metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.4M4(2), Dic5.5Dic6, C15⋊1(C4⋊C8), Dic3⋊(C5⋊C8), C10.9(S3×C8), C30.8(C2×C8), C3⋊2(C20⋊C8), C30.6(C4⋊C4), C5⋊1(Dic3⋊C8), (C5×Dic3)⋊2C8, C6.13(C4⋊F5), C6.3(C4.F5), (C3×Dic5).5Q8, (C2×Dic3).4F5, C10.3(C8⋊S3), C22.15(S3×F5), C2.2(Dic3⋊F5), (C2×Dic15).7C4, (C10×Dic3).6C4, (C3×Dic5).34D4, C10.6(Dic3⋊C4), (C2×Dic5).145D6, C2.3(Dic3.F5), (Dic3×Dic5).21C2, Dic5.18(C3⋊D4), (C6×Dic5).140C22, C6.5(C2×C5⋊C8), C2.5(S3×C5⋊C8), (C6×C5⋊C8).3C2, (C2×C5⋊C8).2S3, (C2×C6).16(C2×F5), (C2×C15⋊C8).3C2, (C2×C30).10(C2×C4), (C2×C10).12(C4×S3), SmallGroup(480,252)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.4M4(2)
G = < a,b,c | a30=b8=1, c2=a15, bab-1=a17, cac-1=a-1, cbc-1=a15b5 >
Subgroups: 308 in 76 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C30, C4⋊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, Dic3⋊C8, C3×C5⋊C8, C15⋊C8, C6×Dic5, C10×Dic3, C2×Dic15, C20⋊C8, Dic3×Dic5, C6×C5⋊C8, C2×C15⋊C8, C30.4M4(2)
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, D6, C4⋊C4, C2×C8, M4(2), F5, Dic6, C4×S3, C3⋊D4, C4⋊C8, C5⋊C8, C2×F5, S3×C8, C8⋊S3, Dic3⋊C4, C4.F5, C4⋊F5, C2×C5⋊C8, Dic3⋊C8, S3×F5, C20⋊C8, Dic3⋊F5, S3×C5⋊C8, Dic3.F5, C30.4M4(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)
(1 465 397 57 215 272 109 83)(2 458 416 44 216 295 98 70)(3 451 405 31 217 288 117 87)(4 474 394 48 218 281 106 74)(5 467 413 35 219 274 95 61)(6 460 402 52 220 297 114 78)(7 453 391 39 221 290 103 65)(8 476 410 56 222 283 92 82)(9 469 399 43 223 276 111 69)(10 462 418 60 224 299 100 86)(11 455 407 47 225 292 119 73)(12 478 396 34 226 285 108 90)(13 471 415 51 227 278 97 77)(14 464 404 38 228 271 116 64)(15 457 393 55 229 294 105 81)(16 480 412 42 230 287 94 68)(17 473 401 59 231 280 113 85)(18 466 420 46 232 273 102 72)(19 459 409 33 233 296 91 89)(20 452 398 50 234 289 110 76)(21 475 417 37 235 282 99 63)(22 468 406 54 236 275 118 80)(23 461 395 41 237 298 107 67)(24 454 414 58 238 291 96 84)(25 477 403 45 239 284 115 71)(26 470 392 32 240 277 104 88)(27 463 411 49 211 300 93 75)(28 456 400 36 212 293 112 62)(29 479 419 53 213 286 101 79)(30 472 408 40 214 279 120 66)(121 172 346 440 312 384 187 268)(122 165 335 427 313 377 206 255)(123 158 354 444 314 370 195 242)(124 151 343 431 315 363 184 259)(125 174 332 448 316 386 203 246)(126 167 351 435 317 379 192 263)(127 160 340 422 318 372 181 250)(128 153 359 439 319 365 200 267)(129 176 348 426 320 388 189 254)(130 169 337 443 321 381 208 241)(131 162 356 430 322 374 197 258)(132 155 345 447 323 367 186 245)(133 178 334 434 324 390 205 262)(134 171 353 421 325 383 194 249)(135 164 342 438 326 376 183 266)(136 157 331 425 327 369 202 253)(137 180 350 442 328 362 191 270)(138 173 339 429 329 385 210 257)(139 166 358 446 330 378 199 244)(140 159 347 433 301 371 188 261)(141 152 336 450 302 364 207 248)(142 175 355 437 303 387 196 265)(143 168 344 424 304 380 185 252)(144 161 333 441 305 373 204 269)(145 154 352 428 306 366 193 256)(146 177 341 445 307 389 182 243)(147 170 360 432 308 382 201 260)(148 163 349 449 309 375 190 247)(149 156 338 436 310 368 209 264)(150 179 357 423 311 361 198 251)
