direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C30, C60.200D4, C60.294C23, C120.106C22, (C2×C8).4C30, C8.5(C2×C30), C4.8(D4×C15), (C2×C40).14C6, C40.27(C2×C6), C10.76(C6×D4), C20.43(C3×D4), C2.13(D4×C30), C12.43(C5×D4), C6.76(D4×C10), (C2×Q8).6C30, Q8.4(C2×C30), (C6×Q8).9C10, (C2×C24).10C10, (C2×C120).30C2, C24.22(C2×C10), C30.459(C2×D4), (C2×C30).197D4, C4.3(C22×C30), (Q8×C30).19C2, (Q8×C10).13C6, C20.46(C22×C6), C22.16(D4×C15), (C2×C60).584C22, C12.46(C22×C10), (Q8×C15).55C22, (C2×C6).54(C5×D4), (C2×C4).28(C2×C30), (C2×C10).54(C3×D4), (C5×Q8).20(C2×C6), (C2×C20).130(C2×C6), (C3×Q8).12(C2×C10), (C2×C12).131(C2×C10), SmallGroup(480,939)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C30
G = < a,b,c | a30=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 152 in 120 conjugacy classes, 88 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C10, C10, C12, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, C20, C20, C2×C10, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C30, C30, C2×Q16, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×C24, C3×Q16, C6×Q8, C60, C60, C2×C30, C2×C40, C5×Q16, Q8×C10, C6×Q16, C120, C2×C60, C2×C60, Q8×C15, Q8×C15, C10×Q16, C2×C120, C15×Q16, Q8×C30, Q16×C30
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, Q16, C2×D4, C2×C10, C3×D4, C22×C6, C30, C2×Q16, C5×D4, C22×C10, C3×Q16, C6×D4, C2×C30, C5×Q16, D4×C10, C6×Q16, D4×C15, C22×C30, C10×Q16, C15×Q16, D4×C30, Q16×C30
►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 81 428 158 93 121 230 193)(2 82 429 159 94 122 231 194)(3 83 430 160 95 123 232 195)(4 84 431 161 96 124 233 196)(5 85 432 162 97 125 234 197)(6 86 433 163 98 126 235 198)(7 87 434 164 99 127 236 199)(8 88 435 165 100 128 237 200)(9 89 436 166 101 129 238 201)(10 90 437 167 102 130 239 202)(11 61 438 168 103 131 240 203)(12 62 439 169 104 132 211 204)(13 63 440 170 105 133 212 205)(14 64 441 171 106 134 213 206)(15 65 442 172 107 135 214 207)(16 66 443 173 108 136 215 208)(17 67 444 174 109 137 216 209)(18 68 445 175 110 138 217 210)(19 69 446 176 111 139 218 181)(20 70 447 177 112 140 219 182)(21 71 448 178 113 141 220 183)(22 72 449 179 114 142 221 184)(23 73 450 180 115 143 222 185)(24 74 421 151 116 144 223 186)(25 75 422 152 117 145 224 187)(26 76 423 153 118 146 225 188)(27 77 424 154 119 147 226 189)(28 78 425 155 120 148 227 190)(29 79 426 156 91 149 228 191)(30 80 427 157 92 150 229 192)(31 272 269 312 334 405 374 458)(32 273 270 313 335 406 375 459)(33 274 241 314 336 407 376 460)(34 275 242 315 337 408 377 461)(35 276 243 316 338 409 378 462)(36 277 244 317 339 410 379 463)(37 278 245 318 340 411 380 464)(38 279 246 319 341 412 381 465)(39 280 247 320 342 413 382 466)(40 281 248 321 343 414 383 467)(41 282 249 322 344 415 384 468)(42 283 250 323 345 416 385 469)(43 284 251 324 346 417 386 470)(44 285 252 325 347 418 387 471)(45 286 253 326 348 419 388 472)(46 287 254 327 349 420 389 473)(47 288 255 328 350 391 390 474)(48 289 256 329 351 392 361 475)(49 290 257 330 352 393 362 476)(50 291 258 301 353 394 363 477)(51 292 259 302 354 395 364 478)(52 293 260 303 355 396 365 479)(53 294 261 304 356 397 366 480)(54 295 262 305 357 398 367 451)(55 296 263 306 358 399 368 452)(56 297 264 307 359 400 369 453)(57 298 265 308 360 401 370 454)(58 299 266 309 331 402 371 455)(59 300 267 310 332 403 372 456)(60 271 268 311 333 404 373 457)
