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G = C3×C202Q8order 480 = 25·3·5

Direct product of C3 and C202Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3×C202Q8, C6010Q8, C129Dic10, C12.42D20, C60.172D4, C202(C3×Q8), (C4×C20).2C6, (C4×C60).8C2, C1511(C4⋊Q8), C4.4(C3×D20), C10.1(C6×D4), C2.4(C6×D20), C10.2(C6×Q8), C42(C3×Dic10), C6.72(C2×D20), (C4×C12).10D5, C20.27(C3×D4), C4⋊Dic5.4C6, C30.74(C2×Q8), C42.4(C3×D5), C30.274(C2×D4), C2.4(C6×Dic10), (C2×C12).426D10, (C2×Dic10).3C6, C6.42(C2×Dic10), (C2×C60).506C22, (C2×C30).327C23, (C6×Dic10).14C2, (C6×Dic5).150C22, C51(C3×C4⋊Q8), (C2×C4).74(C6×D5), C22.34(D5×C2×C6), (C2×C20).89(C2×C6), (C3×C4⋊Dic5).18C2, (C2×Dic5).1(C2×C6), (C2×C10).10(C22×C6), (C2×C6).323(C22×D5), SmallGroup(480,662)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C202Q8
C1C5C10C2×C10C2×C30C6×Dic5C6×Dic10 — C3×C202Q8
C5C2×C10 — C3×C202Q8
C1C2×C6C4×C12

Generators and relations for C3×C202Q8
 G = < a,b,c,d | a3=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 384 in 136 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C4 [×4], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], C12 [×6], C12 [×4], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×6], C2×C10, C2×C12, C2×C12 [×2], C2×C12 [×4], C3×Q8 [×4], C30, C30 [×2], C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C4×C12, C3×C4⋊C4 [×4], C6×Q8 [×2], C3×Dic5 [×4], C60 [×6], C2×C30, C4⋊Dic5 [×4], C4×C20, C2×Dic10 [×2], C3×C4⋊Q8, C3×Dic10 [×4], C6×Dic5 [×4], C2×C60, C2×C60 [×2], C202Q8, C3×C4⋊Dic5 [×4], C4×C60, C6×Dic10 [×2], C3×C202Q8
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], Q8 [×4], C23, D5, C2×C6 [×7], C2×D4, C2×Q8 [×2], D10 [×3], C3×D4 [×2], C3×Q8 [×4], C22×C6, C3×D5, C4⋊Q8, Dic10 [×4], D20 [×2], C22×D5, C6×D4, C6×Q8 [×2], C6×D5 [×3], C2×Dic10 [×2], C2×D20, C3×C4⋊Q8, C3×Dic10 [×4], C3×D20 [×2], D5×C2×C6, C202Q8, C6×Dic10 [×2], C6×D20, C3×C202Q8

