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## G = C3×C10.D8order 480 = 25·3·5

### Direct product of C3 and C10.D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C10.D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C6×C5⋊2C8 — C3×C10.D8
 Lower central C5 — C10 — C20 — C3×C10.D8
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×C10.D8
G = < a,b,c,d | a3=b10=c8=1, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 192 in 72 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C24, C2×C12, C2×C12, C30, C2.D8, C52C8, C2×Dic5, C2×C20, C2×C20, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×Dic5, C60, C60, C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C3×C2.D8, C3×C52C8, C6×Dic5, C2×C60, C2×C60, C10.D8, C6×C52C8, C3×C4⋊Dic5, C15×C4⋊C4, C3×C10.D8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, D5, C12, C2×C6, C4⋊C4, D8, Q16, D10, C2×C12, C3×D4, C3×Q8, C3×D5, C2.D8, Dic10, C4×D5, C5⋊D4, C3×C4⋊C4, C3×D8, C3×Q16, C6×D5, C10.D4, D4⋊D5, C5⋊Q16, C3×C2.D8, C3×Dic10, D5×C12, C3×C5⋊D4, C10.D8, C3×C10.D4, C3×D4⋊D5, C3×C5⋊Q16, C3×C10.D8

Smallest permutation representation of C3×C10.D8
Regular action on 480 points
Generators in S480
(1 99 59)(2 100 60)(3 91 51)(4 92 52)(5 93 53)(6 94 54)(7 95 55)(8 96 56)(9 97 57)(10 98 58)(11 109 69)(12 110 70)(13 101 61)(14 102 62)(15 103 63)(16 104 64)(17 105 65)(18 106 66)(19 107 67)(20 108 68)(21 444 404)(22 445 405)(23 446 406)(24 447 407)(25 448 408)(26 449 409)(27 450 410)(28 441 401)(29 442 402)(30 443 403)(31 111 71)(32 112 72)(33 113 73)(34 114 74)(35 115 75)(36 116 76)(37 117 77)(38 118 78)(39 119 79)(40 120 80)(41 121 81)(42 122 82)(43 123 83)(44 124 84)(45 125 85)(46 126 86)(47 127 87)(48 128 88)(49 129 89)(50 130 90)(131 211 171)(132 212 172)(133 213 173)(134 214 174)(135 215 175)(136 216 176)(137 217 177)(138 218 178)(139 219 179)(140 220 180)(141 221 181)(142 222 182)(143 223 183)(144 224 184)(145 225 185)(146 226 186)(147 227 187)(148 228 188)(149 229 189)(150 230 190)(151 231 191)(152 232 192)(153 233 193)(154 234 194)(155 235 195)(156 236 196)(157 237 197)(158 238 198)(159 239 199)(160 240 200)(161 250 201)(162 241 202)(163 242 203)(164 243 204)(165 244 205)(166 245 206)(167 246 207)(168 247 208)(169 248 209)(170 249 210)(251 331 291)(252 332 292)(253 333 293)(254 334 294)(255 335 295)(256 336 296)(257 337 297)(258 338 298)(259 339 299)(260 340 300)(261 341 301)(262 342 302)(263 343 303)(264 344 304)(265 345 305)(266 346 306)(267 347 307)(268 348 308)(269 349 309)(270 350 310)(271 351 311)(272 352 312)(273 353 313)(274 354 314)(275 355 315)(276 356 316)(277 357 317)(278 358 318)(279 359 319)(280 360 320)(281 361 321)(282 362 322)(283 363 323)(284 364 324)(285 365 325)(286 366 326)(287 367 327)(288 368 328)(289 369 329)(290 370 330)(371 451 411)(372 452 412)(373 453 413)(374 454 414)(375 455 415)(376 456 416)(377 457 417)(378 458 418)(379 459 419)(380 460 420)(381 461 421)(382 462 422)(383 463 423)(384 464 424)(385 465 425)(386 466 426)(387 467 427)(388 468 428)(389 469 429)(390 470 430)(391 471 431)(392 472 432)(393 473 433)(394 474 434)(395 475 435)(396 476 436)(397 477 437)(398 478 438)(399 479 439)(400 480 440)
(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)
(1 290 45 273 37 264 18 256)(2 289 46 272 38 263 19 255)(3 288 47 271 39 262 20 254)(4 287 48 280 40 261 11 253)(5 286 49 279 31 270 12 252)(6 285 50 278 32 269 13 251)(7 284 41 277 33 268 14 260)(8 283 42 276 34 267 15 259)(9 282 43 275 35 266 16 258)(10 281 44 274 36 265 17 257)(21 250 460 215 468 222 477 233)(22 249 451 214 469 221 478 232)(23 248 452 213 470 230 479 231)(24 247 453 212 461 229 480 240)(25 246 454 211 462 228 471 239)(26 245 455 220 463 227 472 238)(27 244 456 219 464 226 473 237)(28 243 457 218 465 225 474 236)(29 242 458 217 466 224 475 235)(30 241 459 216 467 223 476 234)(51 328 87 311 79 302 68 294)(52 327 88 320 80 301 69 293)(53 326 89 319 71 310 70 292)(54 325 90 318 72 309 61 291)(55 324 81 317 73 308 62 300)(56 323 82 316 74 307 63 299)(57 322 83 315 75 306 64 298)(58 321 84 314 76 305 65 297)(59 330 85 313 77 304 66 296)(60 329 86 312 78 303 67 295)(91 368 127 351 119 342 108 334)(92 367 128 360 120 341 109 333)(93 366 129 359 111 350 110 332)(94 365 130 358 112 349 101 331)(95 364 121 357 113 348 102 340)(96 363 122 356 114 347 103 339)(97 362 123 355 115 346 104 338)(98 361 124 354 116 345 105 337)(99 370 125 353 117 344 106 336)(100 369 126 352 118 343 107 335)(131 382 148 391 159 408 167 374)(132 381 149 400 160 407 168 373)(133 390 150 399 151 406 169 372)(134 389 141 398 152 405 170 371)(135 388 142 397 153 404 161 380)(136 387 143 396 154 403 162 379)(137 386 144 395 155 402 163 378)(138 385 145 394 156 401 164 377)(139 384 146 393 157 410 165 376)(140 383 147 392 158 409 166 375)(171 422 188 431 199 448 207 414)(172 421 189 440 200 447 208 413)(173 430 190 439 191 446 209 412)(174 429 181 438 192 445 210 411)(175 428 182 437 193 444 201 420)(176 427 183 436 194 443 202 419)(177 426 184 435 195 442 203 418)(178 425 185 434 196 441 204 417)(179 424 186 433 197 450 205 416)(180 423 187 432 198 449 206 415)
(1 371 6 376)(2 380 7 375)(3 379 8 374)(4 378 9 373)(5 377 10 372)(11 386 16 381)(12 385 17 390)(13 384 18 389)(14 383 19 388)(15 382 20 387)(21 121 26 126)(22 130 27 125)(23 129 28 124)(24 128 29 123)(25 127 30 122)(31 394 36 399)(32 393 37 398)(33 392 38 397)(34 391 39 396)(35 400 40 395)(41 409 46 404)(42 408 47 403)(43 407 48 402)(44 406 49 401)(45 405 50 410)(51 419 56 414)(52 418 57 413)(53 417 58 412)(54 416 59 411)(55 415 60 420)(61 424 66 429)(62 423 67 428)(63 422 68 427)(64 421 69 426)(65 430 70 425)(71 434 76 439)(72 433 77 438)(73 432 78 437)(74 431 79 436)(75 440 80 435)(81 449 86 444)(82 448 87 443)(83 447 88 442)(84 446 89 441)(85 445 90 450)(91 459 96 454)(92 458 97 453)(93 457 98 452)(94 456 99 451)(95 455 100 460)(101 464 106 469)(102 463 107 468)(103 462 108 467)(104 461 109 466)(105 470 110 465)(111 474 116 479)(112 473 117 478)(113 472 118 477)(114 471 119 476)(115 480 120 475)(131 254 136 259)(132 253 137 258)(133 252 138 257)(134 251 139 256)(135 260 140 255)(141 269 146 264)(142 268 147 263)(143 267 148 262)(144 266 149 261)(145 265 150 270)(151 279 156 274)(152 278 157 273)(153 277 158 272)(154 276 159 271)(155 275 160 280)(161 284 166 289)(162 283 167 288)(163 282 168 287)(164 281 169 286)(165 290 170 285)(171 294 176 299)(172 293 177 298)(173 292 178 297)(174 291 179 296)(175 300 180 295)(181 309 186 304)(182 308 187 303)(183 307 188 302)(184 306 189 301)(185 305 190 310)(191 319 196 314)(192 318 197 313)(193 317 198 312)(194 316 199 311)(195 315 200 320)(201 324 206 329)(202 323 207 328)(203 322 208 327)(204 321 209 326)(205 330 210 325)(211 334 216 339)(212 333 217 338)(213 332 218 337)(214 331 219 336)(215 340 220 335)(221 349 226 344)(222 348 227 343)(223 347 228 342)(224 346 229 341)(225 345 230 350)(231 359 236 354)(232 358 237 353)(233 357 238 352)(234 356 239 351)(235 355 240 360)(241 363 246 368)(242 362 247 367)(243 361 248 366)(244 370 249 365)(245 369 250 364)