(1 333 93 60)(2 334 94 31)(3 335 95 32)(4 336 96 33)(5 337 97 34)(6 338 98 35)(7 339 99 36)(8 340 100 37)(9 341 101 38)(10 342 102 39)(11 343 103 40)(12 344 104 41)(13 345 105 42)(14 346 106 43)(15 347 107 44)(16 348 108 45)(17 349 109 46)(18 350 110 47)(19 351 111 48)(20 352 112 49)(21 353 113 50)(22 354 114 51)(23 355 115 52)(24 356 116 53)(25 357 117 54)(26 358 118 55)(27 359 119 56)(28 360 120 57)(29 331 91 58)(30 332 92 59)(61 321 131 467)(62 322 132 468)(63 323 133 469)(64 324 134 470)(65 325 135 471)(66 326 136 472)(67 327 137 473)(68 328 138 474)(69 329 139 475)(70 330 140 476)(71 301 141 477)(72 302 142 478)(73 303 143 479)(74 304 144 480)(75 305 145 451)(76 306 146 452)(77 307 147 453)(78 308 148 454)(79 309 149 455)(80 310 150 456)(81 311 121 457)(82 312 122 458)(83 313 123 459)(84 314 124 460)(85 315 125 461)(86 316 126 462)(87 317 127 463)(88 318 128 464)(89 319 129 465)(90 320 130 466)(151 294 186 397)(152 295 187 398)(153 296 188 399)(154 297 189 400)(155 298 190 401)(156 299 191 402)(157 300 192 403)(158 271 193 404)(159 272 194 405)(160 273 195 406)(161 274 196 407)(162 275 197 408)(163 276 198 409)(164 277 199 410)(165 278 200 411)(166 279 201 412)(167 280 202 413)(168 281 203 414)(169 282 204 415)(170 283 205 416)(171 284 206 417)(172 285 207 418)(173 286 208 419)(174 287 209 420)(175 288 210 391)(176 289 181 392)(177 290 182 393)(178 291 183 394)(179 292 184 395)(180 293 185 396)(211 384 439 249)(212 385 440 250)(213 386 441 251)(214 387 442 252)(215 388 443 253)(216 389 444 254)(217 390 445 255)(218 361 446 256)(219 362 447 257)(220 363 448 258)(221 364 449 259)(222 365 450 260)(223 366 421 261)(224 367 422 262)(225 368 423 263)(226 369 424 264)(227 370 425 265)(228 371 426 266)(229 372 427 267)(230 373 428 268)(231 374 429 269)(232 375 430 270)(233 376 431 241)(234 377 432 242)(235 378 433 243)(236 379 434 244)(237 380 435 245)(238 381 436 246)(239 382 437 247)(240 383 438 248)
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), (1,81,428,158,93,121,230,193)(2,82,429,159,94,122,231,194)(3,83,430,160,95,123,232,195)(4,84,431,161,96,124,233,196)(5,85,432,162,97,125,234,197)(6,86,433,163,98,126,235,198)(7,87,434,164,99,127,236,199)(8,88,435,165,100,128,237,200)(9,89,436,166,101,129,238,201)(10,90,437,167,102,130,239,202)(11,61,438,168,103,131,240,203)(12,62,439,169,104,132,211,204)(13,63,440,170,105,133,212,205)(14,64,441,171,106,134,213,206)(15,65,442,172,107,135,214,207)(16,66,443,173,108,136,215,208)(17,67,444,174,109,137,216,209)(18,68,445,175,110,138,217,210)(19,69,446,176,111,139,218,181)(20,70,447,177,112,140,219,182)(21,71,448,178,113,141,220,183)(22,72,449,179,114,142,221,184)(23,73,450,180,115,143,222,185)(24,74,421,151,116,144,223,186)(25,75,422,152,117,145,224,187)(26,76,423,153