Smallest permutation representation of C3×C202Q8
Regular action on 480 points
Generators in S480
(1 67 474)(2 68 475)(3 69 476)(4 70 477)(5 71 478)(6 72 479)(7 73 480)(8 74 461)(9 75 462)(10 76 463)(11 77 464)(12 78 465)(13 79 466)(14 80 467)(15 61 468)(16 62 469)(17 63 470)(18 64 471)(19 65 472)(20 66 473)(21 150 414)(22 151 415)(23 152 416)(24 153 417)(25 154 418)(26 155 419)(27 156 420)(28 157 401)(29 158 402)(30 159 403)(31 160 404)(32 141 405)(33 142 406)(34 143 407)(35 144 408)(36 145 409)(37 146 410)(38 147 411)(39 148 412)(40 149 413)(41 261 367)(42 262 368)(43 263 369)(44 264 370)(45 265 371)(46 266 372)(47 267 373)(48 268 374)(49 269 375)(50 270 376)(51 271 377)(52 272 378)(53 273 379)(54 274 380)(55 275 361)(56 276 362)(57 277 363)(58 278 364)(59 279 365)(60 280 366)(81 306 101)(82 307 102)(83 308 103)(84 309 104)(85 310 105)(86 311 106)(87 312 107)(88 313 108)(89 314 109)(90 315 110)(91 316 111)(92 317 112)(93 318 113)(94 319 114)(95 320 115)(96 301 116)(97 302 117)(98 303 118)(99 304 119)(100 305 120)(121 204 399)(122 205 400)(123 206 381)(124 207 382)(125 208 383)(126 209 384)(127 210 385)(128 211 386)(129 212 387)(130 213 388)(131 214 389)(132 215 390)(133 216 391)(134 217 392)(135 218 393)(136 219 394)(137 220 395)(138 201 396)(139 202 397)(140 203 398)(161 289 192)(162 290 193)(163 291 194)(164 292 195)(165 293 196)(166 294 197)(167 295 198)(168 296 199)(169 297 200)(170 298 181)(171 299 182)(172 300 183)(173 281 184)(174 282 185)(175 283 186)(176 284 187)(177 285 188)(178 286 189)(179 287 190)(180 288 191)(221 456 243)(222 457 244)(223 458 245)(224 459 246)(225 460 247)(226 441 248)(227 442 249)(228 443 250)(229 444 251)(230 445 252)(231 446 253)(232 447 254)(233 448 255)(234 449 256)(235 450 257)(236 451 258)(237 452 259)(238 453 260)(239 454 241)(240 455 242)(321 426 356)(322 427 357)(323 428 358)(324 429 359)(325 430 360)(326 431 341)(327 432 342)(328 433 343)(329 434 344)(330 435 345)(331 436 346)(332 437 347)(333 438 348)(334 439 349)(335 440 350)(336 421 351)(337 422 352)(338 423 353)(339 424 354)(340 425 355)
(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)
(1 219 41 86)(2 220 42 87)(3 201 43 88)(4 202 44 89)(5 203 45 90)(6 204 46 91)(7 205 47 92)(8 206 48 93)(9 207 49 94)(10 208 50 95)(11 209 51 96)(12 210 52 97)(13 211 53 98)(14 212 54 99)(15 213 55 100)(16 214 56 81)(17 215 57 82)(18 216 58 83)(19 217 59 84)(20 218 60 85)(21 349 197 456)(22 350 198 457)(23 351 199 458)(24 352 200 459)(25 353 181 460)(26 354 182 441)(27 355 183 442)(28 356 184 443)(29 357 185 444)(30 358 186 445)(31 359 187 446)(32 360 188 447)(33 341 189 448)(34 342 190 449)(35 343 191 450)(36 344 192 451)(37 345 193 452)(38 346 194 453)(39 347 195 454)(40 348 196 455)(61 388 275 305)(62 389 276 306)(63 390 277 307)(64 391 278 308)(65 392 279 309)(66 393 280 310)(67 394 261 311)(68 395 262 312)(69 396 263 313)(70 397 264 314)(71 398 265 315)(72 399 266 316)(73 400 267 317)(74 381 268 318)(75 382 269 319)(76 383 270 320)(77 384 271 301)(78 385 272 302)(79 386 273 303)(80 387 274 304)(101 469 131 362)(102 470 132 363)(103 471 133 364)(104 472 134 365)(105 473 135 366)(106 474 136 367)(107 475 137 368)(108 476 138 369)(109 477 139 370)(110 478 140 371)(111 479 121 372)(112 480 122 373)(113 461 123 374)(114 462 124 375)(115 463 125 376)(116 464 126 377)(117 465 127 378)(118 466 128 379)(119 467 129 380)(120 468 130 361)(141 325 177 254)(142 326 178 255)(143 327 179 256)(144 328 180 257)(145 329 161 258)(146 330 162 259)(147 331 163 260)(148 332 164 241)(149 333 165 242)(150 334 166 243)(151 335 167 244)(152 336 168 245)(153 337 169 246)(154 338 170 247)(155 339 171 248)(156 340 172 249)(157 321 173 250)(158 322 174 251)(159 323 175 252)(160 324 176 253)(221 414 439 294)(222 415 440 295)(223 416 421 296)(224 417 422 297)(225 418 423 298)(226 419 424 299)(227 420 425 300)(228 401 426 281)(229 402 427 282)(230 403 428 283)(231 404 429 284)(232 405 430 285)(233 406 431 286)(234 407 432 287)(235 408 433 288)(236 409 434 289)(237 410 435 290)(238 411 436 291)(239 412 437 292)(240 413 438 293)
(1 430 41 232)(2 429 42 231)(3 428 43 230)(4 427 44 229)(5 426 45 228)(6 425 46 227)(7 424 47 226)(8 423 48 225)(9 422 49 224)(10 421 50 223)(11 440 51 222)(12 439 52 221)(13 438 53 240)(14 437 54 239)(15 436 55 238)(16 435 56 237)(17 434 57 236)(18 433 58 235)(19 432 59 234)(20 431 60 233)(21 302 197 385)(22 301 198 384)(23 320 199 383)(24 319 200 382)(25 318 181 381)(26 317 182 400)(27 316 183 399)(28 315 184 398)(29 314 185 397)(30 313 186 396)(31 312 187 395)(32 311 188 394)(33 310 189 393)(34 309 190 392)(35 308 191 391)(36 307 192 390)(37 306 193 389)(38 305 194 388)(39 304 195 387)(40 303 196 386)(61 346 275 453)(62 345 276 452)(63 344 277 451)(64 343 278 450)(65 342 279 449)(66 341 280 448)(67 360 261 447)(68 359 262 446)(69 358 263 445)(70 357 264 444)(71 356 265 443)(72 355 266 442)(73 354 267 441)(74 353 268 460)(75 352 269 459)(76 351 270 458)(77 350 271 457)(78 349 272 456)(79 348 273 455)(80 347 274 454)(81 290 214 410)(82 289 215 409)(83 288 216 408)(84 287 217 407)(85 286 218 406)(86 285 219 405)(87 284 220 404)(88 283 201 403)(89 282 202 402)(90 281 203 401)(91 300 204 420)(92 299 205 419)(93 298 206 418)(94 297 207 417)(95 296 208 416)(96 295 209 415)(97 294 210 414)(98 293 211 413)(99 292 212 412)(100 291 213 411)(101 162 131 146)(102 161 132 145)(103 180 133 144)(104 179 134 143)(105 178 135 142)(106 177 136 141)(107 176 137 160)(108 175 138 159)(109 174 139 158)(110 173 140 157)(111 172 121 156)(112 171 122 155)(113 170 123 154)(114 169 124 153)(115 168 125 152)(116 167 126 151)(117 166 127 150)(118 165 128 149)(119 164 129 148)(120 163 130 147)(241 467 332 380)(242 466 333 379)(243 465 334 378)(244 464 335 377)(245 463 336 376)(246 462 337 375)(247 461 338 374)(248 480 339 373)(249 479 340 372)(250 478 321 371)(251 477 322 370)(252 476 323 369)(253 475 324 368)(254 474 325 367)(255 473 326 366)(256 472 327 365)(257 471 328 364)(258 470 329 363)(259 469 330 362)(260 468 331 361)