(1 201 16 186)(2 200 17 185)(3 199 18 184)(4 198 19 183)(5 197 20 182)(6 196 21 181)(7 195 22 210)(8 194 23 209)(9 193 24 208)(10 192 25 207)(11 191 26 206)(12 190 27 205)(13 189 28 204)(14 188 29 203)(15 187 30 202)(31 363 46 378)(32 362 47 377)(33 361 48 376)(34 390 49 375)(35 389 50 374)(36 388 51 373)(37 387 52 372)(38 386 53 371)(39 385 54 370)(40 384 55 369)(41 383 56 368)(42 382 57 367)(43 381 58 366)(44 380 59 365)(45 379 60 364)(61 177 76 162)(62 176 77 161)(63 175 78 160)(64 174 79 159)(65 173 80 158)(66 172 81 157)(67 171 82 156)(68 170 83 155)(69 169 84 154)(70 168 85 153)(71 167 86 152)(72 166 87 151)(73 165 88 180)(74 164 89 179)(75 163 90 178)(91 326 106 311)(92 325 107 310)(93 324 108 309)(94 323 109 308)(95 322 110 307)(96 321 111 306)(97 320 112 305)(98 319 113 304)(99 318 114 303)(100 317 115 302)(101 316 116 301)(102 315 117 330)(103 314 118 329)(104 313 119 328)(105 312 120 327)(121 408 136 393)(122 407 137 392)(123 406 138 391)(124 405 139 420)(125 404 140 419)(126 403 141 418)(127 402 142 417)(128 401 143 416)(129 400 144 415)(130 399 145 414)(131 398 146 413)(132 397 147 412)(133 396 148 411)(134 395 149 410)(135 394 150 409)(211 334 226 349)(212 333 227 348)(213 332 228 347)(214 331 229 346)(215 360 230 345)(216 359 231 344)(217 358 232 343)(218 357 233 342)(219 356 234 341)(220 355 235 340)(221 354 236 339)(222 353 237 338)(223 352 238 337)(224 351 239 336)(225 350 240 335)(241 291 256 276)(242 290 257 275)(243 289 258 274)(244 288 259 273)(245 287 260 272)(246 286 261 271)(247 285 262 300)(248 284 263 299)(249 283 264 298)(250 282 265 297)(251 281 266 296)(252 280 267 295)(253 279 268 294)(254 278 269 293)(255 277 270 292)(421 476 436 461)(422 475 437 460)(423 474 438 459)(424 473 439 458)(425 472 440 457)(426 471 441 456)(427 470 442 455)(428 469 443 454)(429 468 444 453)(430 467 445 452)(431 466 446 451)(432 465 447 480)(433 464 448 479)(434 463 449 478)(435 462 450 477)
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), (1,465,397,57,215,272,109,83)(2,458,416,44,216,295,98,70)(3,451,405,31,217,288,117,87)(4,474,394,48,218,281,106,74)(5,467,413,35,219,274,95,61)(6,460,402,52,220,297,114,78)(7,453,391,39,221,290,103,65)(8,476,410,56,222,283,92,82)(9,469,399,43,223,276,111,69)(10,462,418,60,224,299,100,86)(11,455,407,47,225,292,119,73)(12,478,396,34,226,285,108,90)(13,471,415,51,227,278,97,77)(14,464,404,38,228,271,116,64)(15,457,393,55,229,294,105,81)(16,480,412,42,230,287,94,68)(17,473,401,59,231,280,113,85)(18,466,420,46,232,273,102,72)(19,459,409,33,233,296,91,89)(20,452,398,50,234,289,110,76)(21,475,417,37,235,282,99,63)(22,468,406,54,236,275,118,80)(23,461,395,41,237,298,107,67)(24,454,414,58,238,291,96,84)(25,477,403,45,239,284,115,71)(26,470,392,32,240,277,104,88)(27,463,411,49,211,300,93,75)(28,456,400,36,212,293,112,62)(29,479,419,53,213,286,101,79)(30,472,408,40,214,279,120,66)(121,172,346,440,312,384,187,268)(122,165,335,427,313,377,206,255)(123,158,354,444,314,370,195,242)(124,151,343,431,315,363,184,259)(125,174,332,448,316,386,203,246)(126,167,351,435,317,379,192,263)(127,160,340,422,318,372,181,250)(128,153,359,439,