,118,146,225,188)(27,77,424,154,119,147,226,189)(28,78,425,155,120,148,227,190)(29,79,426,156,91,149,228,191)(30,80,427,157,92,150,229,192)(31,272,269,312,334,405,374,458)(32,273,270,313,335,406,375,459)(33,274,241,314,336,407,376,460)(34,275,242,315,337,408,377,461)(35,276,243,316,338,409,378,462)(36,277,244,317,339,410,379,463)(37,278,245,318,340,411,380,464)(38,279,246,319,341,412,381,465)(39,280,247,320,342,413,382,466)(40,281,248,321,343,414,383,467)(41,282,249,322,344,415,384,468)(42,283,250,323,345,416,385,469)(43,284,251,324,346,417,386,470)(44,285,252,325,347,418,387,471)(45,286,253,326,348,419,388,472)(46,287,254,327,349,420,389,473)(47,288,255,328,350,391,390,474)(48,289,256,329,351,392,361,475)(49,290,257,330,352,393,362,476)(50,291,258,301,353,394,363,477)(51,292,259,302,354,395,364,478)(52,293,260,303,355,396,365,479)(53,294,261,304,356,397,366,480)(54,295,262,305,357,398,367,451)(55,296,263,306,358,399,368,452)(56,297,264,307,359,400,369,453)(57,298,265,308,360,401,370,454)(58,299,266,309,331,402,371,455)(59,300,267,310,332,403,372,456)(60,271,268,311,333,404,373,457), (1,333,93,60)(2,334,94,31)(3,335,95,32)(4,336,96,33)(5,337,97,34)(6,338,98,35)(7,339,99,36)(8,340,100,37)(9,341,101,38)(10,342,102,39)(11,343,103,40)(12,344,104,41)(13,345,105,42)(14,346,106,43)(15,347,107,44)(16,348,108,45)(17,349,109,46)(18,350,110,47)(19,351,111,48)(20,352,112,49)(21,353,113,50)(22,354,114,51)(23,355,115,52)(24,356,116,53)(25,357,117,54)(26,358,118,55)(27,359,119,56)(28,360,120,57)(29,331,91,58)(30,332,92,59)(61,321,131,467)(62,322,132,468)(63,323,133,469)(64,324,134,470)(65,325,135,471)(66,326,136,472)(67,327,137,473)(68,328,138,474)(69,329,139,475)(70,330,140,476)(71,301,141,477)(72,302,142,478)(73,303,143,479)(74,304,144,480)(75,305,145,451)(76,306,146,452)(77,307,147,453)(78,308,148,454)(79,309,149,455)(80,310,150,456)(81,311,121,457)(82,312,122,458)(83,313,123,459)(84,314,124,460)(85,315,125,461)(86,316,126,462)(87,317,127,463)(88,318,128,464)(89,319,129,465)(90,320,130,466)(151,294,186,397)(152,295,187,398)(153,296,188,399)(154,297,189,400)(155,298,190,401)(156,299,191,402)(157,300,192,403)(158,271,193,404)(159,272,194,405)(160,273,195,406)(161,274,196,407)(162,275,197,408)(163,276,198,409)(164,277,199,410)(165,278,200,411)(166,279,201,412)(167,280,202,413)(168,281,203,414)(169,282,204,415)(170,283,205,416)(171,284,206,417)(172,285,207,418)(173,286,208,419)(174,287,209,420)(175,288,210,391)(176,289,181,392)(177,290,182,393)(178,291,183,394)(179,292,184,395)(180,293,185,396)(211,384,439,249)(212,385,440,250)(213,386,441,251)(214,387,442,252)(215,388,443,253)(216,389,444,254)(217,390,445,255)(218,361,446,256)(219,362,447,257)(220,363,448,258)(221,364,449,259)(222,365,450,260)(223,366,421,261)(224,367,422,262)(225,368,423,263)(226,369,424,264)(227,370,425,265)(228,371,426,266)(229,372,427,267)(230,373,428,268)(231,374,429,269)(232,375,430,270)(233,376,431,241)(234,377,432,242)(235,378,433,243)(236,379,434,244)(237,380,435,245)(238,381,436,246)(239,382,437,247)(240,383,438,248)>;
