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## G = C30.M4(2)  order 480 = 25·3·5

### 1st non-split extension by C30 of M4(2) acting via M4(2)/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C30.M4(2)
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C6×Dic5 — C6×C5⋊C8 — C30.M4(2)
 Lower central C15 — C30 — C30.M4(2)
 Upper central C1 — C22

Generators and relations for C30.M4(2)
G = < a,b,c | a30=b8=1, c2=a15, bab-1=a17, cac-1=a-1, cbc-1=b5 >

Subgroups: 308 in 80 conjugacy classes, 40 normal (34 characteristic)
C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C30, C8⋊C4, C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C3×Dic5, Dic15, C2×C30, C4×Dic5, C2×C5⋊C8, C2×C5⋊C8, C24⋊C4, C3×C5⋊C8, C15⋊C8, C6×Dic5, C10×Dic3, C2×Dic15, C10.C42, Dic3×Dic5, C6×C5⋊C8, C2×C15⋊C8, C30.M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, M4(2), F5, C4×S3, C2×Dic3, C8⋊C4, C2×F5, C8⋊S3, C4×Dic3, C4.F5, C4×F5, C22.F5, C24⋊C4, S3×F5, C10.C42, Dic3×F5, D6.F5, Dic3.F5, C30.M4(2)

Smallest permutation representation of C30.M4(2)
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)
(1 397 341 213 140 206 167 388)(2 420 360 230 141 199 156 375)(3 413 349 217 142 192 175 362)(4 406 338 234 143 185 164 379)(5 399 357 221 144 208 153 366)(6 392 346 238 145 201 172 383)(7 415 335 225 146 194 161 370)(8 408 354 212 147 187 180 387)(9 401 343 229 148 210 169 374)(10 394 332 216 149 203 158 361)(11 417 351 233 150 196 177 378)(12 410 340 220 121 189 166 365)(13 403 359 237 122 182 155 382)(14 396 348 224 123 205 174 369)(15 419 337 211 124 198 163 386)(16 412 356 228 125 191 152 373)(17 405 345 215 126 184 171 390)(18 398 334 232 127 207 160 377)(19 391 353 219 128 200 179 364)(20 414 342 236 129 193 168 381)(21 407 331 223 130 186 157 368)(22 400 350 240 131 209 176 385)(23 393 339 227 132 202 165 372)(24 416 358 214 133 195 154 389)(25 409 347 231 134 188 173 376)(26 402 336 218 135 181 162 363)(27 395 355 235 136 204 151 380)(28 418 344 222 137 197 170 367)(29 411 333 239 138 190 159 384)(30 404 352 226 139 183 178 371)(31 86 312 248 477 278 100 450)(32 79 301 265 478 271 119 437)(33 72 320 252 479 294 108 424)(34 65 309 269 480 287 97 441)(35 88 328 256 451 280 116 428)(36 81 317 243 452 273 105 445)(37 74 306 260 453 296 94 432)(38 67 325 247 454 289 113 449)(39 90 314 264 455 282 102 436)(40 83 303 251 456 275 91 423)(41 76 322 268 457 298 110 440)(42 69 311 255 458 291 99 427)(43 62 330 242 459 284 118 444)(44 85 319 259 460 277 107 431)(45 78 308 246 461 300 96 448)(46 71 327 263 462 293 115 435)(47 64 316 250 463 286 104 422)(48 87 305 267 464 279 93 439)(49 80 324 254 465 272 112 426)(50 73 313 241 466 295 101 443)(51 66 302 258 467 288 120 430)(52 89 321 245 468 281 109 447)(53 82 310 262 469 274 98 434)(54 75 329 249 470 297 117 421)(55 68 318 266 471 290 106 438)(56 61 307 253 472 283 95 425)(57 84 326 270 473 276 114 442)(58 77 315 257 474 299 103 429)(59 70 304 244 475 292 92 446)(60 63 323 261 476 285 111 433)
(1 285 16 300)(2 284 17 299)(3 283 18 298)(4 282 19 297)(5 281 20 296)(6 280 21 295)(7 279 22 294)(8 278 23 293)(9 277 24 292)(10 276 25 291)(11 275 26 290)(12 274 27 289)(13 273 28 288)(14 272 29 287)(15 271 30 286)(31 372 46 387)(32 371 47 386)(33 370 48 385)(34 369 49 384)(35 368 50 383)(36 367 51 382)(37 366 52 381)(38 365 53 380)(39 364 54 379)(40 363 55 378)(41 362 56 377)(42 361 57 376)(43 390 58 375)(44 389 59 374)(45 388 60 373)(61 127 76 142)(62 126 77 141)(63 125 78 140)(64 124 79 139)(65 123 80 138)(66 122 81 137)(67 121 82 136)(68 150 83 135)(69 149 84 134)(70 148 85 133)(71 147 86 132)(72 146 87 131)(73 145 88 130)(74 144 89 129)(75 143 90 128)(91 181 106 196)(92 210 107 195)(93 209 108 194)(94 208 109 193)(95 207 110 192)(96 206 111 191)(97 205 112 190)(98 204 113 189)(99 203 114 188)(100 202 115 187)(101 201 116 186)(102 200 117 185)(103 199 118 184)(104 198 119 183)(105 197 120 182)(151 247 166 262)(152 246 167 261)(153 245 168 260)(154 244 169 259)(155 243 170 258)(156 242 171 257)(157 241 172 256)(158 270 173 255)(159 269 174 254)(160 268 175 253)(161 267 176 252)(162 266 177 251)(163 265 178 250)(164 264 179 249)(165 263 180 248)(211 478 226 463)(212 477 227 462)(213 476 228 461)(214 475 229 460)(215 474 230 459)(216 473 231 458)(217 472 232 457)(218 471 233 456)(219 470 234 455)(220 469 235 454)(221 468 236 453)(222 467 237 452)(223 466 238 451)(224 465 239 480)(225 464 240 479)(301 404 316 419)(302 403 317 418)(303 402 318 417)(304 401 319 416)(305 400 320 415)(306 399 321 414)(307 398 322 413)(308 397 323 412)(309 396 324 411)(310 395 325 410)(311 394 326 409)(312 393 327 408)(313 392 328 407)(314 391 329 406)(315 420 330 405)(331 443 346 428)(332 442 347 427)(333 441 348 426)(334 440 349 425)(335 439 350 424)(336 438 351 423)(337 437 352 422)(338 436 353 421)(339 435 354 450)(340 434 355 449)(341 433 356 448)(342 432 357 447)(343 431 358 446)(344 430 359 445)(345 429 360 444)```