319,365,200,267)(129,176,348,426,320,388,189,254)(130,169,337,443,321,381,208,241)(131,162,356,430,322,374,197,258)(132,155,345,447,323,367,186,245)(133,178,334,434,324,390,205,262)(134,171,353,421,325,383,194,249)(135,164,342,438,326,376,183,266)(136,157,331,425,327,369,202,253)(137,180,350,442,328,362,191,270)(138,173,339,429,329,385,210,257)(139,166,358,446,330,378,199,244)(140,159,347,433,301,371,188,261)(141,152,336,450,302,364,207,248)(142,175,355,437,303,387,196,265)(143,168,344,424,304,380,185,252)(144,161,333,441,305,373,204,269)(145,154,352,428,306,366,193,256)(146,177,341,445,307,389,182,243)(147,170,360,432,308,382,201,260)(148,163,349,449,309,375,190,247)(149,156,338,436,310,368,209,264)(150,179,357,423,311,361,198,251), (1,201,16,186)(2,200,17,185)(3,199,18,184)(4,198,19,183)(5,197,20,182)(6,196,21,181)(7,195,22,210)(8,194,23,209)(9,193,24,208)(10,192,25,207)(11,191,26,206)(12,190,27,205)(13,189,28,204)(14,188,29,203)(15,187,30,202)(31,363,46,378)(32,362,47,377)(33,361,48,376)(34,390,49,375)(35,389,50,374)(36,388,51,373)(37,387,52,372)(38,386,53,371)(39,385,54,370)(40,384,55,369)(41,383,56,368)(42,382,57,367)(43,381,58,366)(44,380,59,365)(45,379,60,364)(61,177,76,162)(62,176,77,161)(63,175,78,160)(64,174,79,159)(65,173,80,158)(66,172,81,157)(67,171,82,156)(68,170,83,155)(69,169,84,154)(70,168,85,153)(71,167,86,152)(72,166,87,151)(73,165,88,180)(74,164,89,179)(75,163,90,178)(91,326,106,311)(92,325,107,310)(93,324,108,309)(94,323,109,308)(95,322,110,307)(96,321,111,306)(97,320,112,305)(98,319,113,304)(99,318,114,303)(100,317,115,302)(101,316,116,301)(102,315,117,330)(103,314,118,329)(104,313,119,328)(105,312,120,327)(121,408,136,393)(122,407,137,392)(123,406,138,391)(124,405,139,420)(125,404,140,419)(126,403,141,418)(127,402,142,417)(128,401,143,416)(129,400,144,415)(130,399,145,414)(131,398,146,413)(132,397,147,412)(133,396,148,411)(134,395,149,410)(135,394,150,409)(211,334,226,349)(212,333,227,348)(213,332,228,347)(214,331,229,346)(215,360,230,345)(216,359,231,344)(217,358,232,343)(218,357,233,342)(219,356,234,341)(220,355,235,340)(221,354,236,339)(222,353,237,338)(223,352,238,337)(224,351,239,336)(225,350,240,335)(241,291,256,276)(242,290,257,275)(243,289,258,274)(244,288,259,273)(245,287,260,272)(246,286,261,271)(247,285,262,300)(248,284,263,299)(249,283,264,298)(250,282,265,297)(251,281,266,296)(252,280,267,295)(253,279,268,294)(254,278,269,293)(255,277,270,292)(421,476,436,461)(422,475,437,460)(423,474,438,459)(424,473,439,458)(425,472,440,457)(426,471,441,456)(427,470,442,455)(428,469,443,454)(429,468,444,453)(430,467,445,452)(431,466,446,451)(432,465,447,480)(433,464,448,479)(434,463,449,478)(435,462,450,477)>;
