direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C2×C30, C30.100C24, C60.298C23, (C22×C4).9C30, C2.2(C23×C30), C4.7(C22×C30), (C22×C20).21C6, C10.17(C23×C6), C6.17(C23×C10), C20.50(C22×C6), (C22×C60).37C2, C23.17(C2×C30), (C2×C60).586C22, (C2×C30).464C23, (C22×C12).17C10, C12.50(C22×C10), C22.9(C22×C30), (C22×C30).188C22, (C2×C4).30(C2×C30), (C2×C20).132(C2×C6), (C2×C12).133(C2×C10), (C2×C6).84(C22×C10), (C2×C10).84(C22×C6), (C22×C10).58(C2×C6), (C22×C6).50(C2×C10), SmallGroup(480,1182)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 312, all normal (16 characteristic)
C1, C2, C2 [×6], C3, C4 [×12], C22 [×7], C5, C6, C6 [×6], C2×C4 [×18], Q8 [×16], C23, C10, C10 [×6], C12 [×12], C2×C6 [×7], C15, C22×C4 [×3], C2×Q8 [×12], C20 [×12], C2×C10 [×7], C2×C12 [×18], C3×Q8 [×16], C22×C6, C30, C30 [×6], C22×Q8, C2×C20 [×18], C5×Q8 [×16], C22×C10, C22×C12 [×3], C6×Q8 [×12], C60 [×12], C2×C30 [×7], C22×C20 [×3], Q8×C10 [×12], Q8×C2×C6, C2×C60 [×18], Q8×C15 [×16], C22×C30, Q8×C2×C10, C22×C60 [×3], Q8×C30 [×12], Q8×C2×C30
Quotients:
C1, C2 [×15], C3, C22 [×35], C5, C6 [×15], Q8 [×4], C23 [×15], C10 [×15], C2×C6 [×35], C15, C2×Q8 [×6], C24, C2×C10 [×35], C3×Q8 [×4], C22×C6 [×15], C30 [×15], C22×Q8, C5×Q8 [×4], C22×C10 [×15], C6×Q8 [×6], C23×C6, C2×C30 [×35], Q8×C10 [×6], C23×C10, Q8×C2×C6, Q8×C15 [×4], C22×C30 [×15], Q8×C2×C10, Q8×C30 [×6], C23×C30, Q8×C2×C30
Generators and relations
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 >
►Regular action on 480 points
Generators in S480
(1 139)(2 140)(3 141)(4 142)(5 143)(6 144)(7 145)(8 146)(9 147)(10 148)(11 149)(12 150)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 169)(41 170)(42 171)(43 172)(44 173)(45 174)(46 175)(47 176)(48 177)(49 178)(50 179)(51 180)(52 151)(53 152)(54 153)(55 154)(56 155)(57 156)(58 157)(59 158)(60 159)(61 194)(62 195)(63 196)(64 197)(65 198)(66 199)(67 200)(68 201)(69 202)(70 203)(71 204)(72 205)(73 206)(74 207)(75 208)(76 209)(77 210)(78 181)(79 182)(80 183)(81 184)(82 185)(83 186)(84 187)(85 188)(86 189)(87 190)(88 191)(89 192)(90 193)(91 315)(92 316)(93 317)(94 318)(95 319)(96 320)(97 321)(98 322)(99 323)(100 324)(101 325)(102 326)(103 327)(104 328)(105 329)(106 330)(107 301)(108 302)(109 303)(110 304)(111 305)(112 306)(113 307)(114 308)(115 309)(116 310)(117 311)(118 312)(119 313)(120 314)(211 388)(212 389)(213 390)(214 361)(215 362)(216 363)(217 364)(218 365)(219 366)(220 367)(221 368)(222 369)(223 370)(224 371)(225 372)(226 373)(227 374)(228 375)(229 376)(230 377)(231 378)(232 379)(233 380)(234 381)(235 382)(236 383)(237 384)(238 385)(239 386)(240 387)(241 428)(242 429)(243 430)(244 431)(245 432)(246 433)(247 434)(248 435)(249 436)(250 437)(251 438)(252 439)(253 440)(254 441)(255 442)(256 443)(257 444)(258 445)(259 446)(260 447)(261 448)(262 449)(263 450)(264 421)(265 422)(266 423)(267 424)(268 425)(269 426)(270 427)(271 395)(272 396)(273 397)(274 398)(275 399)(276 400)(277 401)(278 402)(279 403)(280 404)(281 405)(282 406)(283 407)(284 408)(285 409)(286 410)(287 411)(288 412)(289 413)(290 414)(291 415)(292 416)(293 417)(294 418)(295 419)(296 420)(297 391)(298 392)(299 393)(300 394)(331 459)(332 460)(333 461)(334 462)(335 463)(336 464)(337 465)(338 466)(339 467)(340 468)(341 469)(342 470)(343 471)(344 472)(345 473)(346 474)(347 475)(348 476)(349 477)(350 478)(351 479)(352 480)(353 451)(354 452)(355 453)(356 454)(357 455)(358 456)(359 457)(360 458)
