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## G = C3×C20.44D4order 480 = 25·3·5

### Direct product of C3 and C20.44D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C20.44D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C3×C4⋊Dic5 — C3×C20.44D4
 Lower central C5 — C10 — C20 — C3×C20.44D4
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

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 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], C12 [×2], C12 [×3], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5 [×3], C20 [×2], C2×C10, C24, C2×C12, C2×C12 [×2], C3×Q8 [×3], C30 [×3], Q8⋊C4, C40, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20, C3×C4⋊C4, C2×C24, C6×Q8, C3×Dic5 [×3], C60 [×2], C2×C30, C4⋊Dic5, C2×C40, C2×Dic10, C3×Q8⋊C4, C120, C3×Dic10 [×2], C3×Dic10, C6×Dic5 [×2], C2×C60, C20.44D4, C3×C4⋊Dic5, C2×C120, C6×Dic10, C3×C20.44D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, SD16, Q16, D10, C2×C12, C3×D4 [×2], 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

Smallest permutation representation of C3×C20.44D4
Regular action on 480 points
Generators in S480
(1 339 103)(2 340 104)(3 321 105)(4 322 106)(5 323 107)(6 324 108)(7 325 109)(8 326 110)(9 327 111)(10 328 112)(11 329 113)(12 330 114)(13 331 115)(14 332 116)(15 333 117)(16 334 118)(17 335 119)(18 336 120)(19 337 101)(20 338 102)(21 414 362)(22 415 363)(23 416 364)(24 417 365)(25 418 366)(26 419 367)(27 420 368)(28 401 369)(29 402 370)(30 403 371)(31 404 372)(32 405 373)(33 406 374)(34 407 375)(35 408 376)(36 409 377)(37 410 378)(38 411 379)(39 412 380)(40 413 361)(41 91 222)(42 92 223)(43 93 224)(44 94 225)(45 95 226)(46 96 227)(47 97 228)(48 98 229)(49 99 230)(50 100 231)(51 81 232)(52 82 233)(53 83 234)(54 84 235)(55 85 236)(56 86 237)(57 87 238)(58 88 239)(59 89 240)(60 90 221)(61 441 478)(62 442 479)(63 443 480)(64 444 461)(65 445 462)(66 446 463)(67 447 464)(68 448 465)(69 449 466)(70 450 467)(71 451 468)(72 452 469)(73 453 470)(74 454 471)(75 455 472)(76 456 473)(77 457 474)(78 458 475)(79 459 476)(80 460 477)(121 197 394)(122 198 395)(123 199 396)(124 200 397)(125 181 398)(126 182 399)(127 183 400)(128 184 381)(129 185 382)(130 186 383)(131 187 384)(132 188 385)(133 189 386)(134 190 387)(135 191 388)(136 192 389)(137 193 390)(138 194 391)(139 195 392)(140 196 393)(141 171 358)(142 172 359)(143 173 360)(144 174 341)(145 175 342)(146 176 343)(147 177 344)(148 178 345)(149 179 346)(150 180 347)(151 161 348)(152 162 349)(153 163 350)(154 164 351)(155 165 352)(156 166 353)(157 167 354)(158 168 355)(159 169 356)(160 170 357)(201 319 262)(202 320 263)(203 301 264)(204 302 265)(205 303 266)(206 304 267)(207 305 268)(208 306 269)(209 307 270)(210 308 271)(211 309 272)(212 310 273)(213 311 274)(214 312 275)(215 313 276)(216 314 277)(217 315 278)(218 316 279)(219 317 280)(220 318 261)(241 281 427)(242 282 428)(243 283 429)(244 284 430)(245 285 431)(246 286 432)(247 287 433)(248 288 434)(249 289 435)(250 290 436)(251 291 437)(252 292 438)(253 293 439)(254 294 440)(255 295 421)(256 296 422)(257 297 423)(258 298 424)(259 299 425)(260 300 426)
