000
000100
00002954
0000732

G:=sub<GL(6,GF(61))| [60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,60,44],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,6,53,0,0,0,0,12,55],[52,18,0,0,0,0,9,9,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,29,7,0,0,0,0,54,32] >;

C30.24C42 in GAP, Magma, Sage, TeX

C_{30}._{24}C_4^2
% in TeX

G:=Group("C30.24C4^2");
// GroupNames label

G:=SmallGroup(480,70);
// by ID

G=gap.SmallGroup(480,70);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^30=b^4=c^4=1,b*a*b^-1=a^19,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

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