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## G = C30.22C42order 480 = 25·3·5

### 5th non-split extension by C30 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C30.22C42
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C6×C5⋊2C8 — C30.22C42
 Lower central C15 — C30 — C30.22C42
 Upper central C1 — C2×C4

Generators and relations for C30.22C42
G = < a,b,c | a30=c4=1, b4=a15, bab-1=a19, cac-1=a11, cbc-1=a15b >

Subgroups: 220 in 80 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, C20, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C30, C8⋊C4, C52C8, C52C8, C2×C20, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C60, C2×C30, C2×C52C8, C2×C52C8, C4×C20, C24⋊C4, C3×C52C8, C153C8, C10×Dic3, C2×C60, C42.D5, C6×C52C8, Dic3×C20, C2×C153C8, C30.22C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, C42, M4(2), Dic5, D10, C4×S3, C2×Dic3, C8⋊C4, C4×D5, C2×Dic5, C8⋊S3, C4×Dic3, S3×D5, C4.Dic5, C4×Dic5, C24⋊C4, D5×Dic3, S3×Dic5, D30.C2, C42.D5, D6.Dic5, Dic3×Dic5, C30.22C42

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 12A 12B 12C 12D 15A 15B 20A ··· 20H 20I ··· 20X 24A ··· 24H 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 8 8 8 8 10 ··· 10 12 12 12 12 15 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 2 1 1 1 1 6 6 6 6 2 2 2 2 2 10 10 10 10 30 30 30 30 2 ··· 2 2 2 2 2 4 4 2 ··· 2 6 ··· 6 10 ··· 10 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + - + - + + - + - image C1 C2 C2 C2 C4 C4 C4 S3 D5 Dic3 D6 M4(2) Dic5 D10 C4×S3 C4×S3 C4×D5 C8⋊S3 C4.Dic5 S3×D5 D5×Dic3 D30.C2 S3×Dic5 D6.Dic5 kernel C30.22C42 C6×C5⋊2C8 Dic3×C20 C2×C15⋊3C8 C3×C5⋊2C8 C15⋊3C8 C10×Dic3 C2×C5⋊2C8 C4×Dic3 C5⋊2C8 C2×C20 C30 C2×Dic3 C2×C12 C20 C2×C10 C12 C10 C6 C2×C4 C4 C4 C22 C2 # reps 1 1 1 1 4 4 4 1 2 2 1 4 4 2 2 2 8 8 16 2 2 2 2 8

Matrix representation of C30.22C42 in GL5(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 240 51 0 0 0 0 0 2 5 0 0 0 144 240
,
 64 0 0 0 0 0 206 175 0 0 0 77 35 0 0 0 0 0 34 33 0 0 0 131 207
,
 64 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 43 120 0 0 0 85 198

`G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,0,240,0,0,0,1,51,0,0,0,0,0,2,144,0,0,0,5,240],[64,0,0,0,0,0,206,77,0,0,0,175,35,0,0,0,0,0,34,131,0,0,0,33,207],[64,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,43,85,0,0,0,120,198] >;`

C30.22C42 in GAP, Magma, Sage, TeX

`C_{30}._{22}C_4^2`
`% in TeX`

`G:=Group("C30.22C4^2");`
`// GroupNames label`

`G:=SmallGroup(480,29);`
`// by ID`

`G=gap.SmallGroup(480,29);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=c^4=1,b^4=a^15,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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