(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 67 365 313)(2 66 366 312)(3 65 367 311)(4 64 368 310)(5 63 369 309)(6 62 370 308)(7 61 371 307)(8 80 372 306)(9 79 373 305)(10 78 374 304)(11 77 375 303)(12 76 376 302)(13 75 377 301)(14 74 378 320)(15 73 379 319)(16 72 380 318)(17 71 361 317)(18 70 362 316)(19 69 363 315)(20 68 364 314)(21 279 336 450)(22 278 337 449)(23 277 338 448)(24 276 339 447)(25 275 340 446)(26 274 321 445)(27 273 322 444)(28 272 323 443)(29 271 324 442)(30 270 325 441)(31 269 326 460)(32 268 327 459)(33 267 328 458)(34 266 329 457)(35 265 330 456)(36 264 331 455)(37 263 332 454)(38 262 333 453)(39 261 334 452)(40 280 335 451)(41 139 173 299)(42 138 174 298)(43 137 175 297)(44 136 176 296)(45 135 177 295)(46 134 178 294)(47 133 179 293)(48 132 180 292)(49 131 161 291)(50 130 162 290)(51 129 163 289)(52 128 164 288)(53 127 165 287)(54 126 166 286)(55 125 167 285)(56 124 168 284)(57 123 169 283)(58 122 170 282)(59 121 171 281)(60 140 172 300)(81 185 350 435)(82 184 351 434)(83 183 352 433)(84 182 353 432)(85 181 354 431)(86 200 355 430)(87 199 356 429)(88 198 357 428)(89 197 358 427)(90 196 359 426)(91 195 360 425)(92 194 341 424)(93 193 342 423)(94 192 343 422)(95 191 344 421)(96 190 345 440)(97 189 346 439)(98 188 347 438)(99 187 348 437)(100 186 349 436)(101 466 415 217)(102 465 416 216)(103 464 417 215)(104 463 418 214)(105 462 419 213)(106 461 420 212)(107 480 401 211)(108 479 402 210)(109 478 403 209)(110 477 404 208)(111 476 405 207)(112 475 406 206)(113 474 407 205)(114 473 408 204)(115 472 409 203)(116 471 410 202)(117 470 411 201)(118 469 412 220)(119 468 413 219)(120 467 414 218)(141 241 240 394)(142 260 221 393)(143 259 222 392)(144 258 223 391)(145 257 224 390)(146 256 225 389)(147 255 226 388)(148 254 227 387)(149 253 228 386)(150 252 229 385)(151 251 230 384)(152 250 231 383)(153 249 232 382)(154 248 233 381)(155 247 234 400)(156 246 235 399)(157 245 236 398)(158 244 237 397)(159 243 238 396)(160 242 239 395)
(1 49 11 59)(2 48 12 58)(3 47 13 57)(4 46 14 56)(5 45 15 55)(6 44 16 54)(7 43 17 53)(8 42 18 52)(9 41 19 51)(10 60 20 50)(21 351 31 341)(22 350 32 360)(23 349 33 359)(24 348 34 358)(25 347 35 357)(26 346 36 356)(27 345 37 355)(28 344 38 354)(29 343 39 353)(30 342 40 352)(61 292 71 282)(62 291 72 281)(63 290 73 300)(64 289 74 299)(65 288 75 298)(66 287 76 297)(67 286 77 296)(68 285 78 295)(69 284 79 294)(70 283 80 293)(81 327 91 337)(82 326 92 336)(83 325 93 335)(84 324 94 334)(85 323 95 333)(86 322 96 332)(87 321 97 331)(88 340 98 330)(89 339 99 329)(90 338 100 328)(101 232 111 222)(102 231 112 221)(103 230 113 240)(104 229 114 239)(105 228 115 238)(106 227 116 237)(107 226 117 236)(108 225 118 235)(109 224 119 234)(110 223 120 233)(121 308 131 318)(122 307 132 317)(123 306 133 316)(124 305 134 315)(125 304 135 314)(126 303 136 313)(127 302 137 312)(128 301 138 311)(129 320 139 310)(130 319 140 309)(141 417 151 407)(142 416 152 406)(143 415 153 405)(144 414 154 404)(145 413 155 403)(146 412 156 402)(147 411 157 401)(148 410 158 420)(149 409 159 419)(150 408 160 418)(161 375 171 365)(162 374 172 364)(163 373 173 363)(164 372 174 362)(165 371 175 361)(166 370 176 380)(167 369 177 379)(168 368 178 378)(169 367 179 377)(170 366 180 376)(181 267 191 277)(182 266 192 276)(183 265 193 275)(184 264 194 274)(185 263 195 273)(186 262 196 272)(187 261 197 271)(188 280 198 270)(189 279 199 269)(190 278 200 268)(201 393 211 383)(202 392 212 382)(203 391 213 381)(204 390 214 400)(205 389 215 399)(206 388 216 398)(207 387 217 397)(208 386 218 396)(209 385 219 395)(210 384 220 394)(241 479 251 469)(242 478 252 468)(243 477 253 467)(244 476 254 466)(245 475 255 465)(246 474 256 464)(247 473 257 463)(248 472 258 462)(249 471 259 461)(250 470 260 480)(421 448 431 458)(422 447 432 457)(423 446 433 456)(424 445 434 455)(425 444 435 454)(426 443 436 453)(427 442 437 452)(428 441 438 451)(429 460 439 450)(430 459 440 449)

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

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

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

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

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