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## G = C3×C20.8Q8order 480 = 25·3·5

### Direct product of C3 and C20.8Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C20.8Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C12×Dic5 — C3×C20.8Q8
 Lower central C5 — C10 — C3×C20.8Q8
 Upper central C1 — C2×C12 — C2×C24

Generators and relations for C3×C20.8Q8
G = < a,b,c,d | a3=b20=1, c4=b10, d2=b15c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b9, dcd-1=b15c3 >

Subgroups: 176 in 76 conjugacy classes, 50 normal (46 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C42, C2×C8, C2×C8, Dic5, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C30, C4⋊C8, C52C8, C40, C2×Dic5, C2×C20, C4×C12, C2×C24, C2×C24, C3×Dic5, C3×Dic5, C60, C2×C30, C2×C52C8, C4×Dic5, C2×C40, C3×C4⋊C8, C3×C52C8, C120, C6×Dic5, C2×C60, C20.8Q8, C6×C52C8, C12×Dic5, C2×C120, C3×C20.8Q8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, Q8, D5, C12, C2×C6, C4⋊C4, C2×C8, M4(2), D10, C24, C2×C12, C3×D4, C3×Q8, C3×D5, C4⋊C8, Dic10, C4×D5, C5⋊D4, C3×C4⋊C4, C2×C24, C3×M4(2), C6×D5, C8×D5, C8⋊D5, C10.D4, C3×C4⋊C8, C3×Dic10, D5×C12, C3×C5⋊D4, C20.8Q8, D5×C24, C3×C8⋊D5, C3×C10.D4, C3×C20.8Q8

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

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

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

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

156 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A ··· 6F 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 12A ··· 12H 12I ··· 12P 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 24I ··· 24P 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 ··· 6 8 8 8 8 8 8 8 8 10 ··· 10 12 ··· 12 12 ··· 12 15 15 15 15 20 ··· 20 24 ··· 24 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 1 1 1 1 1 1 10 10 10 10 2 2 1 ··· 1 2 2 2 2 10 10 10 10 2 ··· 2 1 ··· 1 10 ··· 10 2 2 2 2 2 ··· 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

156 irreducible representations

 dim 1 1 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 type + + + + + - + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C8 C12 C24 D4 Q8 D5 M4(2) D10 C3×D4 C3×Q8 C3×D5 Dic10 C5⋊D4 C4×D5 C3×M4(2) C6×D5 C8×D5 C8⋊D5 C3×Dic10 C3×C5⋊D4 D5×C12 D5×C24 C3×C8⋊D5 kernel C3×C20.8Q8 C6×C5⋊2C8 C12×Dic5 C2×C120 C20.8Q8 C6×Dic5 C2×C5⋊2C8 C4×Dic5 C2×C40 C3×Dic5 C2×Dic5 Dic5 C60 C60 C2×C24 C30 C2×C12 C20 C20 C2×C8 C12 C12 C2×C6 C10 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 8 16 1 1 2 2 2 2 2 4 4 4 4 4 4 8 8 8 8 8 16 16

Matrix representation of C3×C20.8Q8 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 225 0 0 0 0 225
,
 177 0 0 0 0 177 0 0 0 0 177 64 0 0 223 195
,
 0 1 0 0 64 0 0 0 0 0 8 0 0 0 0 8
,
 147 163 0 0 172 94 0 0 0 0 20 208 0 0 34 221
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[177,0,0,0,0,177,0,0,0,0,177,223,0,0,64,195],[0,64,0,0,1,0,0,0,0,0,8,0,0,0,0,8],[147,172,0,0,163,94,0,0,0,0,20,34,0,0,208,221] >;

C3×C20.8Q8 in GAP, Magma, Sage, TeX

C_3\times C_{20}._8Q_8
% in TeX

G:=Group("C3xC20.8Q8");
// GroupNames label

G:=SmallGroup(480,92);
// by ID

G=gap.SmallGroup(480,92);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,168,365,92,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=1,c^4=b^10,d^2=b^15*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

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