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

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

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

138 conjugacy classes

class 1 2A2B2C3A3B4A···4F4G4H4I4J5A5B6A···6F10A···10F12A···12L12M···12T15A15B15C15D20A···20X30A···30L60A···60AV
order1222334···44444556···610···1012···1212···121515151520···2030···3060···60
size1111112···220202020221···12···22···220···2022222···22···22···2

138 irreducible representations

dim11111111222222222222
type+++++-++-+
imageC1C2C2C2C3C6C6C6D4Q8D5D10C3×D4C3×Q8C3×D5Dic10D20C6×D5C3×Dic10C3×D20
kernelC3×C202Q8C3×C4⋊Dic5C4×C60C6×Dic10C202Q8C4⋊Dic5C4×C20C2×Dic10C60C60C4×C12C2×C12C20C20C42C12C12C2×C4C4C4
# reps141228242426484168123216

Matrix representation of C3×C202Q8 in GL6(𝔽61)

100000
010000
0047000
0004700
0000130
0000013
,
0600000
100000
0060100
00164400
0000600
0000060
,
0600000
100000
001000
000100
0000272
0000134
,
16320000
32450000
00272800
00353400
00002657
00003235

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,60,16,0,0,0,0,1,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,1,0,0,0,0,2,34],[16,32,0,0,0,0,32,45,0,0,0,0,0,0,27,35,0,0,0,0,28,34,0,0,0,0,0,0,26,32,0,0,0,0,57,35] >;

C3×C202Q8 in GAP, Magma, Sage, TeX

C_3\times C_{20}\rtimes_2Q_8
% in TeX

G:=Group("C3xC20:2Q8");
// GroupNames label

G:=SmallGroup(480,662);
// by ID

G=gap.SmallGroup(480,662);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,701,344,590,142,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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