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

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

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

102 conjugacy classes

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

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + + + - + - + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 Q8 D4 D5 D8 Q16 D10 C3×Q8 C3×D4 C3×D5 Dic10 C4×D5 C5⋊D4 C3×D8 C3×Q16 C6×D5 C3×Dic10 D5×C12 C3×C5⋊D4 D4⋊D5 C5⋊Q16 C3×D4⋊D5 C3×C5⋊Q16 kernel C3×C10.D8 C6×C5⋊2C8 C3×C4⋊Dic5 C15×C4⋊C4 C10.D8 C3×C5⋊2C8 C2×C5⋊2C8 C4⋊Dic5 C5×C4⋊C4 C5⋊2C8 C60 C2×C30 C3×C4⋊C4 C30 C30 C2×C12 C20 C2×C10 C4⋊C4 C12 C12 C2×C6 C10 C10 C2×C4 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 2 2 4 4

Matrix representation of C3×C10.D8 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 51 189 0 0 0 0 51 0
,
 11 11 0 0 0 0 230 11 0 0 0 0 0 0 208 35 0 0 0 0 141 33 0 0 0 0 0 0 133 185 0 0 0 0 6 108
,
 164 196 0 0 0 0 196 77 0 0 0 0 0 0 81 82 0 0 0 0 208 160 0 0 0 0 0 0 133 185 0 0 0 0 6 108

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,51,51,0,0,0,0,189,0],[11,230,0,0,0,0,11,11,0,0,0,0,0,0,208,141,0,0,0,0,35,33,0,0,0,0,0,0,133,6,0,0,0,0,185,108],[164,196,0,0,0,0,196,77,0,0,0,0,0,0,81,208,0,0,0,0,82,160,0,0,0,0,0,0,133,6,0,0,0,0,185,108] >;

C3×C10.D8 in GAP, Magma, Sage, TeX

C_3\times C_{10}.D_8
% in TeX

G:=Group("C3xC10.D8");
// GroupNames label

G:=SmallGroup(480,85);
// by ID

G=gap.SmallGroup(480,85);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,168,365,92,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^10=c^8=1,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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