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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 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], C12 [×2], C12 [×3], C2×C6, C15, C42, C2×C8, C2×C8, Dic5 [×2], Dic5, C20 [×2], C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4⋊C8, C52C8, C40, C2×Dic5 [×2], C2×C20, C4×C12, C2×C24, C2×C24, C3×Dic5 [×2], C3×Dic5, C60 [×2], C2×C30, C2×C52C8, C4×Dic5, C2×C40, C3×C4⋊C8, C3×C52C8, C120, C6×Dic5 [×2], C2×C60, C20.8Q8, C6×C52C8, C12×Dic5, C2×C120, C3×C20.8Q8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C8 [×2], C2×C4, D4, Q8, D5, C12 [×2], C2×C6, C4⋊C4, C2×C8, M4(2), D10, C24 [×2], 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
(1 248 211)(2 249 212)(3 250 213)(4 251 214)(5 252 215)(6 253 216)(7 254 217)(8 255 218)(9 256 219)(10 257 220)(11 258 201)(12 259 202)(13 260 203)(14 241 204)(15 242 205)(16 243 206)(17 244 207)(18 245 208)(19 246 209)(20 247 210)(21 424 221)(22 425 222)(23 426 223)(24 427 224)(25 428 225)(26 429 226)(27 430 227)(28 431 228)(29 432 229)(30 433 230)(31 434 231)(32 435 232)(33 436 233)(34 437 234)(35 438 235)(36 439 236)(37 440 237)(38 421 238)(39 422 239)(40 423 240)(41 406 153)(42 407 154)(43 408 155)(44 409 156)(45 410 157)(46 411 158)(47 412 159)(48 413 160)(49 414 141)(50 415 142)(51 416 143)(52 417 144)(53 418 145)(54 419 146)(55 420 147)(56 401 148)(57 402 149)(58 403 150)(59 404 151)(60 405 152)(61 337 307)(62 338 308)(63 339 309)(64 340 310)(65 321 311)(66 322 312)(67 323 313)(68 324 314)(69 325 315)(70 326 316)(71 327 317)(72 328 318)(73 329 319)(74 330 320)(75 331 301)(76 332 302)(77 333 303)(78 334 304)(79 335 305)(80 336 306)(81 449 195)(82 450 196)(83 451 197)(84 452 198)(85 453 199)(86 454 200)(87 455 181)(88 456 182)(89 457 183)(90 458 184)(91 459 185)(92 460 186)(93 441 187)(94 442 188)(95 443 189)(96 444 190)(97 445 191)(98 446 192)(99 447 193)(100 448 194)(101 166 278)(102 167 279)(103 168 280)(104 169 261)(105 170 262)(106 171 263)(107 172 264)(108 173 265)(109 174 266)(110 175 267)(111 176 268)(112 177 269)(113 178 270)(114 179 271)(115 180 272)(116 161 273)(117 162 274)(118 163 275)(119 164 276)(120 165 277)(121 480 380)(122 461 361)(123 462 362)(124 463 363)(125 464 364)(126 465 365)(127 466 366)(128 467 367)(129 468 368)(130 469 369)(131 470 370)(132 471 371)(133 472 372)(134 473 373)(135 474 374)(136 475 375)(137 476 376)(138 477 377)(139 478 378)(140 479 379)(281 353 389)(282 354 390)(283 355 391)(284 356 392)(285 357 393)(286 358 394)(287 359 395)(288 360 396)(289 341 397)(290 342 398)(291 343 399)(292 344 400)(293 345 381)(294 346 382)(295 347 383)(296 348 384)(297 349 385)(298 350 386)(299 351 387)(300 352 388)
(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)
(1 319 28 293 11 309 38 283)(2 320 29 294 12 310 39 284)(3 301 30 295 13 311 40 285)(4 302 31 296 14 312 21 286)(5 303 32 297 15 313 22 287)(6 304 33 298 16 314 23 288)(7 305 34 299 17 315 24 289)(8 306 35 300 18 316 25 290)(9 307 36 281 19 317 26 291)(10 308 37 282 20 318 27 292)(41 139 119 192 51 129 109 182)(42 140 120 193 52 130 110 183)(43 121 101 194 53 131 111 184)(44 122 102 195 54 132 112 185)(45 123 103 196 55 133 113 186)(46 124 104 197 56 134 114 187)(47 125 105 198 57 135 115 188)(48 126 106 199 58 136 116 189)(49 127 107 200 59 137 117 190)(50 128 108 181 60 138 118 191)(61 439 353 246 71 429 343 256)(62 440 354 247 72 430 344 257)(63 421 355 248 