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), (1,465,397,57,215,272,109,83)(2,458,416,44,216,295,98,70)(3,451,405,31,217,288,117,87)(4,474,394,48,218,281,106,74)(5,467,413,35,219,274,95,61)(6,460,402,52,220,297,114,78)(7,453,391,39,221,290,103,65)(8,476,410,56,222,283,92,82)(9,469,399,43,223,276,111,69)(10,462,418,60,224,299,100,86)(11,455,407,47,225,292,119,73)(12,478,396,34,226,285,108,90)(13,471,415,51,227,278,97,77)(14,464,404,38,228,271,116,64)(15,457,393,55,229,294,105,81)(16,480,412,42,230,287,94,68)(17,473,401,59,231,280,113,85)(18,466,420,46,232,273,102,72)(19,459,409,33,233,296,91,89)(20,452,398,50,234,289,110,76)(21,475,417,37,235,282,99,63)(22,468,406,54,236,275,118,80)(23,461,395,41,237,298,107,67)(24,454,414,58,238,291,96,84)(25,477,403,45,239,284,115,71)(26,470,392,32,240,277,104,88)(27,463,411,49,211,300,93,75)(28,456,400,36,212,293,112,62)(29,479,419,53,213,286,101,79)(30,472,408,40,214,279,120,66)(121,172,346,440,312,384,187,268)(122,165,335,427,313,377,206,255)(123,158,354,444,314,370,195,242)(124,151,343,431,315,363,184,259)(125,174,332,448,316,386,203,246)(126,167,351,435,317,379,192,263)(127,160,340,422,318,372,181,250)(128,153,359,439,319,365,200,267)(129,176,348,426,320,388,189,254)(130,169,337,443,321,381,208,241)(131,162,356,430,322,374,197,258)(132,155,345,447,323,367,186,245)(133,178,334,434,324,390,205,262)(134,171,353,421,325,383,194,249)(135,164,342,438,326,376,183,266)(136,157,331,425,327,369,202,253)(137,180,350,442,328,362,191,270)(138,173,339,429,329,385,210,257)(139,166,358,446,330,378,199,244)(140,159,347,433,301,371,188,261)(141,152,336,450,302,364,207,248)(142,175,355,437,303,387,196,265)(143,168,344,424,304,380,185,252)(144,161,333,441,305,373,204,269)(145,154,352,428,306,366,193,256)(146,177,341,445,307,389,182,243)(147,170,360,432,308,382,201,260)(148,163,349,449,309,375,190,247)(149,156,338,436,310,368,209,264)(150,179,357,423,311,361,198,251), (1,201,16,186)(2,200,17,185)(3,199,18,184)(4,198,19,183)(5,197,20,182)(6,196,21,181)(7,195,22,210)(8,194,23,209)(9,193,24,208)(10,192,25,207)(11,191,26,206)(12,190,27,205)(13,189,28,204)(14,188,29,203)(15,187,30,202)(31,363,46,378)(32,362,47,377)(33,361,48,376)(34,390,49,375)(35,389,50,374)(36,388,51,373)(37,387,52,372)(38,386,53,371)(39,385,54,370)(40,384,55,369)(41,383,56,368)(42,382,57,367)(43,381,58,366)(44,380,59,365)(45,379,60,364)(61,177,76,162)(62,176,77,161)(63,175,78,160)(64,174,79,159)(65,173,80,158)(66,172,81,157)(67,171,82,156)(68,170,83,155)(69,169,84,154)(70,168,85,153)(71,167,86,152)(72,166,87,151)(73,165,88,180)(74,164,89,179)(75,163,90,178)(91,326,106,311)(92,325,107,310)(93,324,108,309)(94,323,109,308)(95,322,110,307)(96,321,111,306)(97,320,112,305)(98,319,113,304)(99,318,114,303)(100,317,115,302)(101,316,116,301)(102,315,117,330)(103,314,118,329)(104,313,119,328)(105,312,120,327)(121,408,136,393)(122,407,137,392)(123,406,138,391)(124,405,139,420)(125,404,140,419)(126,403,141,418)(127,402,142,417)(128,401,143,416)(129,400,144,415)(130,399,145,414)(131,398,146,413)(132,397,147,412)(133,396,148,411)(134,395,149,410)(135,394,150,409)(211,334,226,349)(212,333,227,348)(213,332,228,347)(214,331,229,346)(215,360,230,345)(216,359,231,344)(217,358,232,343)(218,357,233,342)(219,356,234,341)(220,355,235,340)(221,354,236,339)(222,353,237,338)(223,352,238,337)(224,351,239,336)(225,350,240,335)(241,291,256,276)(242,290,257,275)(243,289,258,274)(244,288,259,273)(245,287,260,272)(246,286,261,271)(247,285,262,300)(248,284,263,299)(249,283,264,298)(250,282,265,297)(251,281,266,296)(252,280,267,295)(253,279,268,294)(254,278,269,293)(255,277,270,292)(421,476,436,461)(422,475,437,460)(423,474,438,459)(424,473,439,458)(425,472,440,457)(426,471,441,456)(427,470,442,455)(428,469,443,454)(429,468,444,453)(430,467,445,452)(431,466,446,451)(432,465,447,480)(433,464,448,479)(434,463,449,478)(435,462,450,477) );