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), (1,81,428,158,93,121,230,193)(2,82,429,159,94,122,231,194)(3,83,430,160,95,123,232,195)(4,84,431,161,96,124,233,196)(5,85,432,162,97,125,234,197)(6,86,433,163,98,126,235,198)(7,87,434,164,99,127,236,199)(8,88,435,165,100,128,237,200)(9,89,436,166,101,129,238,201)(10,90,437,167,102,130,239,202)(11,61,438,168,103,131,240,203)(12,62,439,169,104,132,211,204)(13,63,440,170,105,133,212,205)(14,64,441,171,106,134,213,206)(15,65,442,172,107,135,214,207)(16,66,443,173,108,136,215,208)(17,67,444,174,109,137,216,209)(18,68,445,175,110,138,217,210)(19,69,446,176,111,139,218,181)(20,70,447,177,112,140,219,182)(21,71,448,178,113,141,220,183)(22,72,449,179,114,142,221,184)(23,73,450,180,115,143,222,185)(24,74,421,151,116,144,223,186)(25,75,422,152,117,145,224,187)(26,76,423,153,118,146,225,188)(27,77,424,154,119,147,226,189)(28,78,425,155,120,148,227,190)(29,79,426,156,91,149,228,191)(30,80,427,157,92,150,229,192)(31,272,269,312,334,405,374,458)(32,273,270,313,335,406,375,459)(33,274,241,314,336,407,376,460)(34,275,242,315,337,408,377,461)(35,276,243,316,338,409,378,462)(36,277,244,317,339,410,379,463)(37,278,245,318,340,411,380,464)(38,279,246,319,341,412,381,465)(39,280,247,320,342,413,382,466)(40,281,248,321,343,414,383,467)(41,282,249,322,344,415,384,468)(42,283,250,323,345,416,385,469)(43,284,251,324,346,417,386,470)(44,285,252,325,347,418,387,471)(45,286,253,326,348,419,388,472)(46,287,254,327,349,420,389,473)(47,288,255,328,350,391,390,474)(48,289,256,329,351,392,361,475)(49,290,257,330,352,393,362,476)(50,291,258,301,353,394,363,477)(51,292,259,302,354,395,364,478)(52,293,260,303,355,396,365,479)(53,294,261,304,356,397,366,480)(54,295,262,305,357,398,367,451)(55,296,263,306,358,399,368,452)(56,297,264,307,359,400,369,453)(57,298,265,308,360,401,370,454)(58,299,266,309,331,402,371,455)(59,300,267,310,332,403,372,456)(60,271,268,311,333,404,373,457), (1,333,93,60)(2,334,94,31)(3,335,95,32)(4,336,96,33)(5,337,97,34)(6,338,98,35)(7,339,99,36)(8,340,100,37)(9,341,101,38)(10,342,102,39)(11,343,103,40)(12,344,104,41)(13,345,105,42)(14,346,106,43)(15,347,107,44)(16,348,108,45)(17,349,109,46)(18,350,110,47)(19,351,111,48)(20,352,112,49)(21,353,113,50)(22,354,114,51)(23,355,115,52)(24,356,116,53)(25,357,117,54)(26,358,118,55)(27,359,119,56)(28,360,120,57)(29,331,91,58)(30,332,92,59)(61,321,131,467)(62,322,132,468)(63,323,133,469)(64,324,134,470)(65,325,135,471)(66,326,136,472)(67,327,137,473)(68,328,138,474)(69,329,139,475)(70,330,140,476)(71,301,141,477)(72,302,142,478)(73,303,143,479)(74,304,144,480)(75,305,145,451)(76,306,146,452)(77,307,147,453)(78,308,148,454)(79,309,149,455)(80,310,150,456)(81,311,121,457)(82,312,122,458)(83,313,123,459)(84,314,124,460)(85,315,125,461)(86,316,126,462)(87,317,127,463)(88,318,128,464)(89,319,129,465)(90,320,130,466)(151,294,186,397)(152,295,187,398)(153,296,188,399)(154,297,189,400)(155,298,190,401)(156,299,191,402)(157,300,192,403)(158,271,193,404)(159,272,194,405)(160,273,195,406)(161,274,196,407)(162,275,197,408)(163,276,198,409)(164,277,199,410)(165,278,200,411)(166