direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.44D4, C60.215D4, C30.12Q16, C6.6Dic20, Dic10⋊5C12, C30.19SD16, (C2×C40).2C6, (C2×C24).2D5, C4.7(D5×C12), (C2×C120).2C2, C20.44(C3×D4), C12.64(C4×D5), (C2×C6).48D20, C4⋊Dic5.1C6, C10.1(C3×Q16), C20.38(C2×C12), C60.199(C2×C4), (C2×C30).108D4, C6.9(C40⋊C2), C10.1(C3×SD16), C2.1(C3×Dic20), C22.7(C3×D20), (C3×Dic10)⋊14C4, C15⋊15(Q8⋊C4), (C2×C12).421D10, (C2×Dic10).1C6, C12.112(C5⋊D4), C30.84(C22⋊C4), (C2×C60).502C22, (C6×Dic10).10C2, C6.37(D10⋊C4), C5⋊3(C3×Q8⋊C4), (C2×C8).2(C3×D5), (C2×C4).68(C6×D5), C2.1(C3×C40⋊C2), C4.19(C3×C5⋊D4), (C2×C20).85(C2×C6), (C2×C10).12(C3×D4), C2.7(C3×D10⋊C4), C10.16(C3×C22⋊C4), (C3×C4⋊Dic5).13C2, SmallGroup(480,94)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.44D4
G = < a,b,c,d | a3=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c-1 >
Subgroups: 272 in 84 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C3×Q8, C30, Q8⋊C4, C40, Dic10, Dic10, C2×Dic5, C2×C20, C3×C4⋊C4, C2×C24, C6×Q8, C3×Dic5, C60, C2×C30, C4⋊Dic5, C2×C40, C2×Dic10, C3×Q8⋊C4, C120, C3×Dic10, C3×Dic10, C6×Dic5, C2×C60, C20.44D4, C3×C4⋊Dic5, C2×C120, C6×Dic10, C3×C20.44D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, SD16, Q16, D10, C2×C12, C3×D4, C3×D5, Q8⋊C4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C3×SD16, C3×Q16, C6×D5, C40⋊C2, Dic20, D10⋊C4, C3×Q8⋊C4, D5×C12, C3×D20, C3×C5⋊D4, C20.44D4, C3×C40⋊C2, C3×Dic20, C3×D10⋊C4, C3×C20.44D4
►Regular action on 480 points
Generators in S480
(1 315 371)(2 316 372)(3 317 373)(4 318 374)(5 319 375)(6 320 376)(7 301 377)(8 302 378)(9 303 379)(10 304 380)(11 305 361)(12 306 362)(13 307 363)(14 308 364)(15 309 365)(16 310 366)(17 311 367)(18 312 368)(19 313 369)(20 314 370)(21 189 71)(22 190 72)(23 191 73)(24 192 74)(25 193 75)(26 194 76)(27 195 77)(28 196 78)(29 197 79)(30 198 80)(31 199 61)(32 200 62)(33 181 63)(34 182 64)(35 183 65)(36 184 66)(37 185 67)(38 186 68)(39 187 69)(40 188 70)(41 277 241)(42 278 242)(43 279 243)(44 280 244)(45 261 245)(46 262 246)(47 263 247)(48 264 248)(49 265 249)(50 266 250)(51 267 251)(52 268 252)(53 269 253)(54 270 254)(55 271 255)(56 272 256)(57 273 257)(58 274 258)(59 275 259)(60 276 260)(81 322 161)(82 323 162)(83 324 163)(84 325 164)(85 326 165)(86 327 166)(87 328 167)(88 329 168)(89 330 169)(90 331 170)(91 332 171)(92 333 172)(93 334 173)(94 335 174)(95 336 175)(96 337 176)(97 338 177)(98 339 178)(99 340 179)(100 321 180)(101 291 383)(102 292 384)(103 293 385)(104 294 386)(105 295 387)(106 296 388)(107 297 389)(108 298 390)(109 299 391)(110 300 392)(111 281 393)(112 282 394)(113 283 395)(114 284 396)(115 285 397)(116 286 398)(117 287 399)(118 288 400)(119 289 381)(120 290 382)(121 448 149)(122 449 150)(123 450 151)(124 451 152)(125 452 153)(126 453 154)(127 454 155)(128 455 156)(129 456 157)(130 457 158)(131 458 159)(132 459 160)(133 460 141)(134 441 142)(135 442 143)(136 443 144)(137 444 145)(138 445 146)(139 446 147)(140 447 148)(201 341 432)(202 342 433)(203 343 434)(204 344 435)(205 345 436)(206 346 437)(207 347 438)(208 348 439)(209 349 440)(210 350 421)(211 351 422)(212 352 423)(213 353 424)(214 354 425)(215 355 426)(216 356 427)(217 357 428)(218 358 429)(219 359 430)(220 360 431)(221 479 419)(222 480 420)(223 461 401)(224 462 402)(225 463 403)(226 464 404)(227 465 405)(228 466 406)(229 467 407)(230 468 408)(231 469 409)(232 470 410)(233 471 411)(234 472 412)(235 473 413)(236 474 414)(237 475 415)(238 476 416)(239 477 417)(240 478 418)