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)], [(1,465,397,57,215,272,109,83),(2,458,416,44,216,295,98,70),(3,451,405,31,217,288,117,87),(4,474,394,48,218,281,106,74),(5,467,413,35,219,274,95,61),(6,460,402,52,220,297,114,78),(7,453,391,39,221,290,103,65),(8,476,410,56,222,283,92,82),(9,469,399,43,223,276,111,69),(10,462,418,60,224,299,100,86),(11,455,407,47,225,292,119,73),(12,478,396,34,226,285,108,90),(13,471,415,51,227,278,97,77),(14,464,404,38,228,271,116,64),(15,457,393,55,229,294,105,81),(16,480,412,42,230,287,94,68),(17,473,401,59,231,280,113,85),(18,466,420,46,232,273,102,72),(19,459,409,33,233,296,91,89),(20,452,398,50,234,289,110,76),(21,475,417,37,235,282,99,63),(22,468,406,54,236,275,118,80),(23,461,395,41,237,298,107,67),(24,454,414,58,238,291,96,84),(25,477,403,45,239,284,115,71),(26,470,392,32,240,277,104,88),(27,463,411,49,211,300,93,75),(28,456,400,36,212,293,112,62),(29,479,419,53,213,286,101,79),(30,472,408,40,214,279,120,66),(121,172,346,440,312,384,187,268),(122,165,335,427,313,377,206,255),(123,158,354,444,314,370,195,242),(124,151,343,431,315,363,184,259),(125,174,332,448,316,386,203,246),(126,167,351,435,317,379,192,263),(127,160,340,422,318,372,181,250),(128,153,359,439,319,365,200,267),(129,176,348,426,320,388,189,254),(130,169,337,443,321,381,208,241),(131,162,356,430,322,374,197,258),(132,155,345,447,323,367,186,245),(133,178,334,434,324,390,205,262),(134,171,353,421,325,383,194,249),(135,164,342,438,326,376,183,266),(136,157,331,425,327,369,202,253),(137,180,350,442,328,362,191,270),(138,173,339,429,329,385,210,257),(139,166,358,446,330,378,199,244),(140,159,347,433,301,371,188,261),(141,152,336,450,302,364,207,248),(142,175,355,437,303,387,196,265),(143,168,344,424,304,380,185,252),(144,161,333,441,305,373,204,269),(145,154,352,428,306,366,193,256),(146,177,341,445,307,389,182,243),(147,170,360,432,308,382,201,260),(148,163,349,449,309,375,190,247),(149,156,338,436,310,368,209,264),(150,179,357,423,311,361,198,251)], [(1,201,16,186),(2,200,17,185),(3,199,18,184),(4,198,19,183),(5,197,20,182),(6,196,21,181),(7,195,22,210),(8,194,23,209),(9,193,24,208),(10,192,25,207),(11,191,26,206),(12,190,27,205),(13,189,28,204),(14,188,29,203),(15,187,30,202),(31,363,46,378),(32,362,47,377),(33,361,48,376),(34,390,49,375),(35,389,50,374),(36,388,51,373),(37,387,52,372),(38,386,53,371),(39,385,54,370),(40,384,55,369),(41,383,56,368),(42,382,57,367),(43,381,58,366),(44,380,59,365),(45,379,60,364),(61,177,76,162),(62,176,77,161),(63,175,78,160),(64,174,79,159),(65,173,80,158),(66,172,81,157),(67,171,82,156),(68,170,83,155),(69,169,84,154),(70,168,85,153),(71,167,86,152),(72,166,87,151),(73,165,88,180),(74,164,89,179),(75,163,90,178),(91,326,106,311),(92,325,107,310),(93,324,108,309),(94,323,109,308),(95,322,110,307),(96,321,111,306),(97,320,112,305),(98,319,