73 431 345 258)(64 422 356 249 74 432 346 259)(65 423 357 250 75 433 347 260)(66 424 358 251 76 434 348 241)(67 425 359 252 77 435 349 242)(68 426 360 253 78 436 350 243)(69 427 341 254 79 437 351 244)(70 428 342 255 80 438 352 245)(81 419 471 177 91 409 461 167)(82 420 472 178 92 410 462 168)(83 401 473 179 93 411 463 169)(84 402 474 180 94 412 464 170)(85 403 475 161 95 413 465 171)(86 404 476 162 96 414 466 172)(87 405 477 163 97 415 467 173)(88 406 478 164 98 416 468 174)(89 407 479 165 99 417 469 175)(90 408 480 166 100 418 470 176)(141 366 264 454 151 376 274 444)(142 367 265 455 152 377 275 445)(143 368 266 456 153 378 276 446)(144 369 267 457 154 379 277 447)(145 370 268 458 155 380 278 448)(146 371 269 459 156 361 279 449)(147 372 270 460 157 362 280 450)(148 373 271 441 158 363 261 451)(149 374 272 442 159 364 262 452)(150 375 273 443 160 365 263 453)(201 339 238 391 211 329 228 381)(202 340 239 392 212 330 229 382)(203 321 240 393 213 331 230 383)(204 322 221 394 214 332 231 384)(205 323 222 395 215 333 232 385)(206 324 223 396 216 334 233 386)(207 325 224 397 217 335 234 387)(208 326 225 398 218 336 235 388)(209 327 226 399 219 337 236 389)(210 328 227 400 220 338 237 390)
(1 269 23 151)(2 278 24 160)(3 267 25 149)(4 276 26 158)(5 265 27 147)(6 274 28 156)(7 263 29 145)(8 272 30 154)(9 261 31 143)(10 270 32 152)(11 279 33 141)(12 268 34 150)(13 277 35 159)(14 266 36 148)(15 275 37 157)(16 264 38 146)(17 273 39 155)(18 262 40 144)(19 271 21 153)(20 280 22 142)(41 246 114 424)(42 255 115 433)(43 244 116 422)(44 253 117 431)(45 242 118 440)(46 251 119 429)(47 260 120 438)(48 249 101 427)(49 258 102 436)(50 247 103 425)(51 256 104 434)(52 245 105 423)(53 254 106 432)(54 243 107 421)(55 252 108 430)(56 241 109 439)(57 250 110 428)(58 259 111 437)(59 248 112 426)(60 257 113 435)(61 129 348 197)(62 138 349 186)(63 127 350 195)(64 136 351 184)(65 125 352 193)(66 134 353 182)(67 123 354 191)(68 132 355 200)(69 121 356 189)(70 130 357 198)(71 139 358 187)(72 128 359 196)(73 137 360 185)(74 126 341 194)(75 135 342 183)(76 124 343 192)(77 133 344 181)(78 122 345 190)(79 131 346 199)(80 140 347 188)(81 339 466 386)(82 328 467 395)(83 337 468 384)(84 326 469 393)(85 335 470 382)(86 324 471 391)(87 333 472 400)(88 322 473 389)(89 331 474 398)(90 340 475 387)(91 329 476 396)(92 338 477 385)(93 327 478 394)(94 336 479 383)(95 325 480 392)(96 334 461 381)(97 323 462 390)(98 332 463 399)(99 321 464 388)(100 330 465 397)(161 239 408 207)(162 228 409 216)(163 237 410 205)(164 226 411 214)(165 235 412 203)(166 224 413 212)(167 233 414 201)(168 222 415 210)(169 231 416 219)(170 240 417 208)(171 229 418 217)(172 238 419 206)(173 227 420 215)(174 236 401 204)(175 225 402 213)(176 234 403 202)(177 223 404 211)(178 232 405 220)(179 221 406 209)(180 230 407 218)(281 456 312 373)(282 445 313 362)(283 454 314 371)(284 443 315 380)(285 452 316 369)(286 441 317 378)(287 450 318 367)(288 459 319 376)(289 448 320 365)(290 457 301 374)(291 446 302 363)(292 455 303 372)(293 444 304 361)(294 453 305 370)(295 442 306 379)(296 451 307 368)(297 460 308 377)(298 449 309 366)(299 458 310 375)(300 447 311 364)

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

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

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

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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