,279,201,412)(167,280,202,413)(168,281,203,414)(169,282,204,415)(170,283,205,416)(171,284,206,417)(172,285,207,418)(173,286,208,419)(174,287,209,420)(175,288,210,391)(176,289,181,392)(177,290,182,393)(178,291,183,394)(179,292,184,395)(180,293,185,396)(211,384,439,249)(212,385,440,250)(213,386,441,251)(214,387,442,252)(215,388,443,253)(216,389,444,254)(217,390,445,255)(218,361,446,256)(219,362,447,257)(220,363,448,258)(221,364,449,259)(222,365,450,260)(223,366,421,261)(224,367,422,262)(225,368,423,263)(226,369,424,264)(227,370,425,265)(228,371,426,266)(229,372,427,267)(230,373,428,268)(231,374,429,269)(232,375,430,270)(233,376,431,241)(234,377,432,242)(235,378,433,243)(236,379,434,244)(237,380,435,245)(238,381,436,246)(239,382,437,247)(240,383,438,248) );
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)], [(1,81,428,158,93,121,230,193),(2,82,429,159,94,122,231,194),(3,83,430,160,95,123,232,195),(4,84,431,161,96,124,233,196),(5,85,432,162,97,125,234,197),(6,86,433,163,98,126,235,198),(7,87,434,164,99,127,236,199),(8,88,435,165,100,128,237,200),(9,89,436,166,101,129,238,201),(10,90,437,167,102,130,239,202),(11,61,438,168,103,131,240,203),(12,62,439,169,104,132,211,204),(13,63,440,170,105,133,212,205),(14,64,441,171,106,134,213,206),(15,65,442,172,107,135,214,207),(16,66,443,173,108,136,215,208),(17,67,444,174,109,137,216,209),(18,68,445,175,110,138,217,210),(19,69,446,176,111,139,218,181),(20,70,447,177,112,140,219,182),(21,71,448,178,113,141,220,183),(22,72,449,179,114,142,221,184),(23,73,450,180,115,143,222,185),(24,74,421,151,116,144,223,186),(25,75,422,152,117,145,224,187),(26,76,423,153,118,146,225,188),(27,77,424,154,119,147,226,189),(28,78,425,155,120,148,227,190),(29,79,426,156,91,149,228,191),(30,80,427,157,92,150,229,192),(31,272,269,312,334,405,374,458),(32,273,270,313,335,406,375,459),(33,274,241,314,336,407,376,460),(34,275,242,315,337,408,377,461),(35,276,243,316,338,409,378,462),(36,277,244,317,339,410,379,463),(37,278,245,318,340,411,380,464),(38,279,246,319,341,412,381,465),(39,280,247,320,342,413,382,466),(40,281,248,321,343,414,383,467),(41,282,249,322,344,415,384,468),(42,283,250,323,345,416,385,469),(43,284,251,324,346,417,386,470),(44,285,252,325,347,418,387,471),(45,286,253,326,348,419,388,472),(46,287,254,327,349,420,389,473),(47,288,255,328,350,391,390,474),(48,289,256,329,351,392,361,475),(49,290,257,330,352,393,362,476),(50,291,258,301,353,394,363,477),(51,292,259,302,354,395,364,478),(52,293,260,303,355,396,365,479),(53,294,261,304,356,397,366,480),(54,295,262,305,357,398,367,451),(55,296,263,306,358,399,368,452),(56,297,264,307,359,400,369,453),(57,298,265,308,360,401,370,454),(58,299,266,309,331,402,371,455),(59,300,267,310,332,403,372,456),(60,271,268,311,333,404,373,457)], [(1,333,93,60),(2,334,94,31),(3,335,95,32),(4,336,96,33),(5,337,97,34),(6,338,98,35),(7,339,99,36),(8,340,100,37),(9,341,101,38),(10,342,102,39),(11,343,103,40),(12,344,104,41),(13,345,105,42),(14,346,106,43),(15,347,107,44),(16,348,108,45),(17,349,109,46),(18,350,110,47),(19,351,111,48),(20,352,112,49),(21