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## G = Q16×C30order 480 = 25·3·5

### Direct product of C30 and Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — Q16×C30
 Chief series C1 — C2 — C4 — C20 — C60 — Q8×C15 — C15×Q16 — Q16×C30
 Lower central C1 — C2 — C4 — Q16×C30
 Upper central C1 — C2×C30 — C2×C60 — Q16×C30

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 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×4], Q8 [×2], C10, C10 [×2], C12 [×2], C12 [×4], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], C20 [×2], C20 [×4], C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C3×Q8 [×2], C30, C30 [×2], C2×Q16, C40 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], C2×C24, C3×Q16 [×4], C6×Q8 [×2], C60 [×2], C60 [×4], C2×C30, C2×C40, C5×Q16 [×4], Q8×C10 [×2], C6×Q16, C120 [×2], C2×C60, C2×C60 [×2], Q8×C15 [×4], Q8×C15 [×2], C10×Q16, C2×C120, C15×Q16 [×4], Q8×C30 [×2], Q16×C30
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], C23, C10 [×7], C2×C6 [×7], C15, Q16 [×2], C2×D4, C2×C10 [×7], C3×D4 [×2], C22×C6, C30 [×7], C2×Q16, C5×D4 [×2], C22×C10, C3×Q16 [×2], C6×D4, C2×C30 [×7], C5×Q16 [×2], D4×C10, C6×Q16, D4×C15 [×2], C22×C30, C10×Q16, C15×Q16 [×2], D4×C30, Q16×C30

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

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

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

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

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

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