(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 76 438 164)(2 75 439 163)(3 74 440 162)(4 73 421 161)(5 72 422 180)(6 71 423 179)(7 70 424 178)(8 69 425 177)(9 68 426 176)(10 67 427 175)(11 66 428 174)(12 65 429 173)(13 64 430 172)(14 63 431 171)(15 62 432 170)(16 61 433 169)(17 80 434 168)(18 79 435 167)(19 78 436 166)(20 77 437 165)(21 212 99 320)(22 211 100 319)(23 210 81 318)(24 209 82 317)(25 208 83 316)(26 207 84 315)(27 206 85 314)(28 205 86 313)(29 204 87 312)(30 203 88 311)(31 202 89 310)(32 201 90 309)(33 220 91 308)(34 219 92 307)(35 218 93 306)(36 217 94 305)(37 216 95 304)(38 215 96 303)(39 214 97 302)(40 213 98 301)(41 120 459 405)(42 119 460 404)(43 118 441 403)(44 117 442 402)(45 116 443 401)(46 115 444 420)(47 114 445 419)(48 113 446 418)(49 112 447 417)(50 111 448 416)(51 110 449 415)(52 109 450 414)(53 108 451 413)(54 107 452 412)(55 106 453 411)(56 105 454 410)(57 104 455 409)(58 103 456 408)(59 102 457 407)(60 101 458 406)(121 476 250 393)(122 475 251 392)(123 474 252 391)(124 473 253 390)(125 472 254 389)(126 471 255 388)(127 470 256 387)(128 469 257 386)(129 468 258 385)(130 467 259 384)(131 466 260 383)(132 465 241 382)(133 464 242 381)(134 463 243 400)(135 462 244 399)(136 461 245 398)(137 480 246 397)(138 479 247 396)(139 478 248 395)(140 477 249 394)(141 226 278 289)(142 225 279 288)(143 224 280 287)(144 223 261 286)(145 222 262 285)(146 221 263 284)(147 240 264 283)(148 239 265 282)(149 238 266 281)(150 237 267 300)(151 236 268 299)(152 235 269 298)(153 234 270 297)(154 233 271 296)(155 232 272 295)(156 231 273 294)(157 230 274 293)(158 229 275 292)(159 228 276 291)(160 227 277 290)(181 360 332 364)(182 359 333 363)(183 358 334 362)(184 357 335 361)(185 356 336 380)(186 355 337 379)(187 354 338 378)(188 353 339 377)(189 352 340 376)(190 351 321 375)(191 350 322 374)(192 349 323 373)(193 348 324 372)(194 347 325 371)(195 346 326 370)(196 345 327 369)(197 344 328 368)(198 343 329 367)(199 342 330 366)(200 341 331 365)
(1 480 11 470)(2 479 12 469)(3 478 13 468)(4 477 14 467)(5 476 15 466)(6 475 16 465)(7 474 17 464)(8 473 18 463)(9 472 19 462)(10 471 20 461)(21 444 31 454)(22 443 32 453)(23 442 33 452)(24 441 34 451)(25 460 35 450)(26 459 36 449)(27 458 37 448)(28 457 38 447)(29 456 39 446)(30 455 40 445)(41 94 51 84)(42 93 52 83)(43 92 53 82)(44 91 54 81)(45 90 55 100)(46 89 56 99)(47 88 57 98)(48 87 58 97)(49 86 59 96)(50 85 60 95)(61 127 71 137)(62 126 72 136)(63 125 73 135)(64 124 74 134)(65 123 75 133)(66 122 76 132)(67 121 77 131)(68 140 78 130)(69 139 79 129)(70 138 80 128)(101 211 111 201)(102 210 112 220)(103 209 113 219)(104 208 114 218)(105 207 115 217)(106 206 116 216)(107 205 117 215)(108 204 118 214)(109 203 119 213)(110 202 120 212)(141 183 151 193)(142 182 152 192)(143 181 153 191)(144 200 154 190)(145 199 155 189)(146 198 156 188)(147 197 157 187)(148 196 158 186)(149 195 159 185)(150 194 160 184)(161 244 171 254)(162 243 172 253)(163 242 173 252)(164 241 174 251)(165 260 175 250)(166 259 176 249)(167 258 177 248)(168 257 178 247)(169 256 179 246)(170 255 180 245)(221 362 231 372)(222 361 232 371)(223 380 233 370)(224 379 234 369)(225 378 235 368)(226 377 236 367)(227 376 237 366)(228 375 238 365)(229 374 239 364)(230 373 240 363)(261 331 