,353,113,50),(22,354,114,51),(23,355,115,52),(24,356,116,53),(25,357,117,54),(26,358,118,55),(27,359,119,56),(28,360,120,57),(29,331,91,58),(30,332,92,59),(61,321,131,467),(62,322,132,468),(63,323,133,469),(64,324,134,470),(65,325,135,471),(66,326,136,472),(67,327,137,473),(68,328,138,474),(69,329,139,475),(70,330,140,476),(71,301,141,477),(72,302,142,478),(73,303,143,479),(74,304,144,480),(75,305,145,451),(76,306,146,452),(77,307,147,453),(78,308,148,454),(79,309,149,455),(80,310,150,456),(81,311,121,457),(82,312,122,458),(83,313,123,459),(84,314,124,460),(85,315,125,461),(86,316,126,462),(87,317,127,463),(88,318,128,464),(89,319,129,465),(90,320,130,466),(151,294,186,397),(152,295,187,398),(153,296,188,399),(154,297,189,400),(155,298,190,401),(156,299,191,402),(157,300,192,403),(158,271,193,404),(159,272,194,405),(160,273,195,406),(161,274,196,407),(162,275,197,408),(163,276,198,409),(164,277,199,410),(165,278,200,411),(166,279,201,412),(167,280,202,413),(168,281,203,414),(169,282,204,415),(170,283,205,416),(171,284,206,417),(172,285,207,418),(173,286,208,419),(174,287,209,420),(175,288,210,391),(176,289,181,392),(177,290,182,393),(178,291,183,394),(179,292,184,395),(180,293,185,396),(211,384,439,249),(212,385,440,250),(213,386,441,251),(214,387,442,252),(215,388,443,253),(216,389,444,254),(217,390,445,255),(218,361,446,256),(219,362,447,257),(220,363,448,258),(221,364,449,259),(222,365,450,260),(223,366,421,261),(224,367,422,262),(225,368,423,263),(226,369,424,264),(227,370,425,265),(228,371,426,266),(229,372,427,267),(230,373,428,268),(231,374,429,269),(232,375,430,270),(233,376,431,241),(234,377,432,242),(235,378,433,243),(236,379,434,244),(237,380,435,245),(238,381,436,246),(239,382,437,247),(240,383,438,248)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20X | 24A | ··· | 24H | 30A | ··· | 30X | 40A | ··· | 40P | 60A | ··· | 60P | 60Q | ··· | 60AV | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | D4 | D4 | Q16 | C3×D4 | C3×D4 | C5×D4 | C5×D4 | C3×Q16 | C5×Q16 | D4×C15 | D4×C15 | C15×Q16 |
| kernel | Q16×C30 | C2×C120 | C15×Q16 | Q8×C30 | C10×Q16 | C6×Q16 | C2×C40 | C5×Q16 | Q8×C10 | C2×C24 | C3×Q16 | C6×Q8 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C60 | C2×C30 | C30 | C20 | C2×C10 | C12 | C2×C6 | C10 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 8 | 8 | 8 | 32 | 16 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 8 | 16 | 8 | 8 | 32 |
Matrix representation of Q16×C30 ►in GL3(𝔽241) generated by
| 226 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 0 | 0 |
| 0 | 11 | 230 |
| 0 | 11 | 11 |
| 1 | 0 | 0 |
| 0 | 105 | 209 |
| 0 | 209 | 136 |
G:=sub<GL(3,GF(241))| [226,0,0,0,36,0,0,0,36],[1,0,0,0,11,11,0,230,11],[1,0,0,0,105,209,0,209,136] >;
Q16×C30 in GAP, Magma, Sage, TeX
Q_{16}\times C_{30} % in TeX
G:=Group("Q16xC30"); // GroupNames label
G:=SmallGroup(480,939);
// by ID
G=gap.SmallGroup(480,939);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709,1688,15125,7572,124]);
// Polycyclic
G:=Group<a,b,c|a^30=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations