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## G = Q8×C2×C30order 480 = 25·3·5

### Direct product of C2×C30 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C2×C30
 Chief series C1 — C2 — C10 — C30 — C60 — Q8×C15 — Q8×C30 — Q8×C2×C30
 Lower central C1 — C2 — Q8×C2×C30
 Upper central C1 — C22×C30 — Q8×C2×C30

Generators and relations for Q8×C2×C30
G = < a,b,c,d | a2=b30=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 312, all normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, Q8, C23, C10, C10, C12, C2×C6, C15, C22×C4, C2×Q8, C20, C2×C10, C2×C12, C3×Q8, C22×C6, C30, C30, C22×Q8, C2×C20, C5×Q8, C22×C10, C22×C12, C6×Q8, C60, C2×C30, C22×C20, Q8×C10, Q8×C2×C6, C2×C60, Q8×C15, C22×C30, Q8×C2×C10, C22×C60, Q8×C30, Q8×C2×C30
Quotients: C1, C2, C3, C22, C5, C6, Q8, C23, C10, C2×C6, C15, C2×Q8, C24, C2×C10, C3×Q8, C22×C6, C30, C22×Q8, C5×Q8, C22×C10, C6×Q8, C23×C6, C2×C30, Q8×C10, C23×C10, Q8×C2×C6, Q8×C15, C22×C30, Q8×C2×C10, Q8×C30, C23×C30, Q8×C2×C30

Smallest permutation representation of Q8×C2×C30
Regular action on 480 points
Generators in S480
(1 288)(2 289)(3 290)(4 291)(5 292)(6 293)(7 294)(8 295)(9 296)(10 297)(11 298)(12 299)(13 300)(14 271)(15 272)(16 273)(17 274)(18 275)(19 276)(20 277)(21 278)(22 279)(23 280)(24 281)(25 282)(26 283)(27 284)(28 285)(29 286)(30 287)(31 428)(32 429)(33 430)(34 431)(35 432)(36 433)(37 434)(38 435)(39 436)(40 437)(41 438)(42 439)(43 440)(44 441)(45 442)(46 443)(47 444)(48 445)(49 446)(50 447)(51 448)(52 449)(53 450)(54 421)(55 422)(56 423)(57 424)(58 425)(59 426)(60 427)(61 347)(62 348)(63 349)(64 350)(65 351)(66 352)(67 353)(68 354)(69 355)(70 356)(71 357)(72 358)(73 359)(74 360)(75 331)(76 332)(77 333)(78 334)(79 335)(80 336)(81 337)(82 338)(83 339)(84 340)(85 341)(86 342)(87 343)(88 344)(89 345)(90 346)(91 250)(92 251)(93 252)(94 253)(95 254)(96 255)(97 256)(98 257)(99 258)(100 259)(101 260)(102 261)(103 262)(104 263)(105 264)(106 265)(107 266)(108 267)(109 268)(110 269)(111 270)(112 241)(113 242)(114 243)(115 244)(116 245)(117 246)(118 247)(119 248)(120 249)(121 470)(122 471)(123 472)(124 473)(125 474)(126 475)(127 476)(128 477)(129 478)(130 479)(131 480)(132 451)(133 452)(134 453)(135 454)(136 455)(137 456)(138 457)(139 458)(140 459)(141 460)(142 461)(143 462)(144 463)(145 464)(146 465)(147 466)(148 467)(149 468)(150 469)(151 326)(152 327)(153 328)(154 329)(155 330)(156 301)(157 302)(158 303)(159 304)(160 305)(161 306)(162 307)(163 308)(164 309)(165 310)(166 311)(167 312)(168 313)(169 314)(170 315)(171 316)(172 317)(173 318)(174 319)(175 320)(176 321)(177 322)(178 323)(179 324)(180 325)(181 362)(182 363)(183 364)(184 365)(185 366)(186 367)(187 368)(188 369)(189 370)(190 371)(191 372)(192 373)(193 374)(194 375)(195 376)(196 377)(197 378)(198 379)(199 380)(200 381)(201 382)(202 383)(203 384)(204 385)(205 386)(206 387)(207 388)(208 389)(209 390)(210 361)(211 396)(212 397)(213 398)(214 399)(215 400)(216 401)(217 402)(218 403)(219 404)(220 405)(221 406)(222 407)(223 408)(224 409)(225 410)(226 411)(227 412)(228 413)(229 414)(230 415)(231 416)(232 417)(233 418)(234 419)(235 420)(236 391)(237 392)(238 393)(239 394)(240 395)
(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)
(1 350 465 303)(2 351 466 304)(3 352 467 305)(4 353 468 306)(5 354 469 307)(6 355 470 308)(7 356 471 309)(8 357 472 310)(9 358 473 311)(10 359 474 312)(11 360 475 313)(12 331 476 314)(13 332 477 315)(14 333 478 316)(15 334 479 317)(16 335 480 318)(17 336 451 319)(18 337 452 320)(19 338 453 321)(20 339 454 322)(21 340 455 323)(22 341 456 324)(23 342 457 325)(24 343 458 326)(25 344 459 327)(26 345 460 328)(27 346 461 329)(28 347 462 330)(29 348 463 301)(30 349 464 302)(31 118 379 232)(32 119 380 233)(33 120 381 234)(34 91 382 235)(35 92 383 236)(36 93 384 237)(37 94 385 238)(38 95 386 239)(39 96 387 240)(40 97 388 211)(41 98 389 212)(42 99 390 213)(43 100 361 214)(44 101 362 215)(45 102 363 216)(46 103 364 217)(47 104 365 218)(48 105 366 219)(49 106 367 220)(50 107 368 221)(51 108 369 222)(52 109 370 223)(53 110 371 224)(54 111 372 225)(55 112 373 226)(56 113 374 227)(57 114 375 228)(58 115 376 229)(59 116 377 230)(60 117 378 231)(61 143 155 285)(62 144 156 286)(63 145 157 287)(64 146 158 288)(65 147 159 289)(66 148 160 290)(67 149 161 291)(68 150 162 292)(69 121 163 293)(70 122 164 294)(71 123 165 295)(72 124 166 296)(73 125 167 297)(74 126 168 298)(75 127 169 299)(76 128 170 300)(77 129 171 271)(78 130 172 272)(79 131 173 273)(80 132 174 274)(81 133 175 275)(82 134 176 276)(83 135 177 277)(84 136 178 278)(85 137 179 279)(86 138 180 280)(87 139 151 281)(88 140 152 282)(89 141 153 283)(90 142 154 284)(181 400 441 260)(182 401 442 261)(183 402 443 262)(184 403 444 263)(185 404 445 264)(186 405 446 265)(187 406 447 266)(188 407 448 267)(189 408 449 268)(190 409 450 269)(191 410 421 270)(192 411 422 241)(193 412 423 242)(194 413 424 243)(195 414 425 244)(196 415 426 245)(197 416 427 246)(198 417 428 247)(199 418 429 248)(200 419 430 249)(201 420 431 250)(202 391 432 251)(203 392 433 252)(204 393 434 253)(205 394 435 254)(206 395 436 255)(207 396 437 256)(208 397 438 257)(209 398 439 258)(210 399 440 259)
(1 258 465 398)(2 259 466 399)(3 260 467 400)(4 261 468 401)(5 262 469 402)(6 263 470 403)(7 264 471 404)(8 265 472 405)(9 266 473 406)(10 267 474 407)(11 268 475 408)(12 269 476 409)(13 270 477 410)(14 241 478 411)(15 242 479 412)(16 243 480 413)(17 244 451 414)(18 245 452 415)(19 246 453 416)(20 247 454 417)(21 248 455 418)(22 249 456 419)(23 250 457 420)(24 251 458 391)(25 252 459 392)(26 253 460 393)(27 254 461 394)(28 255 462 395)(29 256 463 396)(30 257 464 397)(31 177 379 83)(32 178 380 84)(33 179 381 85)(34 180 382 86)(35 151 383 87)(36 152 384 88)(37 153 385 89)(38 154 386 90)(39 155 387 61)(40 156 388 62)(41 157 389 63)(42 158 390 64)(43 159 361 65)(44 160 362 66)(45 161 363 67)(46 162 364 68)(47 163 365 69)(48 164 366 70)(49 165 367 71)(50 166 368 72)(51 167 369 73)(52 168 370 74)(53 169 371 75)(54 170 372 76)(55 171 373 77)(56 172 374 78)(57 173 375 79)(58 174 376 80)(59 175 377 81)(60 176 378 82)(91 138 235 280)(92 139 236 281)(93 140 237 282)(94 141 238 283)(95 142 239 284)(96 143 240 285)(97 144 211 286)(98 145 212 287)(99 146 213 288)(100 147 214 289)(101 148 215 290)(102 149 216 291)(103 150 217 292)(104 121 218 293)(105 122 219 294)(106 123 220 295)(107 124 221 296)(108 125 222 297)(109 126 223 298)(110 127 224 299)(111 128 225 300)(112 129 226 271)(113 130 227 272)(114 131 228 273)(115 132 229 274)(116 133 230 275)(117 134 231 276)(118 135 232 277)(119 136 233 278)(120 137 234 279)(181 352 441 305)(182 353 442 306)(183 354 443 307)(184 355 444 308)(185 356 445 309)(186 357 446 310)(187 358 447 311)(188 359 448 312)(189 360 449 313)(190 331 450 314)(191 332 421 315)(192 333 422 316)(193 334 423 317)(194 335 424 318)(195 336 425 319)(196 337 426 320)(197 338 427 321)(198 339 428 322)(199 340 429 323)(200 341 430 324)(201 342 431 325)(202 343 432 326)(203 344 433 327)(204 345 434 328)(205 346 435 329)(206 347 436 330)(207 348 437 301)(208 349 438 302)(209 350 439 303)(210 351 440 304)

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

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

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

300 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4L 5A 5B 5C 5D 6A ··· 6N 10A ··· 10AB 12A ··· 12X 15A ··· 15H 20A ··· 20AV 30A ··· 30BD 60A ··· 60CR order 1 2 ··· 2 3 3 4 ··· 4 5 5 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

300 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 Q8 C3×Q8 C5×Q8 Q8×C15 kernel Q8×C2×C30 C22×C60 Q8×C30 Q8×C2×C10 Q8×C2×C6 C22×C20 Q8×C10 C22×C12 C6×Q8 C22×Q8 C22×C4 C2×Q8 C2×C30 C2×C10 C2×C6 C22 # reps 1 3 12 2 4 6 24 12 48 8 24 96 4 8 16 32

Matrix representation of Q8×C2×C30 in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 16 0 0 0 0 16
,
 60 0 0 0 0 1 0 0 0 0 1 59 0 0 1 60
,
 60 0 0 0 0 1 0 0 0 0 45 3 0 0 16 16
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,16,0,0,0,0,16],[60,0,0,0,0,1,0,0,0,0,1,1,0,0,59,60],[60,0,0,0,0,1,0,0,0,0,45,16,0,0,3,16] >;

Q8×C2×C30 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_{30}
% in TeX

G:=Group("Q8xC2xC30");
// GroupNames label

G:=SmallGroup(480,1182);
// by ID

G=gap.SmallGroup(480,1182);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,1680,3389,1688]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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