113,304),(99,318,114,303),(100,317,115,302),(101,316,116,301),(102,315,117,330),(103,314,118,329),(104,313,119,328),(105,312,120,327),(121,408,136,393),(122,407,137,392),(123,406,138,391),(124,405,139,420),(125,404,140,419),(126,403,141,418),(127,402,142,417),(128,401,143,416),(129,400,144,415),(130,399,145,414),(131,398,146,413),(132,397,147,412),(133,396,148,411),(134,395,149,410),(135,394,150,409),(211,334,226,349),(212,333,227,348),(213,332,228,347),(214,331,229,346),(215,360,230,345),(216,359,231,344),(217,358,232,343),(218,357,233,342),(219,356,234,341),(220,355,235,340),(221,354,236,339),(222,353,237,338),(223,352,238,337),(224,351,239,336),(225,350,240,335),(241,291,256,276),(242,290,257,275),(243,289,258,274),(244,288,259,273),(245,287,260,272),(246,286,261,271),(247,285,262,300),(248,284,263,299),(249,283,264,298),(250,282,265,297),(251,281,266,296),(252,280,267,295),(253,279,268,294),(254,278,269,293),(255,277,270,292),(421,476,436,461),(422,475,437,460),(423,474,438,459),(424,473,439,458),(425,472,440,457),(426,471,441,456),(427,470,442,455),(428,469,443,454),(429,468,444,453),(430,467,445,452),(431,466,446,451),(432,465,447,480),(433,464,448,479),(434,463,449,478),(435,462,450,477)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | Q8 | D6 | M4(2) | Dic6 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | F5 | C5⋊C8 | C2×F5 | C4.F5 | C4⋊F5 | S3×F5 | Dic3⋊F5 | S3×C5⋊C8 | Dic3.F5 |
| kernel | C30.4M4(2) | Dic3×Dic5 | C6×C5⋊C8 | C2×C15⋊C8 | C10×Dic3 | C2×Dic15 | C5×Dic3 | C2×C5⋊C8 | C3×Dic5 | C3×Dic5 | C2×Dic5 | C30 | Dic5 | Dic5 | C2×C10 | C10 | C10 | C2×Dic3 | Dic3 | C2×C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 |
Matrix representation of C30.4M4(2) ►in GL8(𝔽241)
| 240 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 73 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 186 | 168 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 169 | 0 | 0 | 0 | 0 |
| 0 | 0 | 169 | 141 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 178 | 58 | 61 | 152 |
| 0 | 0 | 0 | 0 | 89 | 91 | 239 | 210 |
| 0 | 0 | 0 | 0 | 147 | 152 | 150 | 2 |
| 0 | 0 | 0 | 0 | 180 | 89 | 208 | 63 |
| 168 | 223 | 0 | 0 | 0 | 0 | 0 | 0 |
| 55 | 73 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 150 | 83 | 175 | 11 |
| 0 | 0 | 0 | 0 | 233 | 17 | 186 | 74 |
| 0 | 0 | 0 | 0 | 167 | 28 | 8 | 224 |
| 0 | 0 | 0 | 0 | 178 | 91 | 158 | 66 |
G:=sub<GL(8,GF(241))| [240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0],[73,186,0,0,0,0,0,0,18,168,0,0,0,0,0,0,0,0,100,169,0,0,0,0,0,0,169,141,0,0,0,0,0,0,0,0,178,89,147,180,0,0,0,0,58,91,152,89,0,0,0,0,61,239,150,208,0,0,0,0,152,210,2,63],[168,55,0,0,0,0,0,0,223,73,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,150,233,167,178,0,0,0,0,83,17,28,91,0,0,0,0,175,186,8,158,0,0,0,0,11,74,224,66] >;
C30.4M4(2) in GAP, Magma, Sage, TeX
C_{30}._4M_4(2) % in TeX
G:=Group("C30.4M4(2)"); // GroupNames label
G:=SmallGroup(480,252);
// by ID
G=gap.SmallGroup(480,252);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,1356,9414,4724]);
// 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=a^15*b^5>;
// generators/relations