271 321)(262 330 272 340)(263 329 273 339)(264 328 274 338)(265 327 275 337)(266 326 276 336)(267 325 277 335)(268 324 278 334)(269 323 279 333)(270 322 280 332)(281 341 291 351)(282 360 292 350)(283 359 293 349)(284 358 294 348)(285 357 295 347)(286 356 296 346)(287 355 297 345)(288 354 298 344)(289 353 299 343)(290 352 300 342)(301 414 311 404)(302 413 312 403)(303 412 313 402)(304 411 314 401)(305 410 315 420)(306 409 316 419)(307 408 317 418)(308 407 318 417)(309 406 319 416)(310 405 320 415)(381 424 391 434)(382 423 392 433)(383 422 393 432)(384 421 394 431)(385 440 395 430)(386 439 396 429)(387 438 397 428)(388 437 398 427)(389 436 399 426)(390 435 400 425)
G:=sub<Sym(480)| (1,315,371)(2,316,372)(3,317,373)(4,318,374)(5,319,375)(6,320,376)(7,301,377)(8,302,378)(9,303,379)(10,304,380)(11,305,361)(12,306,362)(13,307,363)(14,308,364)(15,309,365)(16,310,366)(17,311,367)(18,312,368)(19,313,369)(20,314,370)(21,189,71)(22,190,72)(23,191,73)(24,192,74)(25,193,75)(26,194,76)(27,195,77)(28,196,78)(29,197,79)(30,198,80)(31,199,61)(32,200,62)(33,181,63)(34,182,64)(35,183,65)(36,184,66)(37,185,67)(38,186,68)(39,187,69)(40,188,70)(41,277,241)(42,278,242)(43,279,243)(44,280,244)(45,261,245)(46,262,246)(47,263,247)(48,264,248)(49,265,249)(50,266,250)(51,267,251)(52,268,252)(53,269,253)(54,270,254)(55,271,255)(56,272,256)(57,273,257)(58,274,258)(59,275,259)(60,276,260)(81,322,161)(82,323,162)(83,324,163)(84,325,164)(85,326,165)(86,327,166)(87,328,167)(88,329,168)(89,330,169)(90,331,170)(91,332,171)(92,333,172)(93,334,173)(94,335,174)(95,336,175)(96,337,176)(97,338,177)(98,339,178)(99,340,179)(100,321,180)(101,291,383)(102,292,384)(103,293,385)(104,294,386)(105,295,387)(106,296,388)(107,297,389)(108,298,390)(109,299,391)(110,300,392)(111,281,393)(112,282,394)(113,283,395)(114,284,396)(115,285,397)(116,286,398)(117,287,399)(118,288,400)(119,289,381)(120,290,382)(121,448,149)(122,449,150)(123,450,151)(124,451,152)(125,452,153)(126,453,154)(127,454,155)(128,455,156)(129,456,157)(130,457,158)(131,458,159)(132,459,160)(133,460,141)(134,441,142)(135,442,143)(136,443,144)(137,444,145)(138,445,146)(139,446,147)(140,447,148)(201,341,432)(202,342,433)(203,343,434)(204,344,435)(205,345,436)(206,346,437)(207,347,438)(208,348,439)(209,349,440)(210,350,421)(211,351,422)(212,352,423)(213,353,424)(214,354,425)(215,355,426)(216,356,427)(217,357,428)(218,358,429)(219,359,430)(220,360,431)(221,479,419)(222,480,420)(223,461,401)(224,462,402)(225,463,403)(226,464,404)(227,465,405)(228,466,406)(229,467,407)(230,468,408)(231,469,409)(232,470,410)(233,471,411)(234,472,412)(235,473,413)(236,474,414)(237,475,415)(238,476,416)(239,477,417)(240,478,418), (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), (1,76,438,164)(2,75,439,163)(3,74,440,162)(4,73,421,161)(5,72,422,180)(6,71,423,179)(7,70,424,178)(8,69,425,177)(9,68,426,176)(10,67,427,175)(11,66,428,174)(12,65,429,173)(13,64,430,172)(14,63,431,171)(15,62,432,170)(16,61,433,169)(17,80,434,168)(18,79,435,167)(19,78,436,166)(20,77,437,165)(21,212,99,320)(22,211,100,319)(23,210,81,318)(24,209,82,317)(25,208,83,316)(26,207,84,315)(27,206,85,314)(28,205,86,313)(29,204,87,312)(30,203,88,311)(31,202,89,310)(32,201,90,309)(33,220,91,308)(34,219,92,307)(35,218,93,306)(36,217,94,305)(37,216,95,304)(38,215,96,303)(39,214,97,302)(40,213,98,301)(41,120,459,405)(42,119,460,404)(43,118,441,403)(44,117,442,402)(45,116,443,401)(46,115,444,420)(47,114,445,419)(48,113,446,418)(49,112,447,417)(50,111,448,416)(51,110,449,415)(52,109,450,414)(53,108,451,413)(54,107,452,412)(55,106,453,411)(56,105,454,410)(57,104,455,409)(58,103,456,408)(59,102,457,407)(60,101,458,406)(121,476,250,393)(122,475,251,392)(123,474,252,391)(124,473,253,390)(125,472,254,389)(126,471,255,388)(127,470,256,387)(128,469,257,386)(129,468,258,385)(130,467,259,384)(131,466,260,383)(132,465,241,382)(133,464,242,381)(134,463,243,400)(135,462,244,399)(136,461,245,398)(137,480,246,397)(138,479,247,396)(139,478,248,395)(140,477,249,394)(141,226,278,289)(142,225,279,288)(143,224,280,287)(144,223,261,286)(145,222,262,285)(146,221,263,284)(147,240,264,283)(148,239,265,282)(149,238,266,281)(150,237,267,300)(151,236,268,299)(152,235,269,298)(153,234,270,297)(154,233,271,296)(155,232,272,295)(156,231,273,294)(157,230,274,293)(158,229,275,292)(159,228,276,291)(160,227,277,290)(181,360,332,364)(182,359,333,363)(183,358,334,362)(184,357,335,361)(185,356,336,380)(186,355,337,379)(187,354,338,378)(188,353,339,377)(189,352,340,376)(190,351,321,375)(191,350,322,374)(192,349,323,373)(193,348,324,372)(194,347,325,371)(195,346,326,370)(196,345,327,369)(197,344,328,368)(198,343,329,367)(199,342,330,366)(200,341,331,365), (1,480,11,470)(2,479,12,469)(3,478,13,468)(4,477,14,467)(5,476,15,466)(6,475,16,465)(7,474,17,464)(8,473,18,463)(9,472,19,462)(10,471,20,461)(21,444,31,454)(22,443,32,453)(23,442,33,452)(24,441,34,451)(25,460,35,450)(26,459,36,449)(27,458,37,448)(28,457,38,447)(29,456,39,446)(30,455,40,445)(41,94,51,84)(42,93,52,83)(43,92,53,82)(44,91,54,81)(45,90,55,100)(46,89,56,99)(47,88,57,98)(48,87,58,97)(49,86,59,96)(50,85,60,95)(61,127,71,137)(62,126,72,136)(63,125,73,135)(64,124,74,134)(65,123,75,133)(66,122,76,132)(67,121,77,131)(68,140,78,130)(69,139,79,129)(70,138,80,128)(101,211,111,201)(102,210,112,220)(103,209,113,219)(104,208,114,218)(105,207,115,217)(106,206,116,216)(107,205,117,215)(108,204,118,214)(109,203,119,213)(110,202,120,212)(141,183,151,193)(142,182,152,192)(143,181,153,191)(144,200,154,190)(145,199,155,189)(146,198,156,188)(147,197,157,187)(148,196,158,186)(149,195,159,185)(150,194,160,184)(161,244,171,254)(162,243,172,253)(163,242,173,252)(164,241,174,251)(165,260,175,250)(166,259,176,249)(167,258,177,248)(168,257,178,247)(169,256,179,246)(170,255,180,245)(221,362,231,372)(222,361,232,371)(223,380,233,370)(224,379,234,369)(225,378,235,368)(226,377,236,367)(227,376,237,366)(228,375,238,365)(229,374,239,364)(230,373,240,363)(261,331,271,321)(262,330,272,340)(263,329,273,339)(264,328,274,338)(265,327,275,337)(266,326,276,336)(267,325,277,335)(268,324,278,334)(269,323,279,333)(270,322,280,332)(281,341,291,351)(282,360,292,350)(283,359,293,349)(284,358,294,348)(285,357,295,347)(286,356,296,346)(287,355,297,345)(288,354,298,344)(289,353,299,343)(290,352,300,342)(301,414,311,404)(302,413,312,403)(303,412,313,402)(304,411,314,401)(305,410,315,420)(306,409,316,419)(307,408,317,418)(308,407,318,417)(309,406,319,416)(310,405,320,415)(381,424,391,434)(382,423,392,433)(383,422,393,432)(384,421,394,431)(385,440,395,430)(386,439,396,429)(387,438,397,428)(388,437,398,427)(389,436,399,426)(390,435,400,425)>;
G:=Group( (1,315,371)(2,316,372)(3,317,373)(4,318,374)(5,319,375)(6,320,376)(7,301,377)(8,302,378)(9,303,379)(10,304,380)(11,305,361)(12,306,362)(13,307,363)(14,308,364)(15,309,365)(16,310,366)(17,311,367)(18,312,368)(19,313,369)(20,314,370)(21,189,71)(22,190,72)(23,191,73)(24,192,74)(25,193,75)(26,194,76)(27,195,77)(28,196,78)(29,197,79)(30,198,80)(31,199,61)(32,200,62)(33,181,63)(34,182,64)(35,183,65)(36,184,66)(37,185,67)(38,186,68)(39,187,69)(40,188,70)(41,277,241)(42,278,242)(43,279,243)(44,280,244)(45,261,245)(46,262,246)(47,263,247)(48,264,248)(49,265,249)(50,266,250)(51,267,251)(52,268,252)(53,269,253)(54,270,254)(55,271,255)(56,272,256)(57,273,257)(58,274,258)(59,275,259)(60,276,260)(81,322,161)(82,323,162)(83,324,163)(84,325,164)(85,326,165)(86,327,166)(87,328,167)(88,329,168)(89,330,169)(90,331,170)(91,332,171)(92,333,172)(93,334,173)(94,335,174)(95,336,175)(96,337,176)(97,338,177)(98,339,178)(99,340,179)(100,321,180)(101,291,383)(102,292,384)(103,293,385)(104,294,386)(105,295,387)(106,296,388)(107,297,389)(108,298,390)(109,299,391)(110,300,392)(111,281,393)(112,282,394)(113,283,395)(114,284,396)(115,285,397)(116,286,398)(117,287,399)(118,288,400)(119,289,381)(120,290,382)(121,448,149)(122,449,150)(123,450,151)(124,451,152)(125,452,153)(126,453,154)(127,454,155)(128,455,156)(129,456,157)(130,457,158)(131,458,159)(132,459,160)(133,460,141)(134,441,142)(135,442,143)(136,443,144)(137,444,145)(138,445,146)(139,446,147)(140,447,148)(201,341,432)(202,342,433)(203,343,434)(204,344,435)(205,345,436)(206,346,437)(207,347,438)(208,348,439)(209,349,440)(210,350,421)(211,351,422)(212,352,423)(213,353,424)(214,354,425)(215,355,426)(216,356,427)(217,357,428)(218,358,429)(219,359,430)(220,360,431)(221,479,419)(222,480,420)(223,461,401)(224,462,402)(225,463,403)(226,464,404)(227,465,405)(228,466,406)(229,467,407)(230,468,408)(231,469,409)(232,470,410)(233,471,411)(234,472,412)(235,473,413)(236,474,414)(237,475,415)(238,476,416)(239,477,417)(240,478,418), (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), (1,76,438,164)(2,75,439,163)(3,74,440,162)(4,73,421,161)(5,72,422,180)(6,71,423,179)(7,70,424,178)(8,69,425,177)(9,68,426,176)(10,67,427,175)(11,66,428,174)(12,65,429,173)(13,64,430,172)(14,63,431,171)(15,62,432,170)(16,61,433,169)(17,80,434,168)(18,79,435,167)(19,78,436,166)(20,77,437,165)(21,212,99,320)(22,211,100,319)(23,210,81,318)(24,209,82,317)(25,208,83,316)(26,207,84,315)(27,206,85,314)(28,205,86,313)(29,204,87,312)(30,203,88,311)(31,202,89,310)(32,201,90,309)(33,220,91,308)(34,219,92,307)(35,218,93,306)(36,217,94,305)(37,216,95,304)(38,215,96,303)(39,214,97,302)(40,213,98,301)(41,120,459,405)(42,119,460,404)(43,118,441,403)(44,117,442,402)(45,116,443,401)(46,115,444,420)(47,114,445,419)(48,113,446,418)(49,112,447,417)(50,111,448,416)(51,110,449,415)(52,109,450,414)(53,108,451,413)(54,107,452,412)(55,106,453,411)(56,105,454,410)(57,104,455,409)(58,103,456,408)(59,102,457,407)(60,101,458,406)(121,476,250,393)(122,475,251,392)(123,474,252,391)(124,473,253,390)(125,472,254,389)(126,471,255,388)(127,470,256,387)(128,469,257,386)(129,468,258,385)(130,467,259,384)(131,466,260,383)(132,465,241,382)(133,464,242,381)(134,463,243,400)(135,462,244,399)(136,461,245,398)(137,480,246,397)(138,479,247,396)(139,478,248,395)(140,477,249,394)(141,226,278,289)(142,225,279,288)(143,224,280,287)(144,223,261,286)(145,222,262,285)(146,221,263,284)(147,240,264,283)(148,239,265,282)(149,238,266,281)(150,237,267,300)(151,236,268,299)(152,235,269,298)(153,234,270,297)(154,233,271,296)(155,232,272,295)(156,231,273,294)(157,230,274,293)(158,229,275,292)(159,228,276,291)(160,227,277,290)(181,360,332,364)(182,359,333,363)(183,358,334,362)(184,357,335,361)(185,356,336,380)(186,355,337,379)(187,354,338,378)(188,353,339,377)(189,352,340,376)(190,351,321,375)(191,350,322,374)(192,349,323,373)(193,348,324,372)(194,347,325,371)(195,346,326,370)(196,345,327,369)(197,344,328,368)(198,343,329,367)(199,342,330,366)(200,341,331,365), (1,480,11,470)(2,479,12,469)(3,478,13,468)(4,477,14,467)(5,476,15,466)(6,475,16,465)(7,474,17,464)(8,473,18,463)(9,472,19,462)(10,471,20,461)(21,444,31,454)(22,443,32,453)(23,442,33,452)(24,441,34,451)(25,460,35,450)(26,459,36,449)(27,458,37,448)(28,457,38,447)(29,456,39,446)(30,455,40,445)(41,94,51,84)(42,93,52,83)(43,92,53,82)(44,91,54,81)(45,90,55,100)(46,89,56,99)(47,88,57,98)(48,87,58,97)(49,86,59,96)(50,85,60,95)(61,127,71,137)(62,126,72,136)(63,125,73,135)(64,124,74,134)(65,123,75,133)(66,122,76,132)(67,121,77,131)(68,140,78,130)(69,139,79,129)(70,138,80,128)(101,211,111,201)(102,210,112,220)(103,209,113,219)(104,208,114,218)(105,207,115,217)(106,206,116,216)(107,205,117,215)(108,204,118,214)(109,203,119,213)(110,202,120,212)(141,183,151,193)(142,182,152,192)(143,181,153,191)(144,200,154,190)(145,199,155,189)(146,198,156,188)(147,197,157,187)(148,196,158,186)(149,195,159,185)(150,194,160,184)(161,244,171,254)(162,243,172,253)(163,242,173,252)(164,241,174,251)(165,260,175,250)(166,259,176,249)(167,258,177,248)(168,257,178,247)(169,256,179,246)(170,255,180,245)(221,362,231,372)(222,361,232,371)(223,380,233,370)(224,379,234,369)(225,378,235,368)(226,377,236,367)(227,376,237,366)(228,375,238,365)(229,374,239,364)(230,373,240,363)(261,331,271,321)(262,330,272,340)(263,329,273,339)(264,328,274,338)(265,327,275,337)(266,326,276,336)(267,325,277,335)(268,324,278,334)(269,323,279,333)(270,322,280,332)(281,341,291,351)(282,360,292,350)(283,359,293,349)(284,358,294,348)(285,357,295,347)(286,356,296,346)(287,355,297,345)(288,354,298,344)(289,353,299,343)(290,352,300,342)(301,414,311,404)(302,413,312,403)(303,412,313,402)(304,411,314,401)(305,410,315,420)(306,409,316,419)(307,408,317,418)(308,407,318,417)(309,406,319,416)(310,405,320,415)(381,424,391,434)(382,423,392,433)(383,422,393,432)(384,421,394,431)(385,440,395,430)(386,439,396,429)(387,438,397,428)(388,437,398,427)(389,436,399,426)(390,435,400,425) );
G=PermutationGroup([[(1,315,371),(2,316,372),(3,317,373),(4,318,374),(5,319,375),(6,320,376),(7,301,377),(8,302,378),(9,303,379),(10,304,380),(11,305,361),(12,306,362),(13,307,363),(14,308,364),(15,309,365),(16,310,366),(17,311,367),(18,312,368),(19,313,369),(20,314,370),(21,189,71),(22,190,72),(23,191,73),(24,192,74),(25,193,75),(26,194,76),(27,195,77),(28,196,78),(29,197,79),(30,198,80),(31,199,61),(32,200,62),(33,181,63),(34,182,64),(35,183,65),(36,184,66),(37,185,67),(38,186,68),(39,187,69),(40,188,70),(41,277,241),(42,278,242),(43,279,243),(44,280,244),(45,261,245),(46,262,246),(47,263,247),(48,264,248),(49,265,249),(50,266,250),(51,267,251),(52,268,252),(53,269,253),(54,270,254),(55,271,255),(56,272,256),(57,273,257),(58,274,258),(59,275,259),(60,276,260),(81,322,161),(82,323,162),(83,324,163),(84,325,164),(85,326,165),(86,327,166),(87,328,167),(88,329,168),(89,330,169),(90,331,170),(91,332,171),(92,333,172),(93,334,173),(94,335,174),(95,336,175),(96,337,176),(97,338,177),(98,339,178),(99,340,179),(100,321,180),(101,291,383),(102,292,384),(103,293,385),(104,294,386),(105,295,387),(106,296,388),(107,297,389),(108,298,390),(109,299,391),(110,300,392),(111,281,393),(112,282,394),(113,283,395),(114,284,396),(115,285,397),(116,286,398),(117,287,399),(118,288,400),(119,289,381),(120,290,382),(121,448,149),(122,449,150),(123,450,151),(124,451,152),(125,452,153),(126,453,154),(127,454,155),(128,455,156),(129,456,157),(130,457,158),(131,458,159),(132,459,160),(133,460,141),(134,441,142),(135,442,143),(136,443,144),(137,444,145),(138,445,146),(139,446,147),(140,447,148),(201,341,432),(202,342,433),(203,343,434),(204,344,435),(205,345,436),(206,346,437),(207,347,438),(208,348,439),(209,349,440),(210,350,421),(211,351,422),(212,352,423),(213,353,424),(214,354,425),(215,355,426),(216,356,427),(217,357,428),(218,358,429),(219,359,430),(220,360,431),(221,479,419),(222,480,420),(223,461,401),(224,462,402),(225,463,403),(226,464,404),(227,465,405),(228,466,406),(229,467,407),(230,468,408),(231,469,409),(232,470,410),(233,471,411),(234,472,412),(235,473,413),(236,474,414),(237,475,415),(238,476,416),(239,477,417),(240,478,418)], [(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)], [(1,76,438,164),(2,75,439,163),(3,74,440,162),(4,73,421,161),(5,72,422,180),(6,71,423,179),(7,70,424,178),(8,69,425,177),(9,68,426,176),(10,67,427,175),(11,66,428,174),(12,65,429,173),(13,64,430,172),(14,63,431,171),(15,62,432,170),(16,61,433,169),(17,80,434,168),(18,79,435,167),(19,78,436,166),(20,77,437,165),(21,212,99,320),(22,211,100,319),(23,210,81,318),(24,209,82,317),(25,208,83,316),(26,207,84,315),(27,206,85,314),(28,205,86,313),(29,204,87,312),(30,203,88,311),(31,202,89,310),(32,201,90,309),(33,220,91,308),(34,219,92,307),(35,218,93,306),(36,217,94,305),(37,216,95,304),(38,215,96,303),(39,214,97,302),(40,213,98,301),(41,120,459,405),(42,119,460,404),(43,118,441,403),(44,117,442,402),(45,116,443,401),(46,115,444,420),(47,114,445,419),(48,113,446,418),(49,112,447,417),(50,111,448,416),(51,110,449,415),(52,109,450,414),(53,108,451,413),(54,107,452,412),(55,106,453,411),(56,105,454,410),(57,104,455,409),(58,103,456,408),(59,102,457,407),(60,101,458,406),(121,476,250,393),(122,475,251,392),(123,474,252,391),(124,473,253,390),(125,472,254,389),(126,471,255,388),(127,470,256,387),(128,469,257,386),(129,468,258,385),(130,467,259,384),(131,466,260,383),(132,465,241,382),(133,464,242,381),(134,463,243,400),(135,462,244,399),(136,461,245,398),(137,480,246,397),(138,479,247,396),(139,478,248,395),(140,477,249,394),(141,226,278,289),(142,225,279,288),(143,224,280,287),(144,223,261,286),(145,222,262,285),(146,221,263,284),(147,240,264,283),(148,239,265,282),(149,238,266,281),(150,237,267,300),(151,236,268,299),(152,235,269,298),(153,234,270,297),(154,233,271,296),(155,232,272,295),(156,231,273,294),(157,230,274,293),(158,229,275,292),(159,228,276,291),(160,227,277,290),(181,360,332,364),(182,359,333,363),(183,358,334,362),(184,357,335,361),(185,356,336,380),(186,355,337,379),(187,354,338,378),(188,353,339,377),(189,352,340,376),(190,351,321,375),(191,350,322,374),(192,349,323,373),(193,348,324,372),(194,347,325,371),(195,346,326,370),(196,345,327,369),(197,344,328,368),(198,343,329,367),(199,342,330,366),(200,341,331,365)], [(1,480,11,470),(2,479,12,469),(3,478,13,468),(4,477,14,467),(5,476,15,466),(6,475,16,465),(7,474,17,464),(8,473,18,463),(9,472,19,462),(10,471,20,461),(21,444,31,454),(22,443,32,453),(23,442,33,452),(24,441,34,451),(25,460,35,450),(26,459,36,449),(27,458,37,448),(28,457,38,447),(29,456,39,446),(30,455,40,445),(41,94,51,84),(42,93,52,83),(43,92,53,82),(44,91,54,81),(45,90,55,100),(46,89,56,99),(47,88,57,98),(48,87,58,97),(49,86,59,96),(50,85,60,95),(61,127,71,137),(62,126,72,136),(63,125,73,135),(64,124,74,134),(65,123,75,133),(66,122,76,132),(67,121,77,131),(68,140,78,130),(69,139,79,129),(70,138,80,128),(101,211,111,201),(102,210,112,220),(103,209,113,219),(104,208,114,218),(105,207,115,217),(106,206,116,216),(107,205,117,215),(108,204,118,214),(109,203,119,213),(110,202,120,212),(141,183,151,193),(142,182,152,192),(143,181,153,191),(144,200,154,190),(145,199,155,189),(146,198,156,188),(147,197,157,187),(148,196,158,186),(149,195,159,185),(150,194,160,184),(161,244,171,254),(162,243,172,253),(163,242,173,252),(164,241,174,251),(165,260,175,250),(166,259,176,249),(167,258,177,248),(168,257,178,247),(169,256,179,246),(170,255,180,245),(221,362,231,372),(222,361,232,371),(223,380,233,370),(224,379,234,369),(225,378,235,368),(226,377,236,367),(227,376,237,366),(228,375,238,365),(229,374,239,364),(230,373,240,363),(261,331,271,321),(262,330,272,340),(263,329,273,339),(264,328,274,338),(265,327,275,337),(266,326,276,336),(267,325,277,335),(268,324,278,334),(269,323,279,333),(270,322,280,332),(281,341,291,351),(282,360,292,350),(283,359,293,349),(284,358,294,348),(285,357,295,347),(286,356,296,346),(287,355,297,345),(288,354,298,344),(289,353,299,343),(290,352,300,342),(301,414,311,404),(302,413,312,403),(303,412,313,402),(304,411,314,401),(305,410,315,420),(306,409,316,419),(307,408,317,418),(308,407,318,417),(309,406,319,416),(310,405,320,415),(381,424,391,434),(382,423,392,433),(383,422,393,432),(384,421,394,431),(385,440,395,430),(386,439,396,429),(387,438,397,428),(388,437,398,427),(389,436,399,426),(390,435,400,425)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
138 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | - | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | D5 | SD16 | Q16 | D10 | C3×D4 | C3×D4 | C3×D5 | C4×D5 | C5⋊D4 | D20 | C3×SD16 | C3×Q16 | C6×D5 | C40⋊C2 | Dic20 | D5×C12 | C3×C5⋊D4 | C3×D20 | C3×C40⋊C2 | C3×Dic20 |
| kernel | C3×C20.44D4 | C3×C4⋊Dic5 | C2×C120 | C6×Dic10 | C20.44D4 | C3×Dic10 | C4⋊Dic5 | C2×C40 | C2×Dic10 | Dic10 | C60 | C2×C30 | C2×C24 | C30 | C30 | C2×C12 | C20 | C2×C10 | C2×C8 | C12 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C3×C20.44D4 ►in GL3(𝔽241) generated by
| 15 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 240 | 0 | 0 |
| 0 | 156 | 200 |
| 0 | 41 | 119 |
| 177 | 0 | 0 |
| 0 | 225 | 88 |
| 0 | 197 | 16 |
| 1 | 0 | 0 |
| 0 | 150 | 186 |
| 0 | 98 | 91 |
G:=sub<GL(3,GF(241))| [15,0,0,0,1,0,0,0,1],[240,0,0,0,156,41,0,200,119],[177,0,0,0,225,197,0,88,16],[1,0,0,0,150,98,0,186,91] >;
C3×C20.44D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}._{44}D_4 % in TeX
G:=Group("C3xC20.44D4"); // GroupNames label
G:=SmallGroup(480,94);
// by ID
G=gap.SmallGroup(480,94);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,168,197,260,1683,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^-1>;
// generators/relations