(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)
(1 391 322 152)(2 392 323 153)(3 393 324 154)(4 394 325 155)(5 395 326 156)(6 396 327 157)(7 397 328 158)(8 398 329 159)(9 399 330 160)(10 400 301 161)(11 401 302 162)(12 402 303 163)(13 403 304 164)(14 404 305 165)(15 405 306 166)(16 406 307 167)(17 407 308 168)(18 408 309 169)(19 409 310 170)(20 410 311 171)(21 411 312 172)(22 412 313 173)(23 413 314 174)(24 414 315 175)(25 415 316 176)(26 416 317 177)(27 417 318 178)(28 418 319 179)(29 419 320 180)(30 420 321 151)(31 147 275 106)(32 148 276 107)(33 149 277 108)(34 150 278 109)(35 121 279 110)(36 122 280 111)(37 123 281 112)(38 124 282 113)(39 125 283 114)(40 126 284 115)(41 127 285 116)(42 128 286 117)(43 129 287 118)(44 130 288 119)(45 131 289 120)(46 132 290 91)(47 133 291 92)(48 134 292 93)(49 135 293 94)(50 136 294 95)(51 137 295 96)(52 138 296 97)(53 139 297 98)(54 140 298 99)(55 141 299 100)(56 142 300 101)(57 143 271 102)(58 144 272 103)(59 145 273 104)(60 146 274 105)(61 368 340 241)(62 369 341 242)(63 370 342 243)(64 371 343 244)(65 372 344 245)(66 373 345 246)(67 374 346 247)(68 375 347 248)(69 376 348 249)(70 377 349 250)(71 378 350 251)(72 379 351 252)(73 380 352 253)(74 381 353 254)(75 382 354 255)(76 383 355 256)(77 384 356 257)(78 385 357 258)(79 386 358 259)(80 387 359 260)(81 388 360 261)(82 389 331 262)(83 390 332 263)(84 361 333 264)(85 362 334 265)(86 363 335 266)(87 364 336 267)(88 365 337 268)(89 366 338 269)(90 367 339 270)(181 238 455 445)(182 239 456 446)(183 240 457 447)(184 211 458 448)(185 212 459 449)(186 213 460 450)(187 214 461 421)(188 215 462 422)(189 216 463 423)(190 217 464 424)(191 218 465 425)(192 219 466 426)(193 220 467 427)(194 221 468 428)(195 222 469 429)(196 223 470 430)(197 224 471 431)(198 225 472 432)(199 226 473 433)(200 227 474 434)(201 228 475 435)(202 229 476 436)(203 230 477 437)(204 231 478 438)(205 232 479 439)(206 233 480 440)(207 234 451 441)(208 235 452 442)(209 236 453 443)(210 237 454 444)
(1 235 322 442)(2 236 323 443)(3 237 324 444)(4 238 325 445)(5 239 326 446)(6 240 327 447)(7 211 328 448)(8 212 329 449)(9 213 330 450)(10 214 301 421)(11 215 302 422)(12 216 303 423)(13 217 304 424)(14 218 305 425)(15 219 306 426)(16 220 307 427)(17 221 308 428)(18 222 309 429)(19 223 310 430)(20 224 311 431)(21 225 312 432)(22 226 313 433)(23 227 314 434)(24 228 315 435)(25 229 316 436)(26 230 317 437)(27 231 318 438)(28 232 319 439)(29 233 320 440)(30 234 321 441)(31 332 275 83)(32 333 276 84)(33 334 277 85)(34 335 278 86)(35 336 279 87)(36 337 280 88)(37 338 281 89)(38 339 282 90)(39 340 283 61)(40 341 284 62)(41 342 285 63)(42 343 286 64)(43 344 287 65)(44 345 288 66)(45 346 289 67)(46 347 290 68)(47 348 291 69)(48 349 292 70)(49 350 293 71)(50 351 294 72)(51 352 295 73)(52 353 296 74)(53 354 297 75)(54 355 298 76)(55 356 299 77)(56 357 300 78)(57 358 271 79)(58 359 272 80)(59 360 273 81)(60 331 274 82)(91 248 132 375)(92 249 133 376)(93 250 134 377)(94 251 135 378)(95 252 136 379)(96 253 137 380)(97 254 138 381)(98 255 139 382)(99 256 140 383)(100 257 141 384)(101 258 142 385)(102 259 143 386)(103 260 144 387)(104 261 145 388)(105 262 146 389)(106 263 147 390)(107 264 148 361)(108 265 149 362)(109 266 150 363)(110 267 121 364)(111 268 122 365)(112 269 123 366)(113 270 124 367)(114 241 125 368)(115 242 126 369)(116 243 127 370)(117 244 128 371)(118 245 129 372)(119 246 130 373)(120 247 131 374)(151 451 420 207)(152 452 391 208)(153 453 392 209)(154 454 393 210)(155 455 394 181)(156 456 395 182)(157 457 396 183)(158 458 397 184)(159 459 398 185)(160 460 399 186)(161 461 400 187)(162 462 401 188)(163 463 402 189)(164 464 403 190)(165 465 404 191)(166 466 405 192)(167 467 406 193)(168 468 407 194)(169 469 408 195)(170 470 409 196)(171 471 410 197)(172 472 411 198)(173 473 412 199)(174 474 413 200)(175 475 414 201)(176 476 415 202)(177 477 416 203)(178 478 417 204)(179 479 418 205)(180 480 419 206)
G:=sub<Sym(480)| (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,169)(41,170)(42,171)(43,172)(44,173)(45,174)(46,175)(47,176)(48,177)(49,178)(50,179)(51,180)(52,151)(53,152)(54,153)(55,154)(56,155)(57,156)(58,157)(59,158)(60,159)(61,194)(62,195)(63,196)(64,197)(65,198)(66,199)(67,200)(68,201)(69,202)(70,203)(71,204)(72,205)(73,206)(74,207)(75,208)(76,209)(77,210)(78,181)(79,182)(80,183)(81,184)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(89,192)(90,193)(91,315)(92,316)(93,317)(94,318)(95,319)(96,320)(97,321)(98,322)(99,323)(100,324)(101,325)(102,326)(103,327)(104,328)(105,329)(106,330)(107,301)(108,302)(109,303)(110,304)(111,305)(112,306)(113,307)(114,308)(115,309)(116,310)(117,311)(118,312)(119,313)(120,314)(211,388)(212,389)(213,390)(214,361)(215,362)(216,363)(217,364)(218,365)(219,366)(220,367)(221,368)(222,369)(223,370)(224,371)(225,372)(226,373)(227,374)(228,375)(229,376)(230,377)(231,378)(232,379)(233,380)(234,381)(235,382)(236,383)(237,384)(238,385)(239,386)(240,387)(241,428)(242,429)(243,430)(244,431)(245,432)(246,433)(247,434)(248,435)(249,436)(250,437)(251,438)(252,439)(253,440)(254,441)(255,442)(256,443)(257,444)(258,445)(259,446)(260,447)(261,448)(262,449)(263,450)(264,421)(265,422)(266,423)(267,424)(268,425)(269,426)(270,427)(271,395)(272,396)(273,397)(274,398)(275,399)(276,400)(277,401)(278,402)(279,403)(280,404)(281,405)(282,406)(283,407)(284,408)(285,409)(286,410)(287,411)(288,412)(289,413)(290,414)(291,415)(292,416)(293,417)(294,418)(295,419)(296,420)(297,391)(298,392)(299,393)(300,394)(331,459)(332,460)(333,461)(334,462)(335,463)(336,464)(337,465)(338,466)(339,467)(340,468)(341,469)(342,470)(343,471)(344,472)(345,473)(346,474)(347,475)(348,476)(349,477)(350,478)(351,479)(352,480)(353,451)(354,452)(355,453)(356,454)(357,455)(358,456)(359,457)(360,458), (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), (1,391,322,152)(2,392,323,153)(3,393,324,154)(4,394,325,155)(5,395,326,156)(6,396,327,157)(7,397,328,158)(8,398,329,159)(9,399,330,160)(10,400,301,161)(11,401,302,162)(12,402,303,163)(13,403,304,164)(14,404,305,165)(15,405,306,166)(16,406,307,167)(17,407,308,168)(18,408,309,169)(19,409,310,170)(20,410,311,171)(21,411,312,172)(22,412,313,173)(23,413,314,174)(24,414,315,175)(25,415,316,176)(26,416,317,177)(27,417,318,178)(28,418,319,179)(29,419,320,180)(30,420,321,151)(31,147,275,106)(32,148,276,107)(33,149,277,108)(34,150,278,109)(35,121,279,110)(36,122,280,111)(37,123,281,112)(38,124,282,113)(39,125,283,114)(40,126,284,115)(41,127,285,116)(42,128,286,117)(43,129,287,118)(44,130,288,119)(45,131,289,120)(46,132,290,91)(47,133,291,92)(48,134,292,93)(49,135,293,94)(50,136,294,95)(51,137,295,96)(52,138,296,97)(53,139,297,98)(54,140,298,99)(55,141,299,100)(56,142,300,101)(57,143,271,102)(58,144,272,103)(59,145,273,104)(60,146,274,105)(61,368,340,241)(62,369,341,242)(63,370,342,243)(64,371,343,244)(65,372,344,245)(66,373,345,246)(67,374,346,247)(68,375,347,248)(69,376,348,249)(70,377,349,250)(71,378,350,251)(72,379,351,252)(73,380,352,253)(74,381,353,254)(75,382,354,255)(76,383,355,256)(77,384,356,257)(78,385,357,258)(79,386,358,259)(80,387,359,260)(81,388,360,261)(82,389,331,262)(83,390,332,263)(84,361,333,264)(85,362,334,265)(86,363,335,266)(87,364,336,267)(88,365,337,268)(89,366,338,269)(90,367,339,270)(181,238,455,445)(182,239,456,446)(183,240,457,447)(184,211,458,448)(185,212,459,449)(186,213,460,450)(187,214,461,421)(188,215,462,422)(189,216,463,423)(190,217,464,424)(191,218,465,425)(192,219,466,426)(193,220,467,427)(194,221,468,428)(195,222,469,429)(196,223,470,430)(197,224,471,431)(198,225,472,432)(199,226,473,433)(200,227,474,434)(201,228,475,435)(202,229,476,436)(203,230,477,437)(204,231,478,438)(205,232,479,439)(206,233,480,440)(207,234,451,441)(208,235,452,442)(209,236,453,443)(210,237,454,444), (1,235,322,442)(2,236,323,443)(3,237,324,444)(4,238,325,445)(5,239,326,446)(6,240,327,447)(7,211,328,448)(8,212,329,449)(9,213,330,450)(10,214,301,421)(11,215,302,422)(12,216,303,423)(13,217,304,424)(14,218,305,425)(15,219,306,426)(16,220,307,427)(17,221,308,428)(18,222,309,429)(19,223,310,430)(20,224,311,431)(21,225,312,432)(22,226,313,433)(23,227,314,434)(24,228,315,435)(25,229,316,436)(26,230,317,437)(27,231,318,438)(28,232,319,439)(29,233,320,440)(30,234,321,441)(31,332,275,83)(32,333,276,84)(33,334,277,85)(34,335,278,86)(35,336,279,87)(36,337,280,88)(37,338,281,89)(38,339,282,90)(39,340,283,61)(40,341,284,62)(41,342,285,63)(42,343,286,64)(43,344,287,65)(44,345,288,66)(45,346,289,67)(46,347,290,68)(47,348,291,69)(48,349,292,70)(49,350,293,71)(50,351,294,72)(51,352,295,73)(52,353,296,74)(53,354,297,75)(54,355,298,76)(55,356,299,77)(56,357,300,78)(57,358,271,79)(58,359,272,80)(59,360,273,81)(60,331,274,82)(91,248,132,375)(92,249,133,376)(93,250,134,377)(94,251,135,378)(95,252,136,379)(96,253,137,380)(97,254,138,381)(98,255,139,382)(99,256,140,383)(100,257,141,384)(101,258,142,385)(102,259,143,386)(103,260,144,387)(104,261,145,388)(105,262,146,389)(106,263,147,390)(107,264,148,361)(108,265,149,362)(109,266,150,363)(110,267,121,364)(111,268,122,365)(112,269,123,366)(113,270,124,367)(114,241,125,368)(115,242,126,369)(116,243,127,370)(117,244,128,371)(118,245,129,372)(119,246,130,373)(120,247,131,374)(151,451,420,207)(152,452,391,208)(153,453,392,209)(154,454,393,210)(155,455,394,181)(156,456,395,182)(157,457,396,183)(158,458,397,184)(159,459,398,185)(160,460,399,186)(161,461,400,187)(162,462,401,188)(163,463,402,189)(164,464,403,190)(165,465,404,191)(166,466,405,192)(167,467,406,193)(168,468,407,194)(169,469,408,195)(170,470,409,196)(171,471,410,197)(172,472,411,198)(173,473,412,199)(174,474,413,200)(175,475,414,201)(176,476,415,202)(177,477,416,203)(178,478,417,204)(179,479,418,205)(180,480,419,206)>;
G:=Group( (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,169)(41,170)(42,171)(43,172)(44,173)(45,174)(46,175)(47,176)(48,177)(49,178)(50,179)(51,180)(52,151)(53,152)(54,153)(55,154)(56,155)(57,156)(58,157)(59,158)(60,159)(61,194)(62,195)(63,196)(64,197)(65,198)(66,199)(67,200)(68,201)(69,202)(70,203)(71,204)(72,205)(73,206)(74,207)(75,208)(76,209)(77,210)(78,181)(79,182)(80,183)(81,184)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(89,192)(90,193)(91,315)(92,316)(93,317)(94,318)(95,319)(96,320)(97,321)(98,322)(99,323)(100,324)(101,325)(102,326)(103,327)(104,328)(105,329)(106,330)(107,301)(108,302)(109,303)(110,304)(111,305)(112,306)(113,307)(114,308)(115,309)(116,310)(117,311)(118,312)(119,313)(120,314)(211,388)(212,389)(213,390)(214,361)(215,362)(216,363)(217,364)(218,365)(219,366)(220,367)(221,368)(222,369)(223,370)(224,371)(225,372)(226,373)(227,374)(228,375)(229,376)(230,377)(231,378)(232,379)(233,380)(234,381)(235,382)(236,383)(237,384)(238,385)(239,386)(240,387)(241,428)(242,429)(243,430)(244,431)(245,432)(246,433)(247,434)(248,435)(249,436)(250,437)(251,438)(252,439)(253,440)(254,441)(255,442)(256,443)(257,444)(258,445)(259,446)(260,447)(261,448)(262,449)(263,450)(264,421)(265,422)(266,423)(267,424)(268,425)(269,426)(270,427)(271,395)(272,396)(273,397)(274,398)(275,399)(276,400)(277,401)(278,402)(279,403)(280,404)(281,405)(282,406)(283,407)(284,408)(285,409)(286,410)(287,411)(288,412)(289,413)(290,414)(291,415)(292,416)(293,417)(294,418)(295,419)(296,420)(297,391)(298,392)(299,393)(300,394)(331,459)(332,460)(333,461)(334,462)(335,463)(336,464)(337,465)(338,466)(339,467)(340,468)(341,469)(342,470)(343,471)(344,472)(345,473)(346,474)(347,475)(348,476)(349,477)(350,478)(351,479)(352,480)(353,451)(354,452)(355,453)(356,454)(357,455)(358,456)(359,457)(360,458), (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), (1,391,322,152)(2,392,323,153)(3,393,324,154)(4,394,325,155)(5,395,326,156)(6,396,327,157)(7,397,328,158)(8,398,329,159)(9,399,330,160)(10,400,301,161)(11,401,302,162)(12,402,303,163)(13,403,304,164)(14,404,305,165)(15,405,306,166)(16,406,307,167)(17,407,308,168)(18,408,309,169)(19,409,310,170)(20,410,311,171)(21,411,312,172)(22,412,313,173)(23,413,314,174)(24,414,315,175)(25,415,316,176)(26,416,317,177)(27,417,318,178)(28,418,319,179)(29,419,320,180)(30,420,321,151)(31,147,275,106)(32,148,276,107)(33,149,277,108)(34,150,278,109)(35,121,279,110)(36,122,280,111)(37,123,281,112)(38,124,282,113)(39,125,283,114)(40,126,284,115)(41,127,285,116)(42,128,286,117)(43,129,287,118)(44,130,288,119)(45,131,289,120)(46,132,290,91)(47,133,291,92)(48,134,292,93)(49,135,293,94)(50,136,294,95)(51,137,295,96)(52,138,296,97)(53,139,297,98)(54,140,298,99)(55,141,299,100)(56,142,300,101)(57,143,271,102)(58,144,272,103)(59,145,273,104)(60,146,274,105)(61,368,340,241)(62,369,341,242)(63,370,342,243)(64,371,343,244)(65,372,344,245)(66,373,345,246)(67,374,346,247)(68,375,347,248)(69,376,348,249)(70,377,349,250)(71,378,350,251)(72,379,351,252)(73,380,352,253)(74,381,353,254)(75,382,354,255)(76,383,355,256)(77,384,356,257)(78,385,357,258)(79,386,358,259)(80,387,359,260)(81,388,360,261)(82,389,331,262)(83,390,332,263)(84,361,333,264)(85,362,334,265)(86,363,335,266)(87,364,336,267)(88,365,337,268)(89,366,338,269)(90,367,339,270)(181,238,455,445)(182,239,456,446)(183,240,457,447)(184,211,458,448)(185,212,459,449)(186,213,460,450)(187,214,461,421)(188,215,462,422)(189,216,463,423)(190,217,464,424)(191,218,465,425)(192,219,466,426)(193,220,467,427)(194,221,468,428)(195,222,469,429)(196,223,470,430)(197,224,471,431)(198,225,472,432)(199,226,473,433)(200,227,474,434)(201,228,475,435)(202,229,476,436)(203,230,477,437)(204,231,478,438)(205,232,479,439)(206,233,480,440)(207,234,451,441)(208,235,452,442)(209,236,453,443)(210,237,454,444), (1,235,322,442)(2,236,323,443)(3,237,324,444)(4,238,325,445)(5,239,326,446)(6,240,327,447)(7,211,328,448)(8,212,329,449)(9,213,330,450)(10,214,301,421)(11,215,302,422)(12,216,303,423)(13,217,304,424)(14,218,305,425)(15,219,306,426)(16,220,307,427)(17,221,308,428)(18,222,309,429)(19,223,310,430)(20,224,311,431)(21,225,312,432)(22,226,313,433)(23,227,314,434)(24,228,315,435)(25,229,316,436)(26,230,317,437)(27,231,318,438)(28,232,319,439)(29,233,320,440)(30,234,321,441)(31,332,275,83)(32,333,276,84)(33,334,277,85)(34,335,278,86)(35,336,279,87)(36,337,280,88)(37,338,281,89)(38,339,282,90)(39,340,283,61)(40,341,284,62)(41,342,285,63)(42,343,286,64)(43,344,287,65)(44,345,288,66)(45,346,289,67)(46,347,290,68)(47,348,291,69)(48,349,292,70)(49,350,293,71)(50,351,294,72)(51,352,295,73)(52,353,296,74)(53,354,297,75)(54,355,298,76)(55,356,299,77)(56,357,300,78)(57,358,271,79)(58,359,272,80)(59,360,273,81)(60,331,274,82)(91,248,132,375)(92,249,133,376)(93,250,134,377)(94,251,135,378)(95,252,136,379)(96,253,137,380)(97,254,138,381)(98,255,139,382)(99,256,140,383)(100,257,141,384)(101,258,142,385)(102,259,143,386)(103,260,144,387)(104,261,145,388)(105,262,146,389)(106,263,147,390)(107,264,148,361)(108,265,149,362)(109,266,150,363)(110,267,121,364)(111,268,122,365)(112,269,123,366)(113,270,124,367)(114,241,125,368)(115,242,126,369)(116,243,127,370)(117,244,128,371)(118,245,129,372)(119,246,130,373)(120,247,131,374)(151,451,420,207)(152,452,391,208)(153,453,392,209)(154,454,393,210)(155,455,394,181)(156,456,395,182)(157,457,396,183)(158,458,397,184)(159,459,398,185)(160,460,399,186)(161,461,400,187)(162,462,401,188)(163,463,402,189)(164,464,403,190)(165,465,404,191)(166,466,405,192)(167,467,406,193)(168,468,407,194)(169,469,408,195)(170,470,409,196)(171,471,410,197)(172,472,411,198)(173,473,412,199)(174,474,413,200)(175,475,414,201)(176,476,415,202)(177,477,416,203)(178,478,417,204)(179,479,418,205)(180,480,419,206) );
G=PermutationGroup([(1,139),(2,140),(3,141),(4,142),(5,143),(6,144),(7,145),(8,146),(9,147),(10,148),(11,149),(12,150),(13,121),(14,122),(15,123),(16,124),(17,125),(18,126),(19,127),(20,128),(21,129),(22,130),(23,131),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,138),(31,160),(32,161),(33,162),(34,163),(35,164),(36,165),(37,166),(38,167),(39,168),(40,169),(41,170),(42,171),(43,172),(44,173),(45,174),(46,175),(47,176),(48,177),(49,178),(50,179),(51,180),(52,151),(53,152),(54,153),(55,154),(56,155),(57,156),(58,157),(59,158),(60,159),(61,194),(62,195),(63,196),(64,197),(65,198),(66,199),(67,200),(68,201),(69,202),(70,203),(71,204),(72,205),(73,206),(74,207),(75,208),(76,209),(77,210),(78,181),(79,182),(80,183),(81,184),(82,185),(83,186),(84,187),(85,188),(86,189),(87,190),(88,191),(89,192),(90,193),(91,315),(92,316),(93,317),(94,318),(95,319),(96,320),(97,321),(98,322),(99,323),(100,324),(101,325),(102,326),(103,327),(104,328),(105,329),(106,330),(107,301),(108,302),(109,303),(110,304),(111,305),(112,306),(113,307),(114,308),(115,309),(116,310),(117,311),(118,312),(119,313),(120,314),(211,388),(212,389),(213,390),(214,361),(215,362),(216,363),(217,364),(218,365),(219,366),(220,367),(221,368),(222,369),(223,370),(224,371),(225,372),(226,373),(227,374),(228,375),(229,376),(230,377),(231,378),(232,379),(233,380),(234,381),(235,382),(236,383),(237,384),(238,385),(239,386),(240,387),(241,428),(242,429),(243,430),(244,431),(245,432),(246,433),(247,434),(248,435),(249,436),(250,437),(251,438),(252,439),(253,440),(254,441),(255,442),(256,443),(257,444),(258,445),(259,446),(260,447),(261,448),(262,449),(263,450),(264,421),(265,422),(266,423),(267,424),(268,425),(269,426),(270,427),(271,395),(272,396),(273,397),(274,398),(275,399),(276,400),(277,401),(278,402),(279,403),(280,404),(281,405),(282,406),(283,407),(284,408),(285,409),(286,410),(287,411),(288,412),(289,413),(290,414),(291,415),(292,416),(293,417),(294,418),(295,419),(296,420),(297,391),(298,392),(299,393),(300,394),(331,459),(332,460),(333,461),(334,462),(335,463),(336,464),(337,465),(338,466),(339,467),(340,468),(341,469),(342,470),(343,471),(344,472),(345,473),(346,474),(347,475),(348,476),(349,477),(350,478),(351,479),(352,480),(353,451),(354,452),(355,453),(356,454),(357,455),(358,456),(359,457),(360,458)], [(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)], [(1,391,322,152),(2,392,323,153),(3,393,324,154),(4,394,325,155),(5,395,326,156),(6,396,327,157),(7,397,328,158),(8,398,329,159),(9,399,330,160),(10,400,301,161),(11,401,302,162),(12,402,303,163),(13,403,304,164),(14,404,305,165),(15,405,306,166),(16,406,307,167),(17,407,308,168),(18,408,309,169),(19,409,310,170),(20,410,311,171),(21,411,312,172),(22,412,313,173),(23,413,314,174),(24,414,315,175),(25,415,316,176),(26,416,317,177),(27,417,318,178),(28,418,319,179),(29,419,320,180),(30,420,321,151),(31,147,275,106),(32,148,276,107),(33,149,277,108),(34,150,278,109),(35,121,279,110),(36,122,280,111),(37,123,281,112),(38,124,282,113),(39,125,283,114),(40,126,284,115),(41,127,285,116),(42,128,286,117),(43,129,287,118),(44,130,288,119),(45,131,289,120),(46,132,290,91),(47,133,291,92),(48,134,292,93),(49,135,293,94),(50,136,294,95),(51,137,295,96),(52,138,296,97),(53,139,297,98),(54,140,298,99),(55,141,299,100),(56,142,300,101),(57,143,271,102),(58,144,272,103),(59,145,273,104),(60,146,274,105),(61,368,340,241),(62,369,341,242),(63,370,342,243),(64,371,343,244),(65,372,344,245),(66,373,345,246),(67,374,346,247),(68,375,347,248),(69,376,348,249),(70,377,349,250),(71,378,350,251),(72,379,351,252),(73,380,352,253),(74,381,353,254),(75,382,354,255),(76,383,355,256),(77,384,356,257),(78,385,357,258),(79,386,358,259),(80,387,359,260),(81,388,360,261),(82,389,331,262),(83,390,332,263),(84,361,333,264),(85,362,334,265),(86,363,335,266),(87,364,336,267),(88,365,337,268),(89,366,338,269),(90,367,339,270),(181,238,455,445),(182,239,456,446),(183,240,457,447),(184,211,458,448),(185,212,459,449),(186,213,460,450),(187,214,461,421),(188,215,462,422),(189,216,463,423),(190,217,464,424),(191,218,465,425),(192,219,466,426),(193,220,467,427),(194,221,468,428),(195,222,469,429),(196,223,470,430),(197,224,471,431),(198,225,472,432),(199,226,473,433),(200,227,474,434),(201,228,475,435),(202,229,476,436),(203,230,477,437),(204,231,478,438),(205,232,479,439),(206,233,480,440),(207,234,451,441),(208,235,452,442),(209,236,453,443),(210,237,454,444)], [(1,235,322,442),(2,236,323,443),(3,237,324,444),(4,238,325,445),(5,239,326,446),(6,240,327,447),(7,211,328,448),(8,212,329,449),(9,213,330,450),(10,214,301,421),(11,215,302,422),(12,216,303,423),(13,217,304,424),(14,218,305,425),(15,219,306,426),(16,220,307,427),(17,221,308,428),(18,222,309,429),(19,223,310,430),(20,224,311,431),(21,225,312,432),(22,226,313,433),(23,227,314,434),(24,228,315,435),(25,229,316,436),(26,230,317,437),(27,231,318,438),(28,232,319,439),(29,233,320,440),(30,234,321,441),(31,332,275,83),(32,333,276,84),(33,334,277,85),(34,335,278,86),(35,336,279,87),(36,337,280,88),(37,338,281,89),(38,339,282,90),(39,340,283,61),(40,341,284,62),(41,342,285,63),(42,343,286,64),(43,344,287,65),(44,345,288,66),(45,346,289,67),(46,347,290,68),(47,348,291,69),(48,349,292,70),(49,350,293,71),(50,351,294,72),(51,352,295,73),(52,353,296,74),(53,354,297,75),(54,355,298,76),(55,356,299,77),(56,357,300,78),(57,358,271,79),(58,359,272,80),(59,360,273,81),(60,331,274,82),(91,248,132,375),(92,249,133,376),(93,250,134,377),(94,251,135,378),(95,252,136,379),(96,253,137,380),(97,254,138,381),(98,255,139,382),(99,256,140,383),(100,257,141,384),(101,258,142,385),(102,259,143,386),(103,260,144,387),(104,261,145,388),(105,262,146,389),(106,263,147,390),(107,264,148,361),(108,265,149,362),(109,266,150,363),(110,267,121,364),(111,268,122,365),(112,269,123,366),(113,270,124,367),(114,241,125,368),(115,242,126,369),(116,243,127,370),(117,244,128,371),(118,245,129,372),(119,246,130,373),(120,247,131,374),(151,451,420,207),(152,452,391,208),(153,453,392,209),(154,454,393,210),(155,455,394,181),(156,456,395,182),(157,457,396,183),(158,458,397,184),(159,459,398,185),(160,460,399,186),(161,461,400,187),(162,462,401,188),(163,463,402,189),(164,464,403,190),(165,465,404,191),(166,466,405,192),(167,467,406,193),(168,468,407,194),(169,469,408,195),(170,470,409,196),(171,471,410,197),(172,472,411,198),(173,473,412,199),(174,474,413,200),(175,475,414,201),(176,476,415,202),(177,477,416,203),(178,478,417,204),(179,479,418,205),(180,480,419,206)])
Matrix representation ►G ⊆ 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] >;
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 |
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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ℤ
𝔽
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