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 8 8 type + + + + + - + + + - + - - + image C1 C2 C2 C2 C4 C4 C4 C4 S3 Dic3 D6 M4(2) C4×S3 C4×S3 C8⋊S3 F5 C2×F5 C4.F5 C4×F5 C22.F5 S3×F5 Dic3×F5 D6.F5 Dic3.F5 kernel C30.M4(2) Dic3×Dic5 C6×C5⋊C8 C2×C15⋊C8 C3×C5⋊C8 C15⋊C8 C10×Dic3 C2×Dic15 C2×C5⋊C8 C5⋊C8 C2×Dic5 C30 Dic5 C2×C10 C10 C2×Dic3 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 4 2 2 1 2 1 4 2 2 8 1 1 2 2 2 1 1 1 1

Matrix representation of C30.M4(2) in GL6(𝔽241)

 0 1 0 0 0 0 240 1 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 1 1 1 1
,
 169 124 0 0 0 0 52 72 0 0 0 0 0 0 150 77 21 140 0 0 119 220 129 56 0 0 168 112 231 91 0 0 101 10 178 122
,
 94 175 0 0 0 0 28 147 0 0 0 0 0 0 84 46 76 27 0 0 203 233 184 157 0 0 30 222 195 38 0 0 192 165 8 211

`G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,240,0,0,1,0,0,0,240,0,1,0,0,0,0,240,1],[169,52,0,0,0,0,124,72,0,0,0,0,0,0,150,119,168,101,0,0,77,220,112,10,0,0,21,129,231,178,0,0,140,56,91,122],[94,28,0,0,0,0,175,147,0,0,0,0,0,0,84,203,30,192,0,0,46,233,222,165,0,0,76,184,195,8,0,0,27,157,38,211] >;`

C30.M4(2) in GAP, Magma, Sage, TeX

`C_{30}.M_4(2)`
`% in TeX`

`G:=Group("C30.M4(2)");`
`// GroupNames label`

`G:=SmallGroup(480,245);`
`// by ID`

`G=gap.SmallGroup(480,245);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,100,1356,9414,4724]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=b^8=1,c^2=a^15,b*a*b^-1=a^17,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

׿
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