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G = C3×C10.D8order 480 = 25·3·5

Direct product of C3 and C10.D8

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

Aliases: C3×C10.D8, C30.44D8, C60.16Q8, C30.19Q16, C12.16Dic10, C52C81C12, C10.6(C3×D8), C158(C2.D8), C20.1(C3×Q8), C4.11(D5×C12), C12.81(C4×D5), C4⋊Dic5.8C6, C30.45(C4⋊C4), C10.3(C3×Q16), C60.156(C2×C4), C20.22(C2×C12), C6.22(D4⋊D5), (C2×C30).154D4, C4.1(C3×Dic10), (C2×C12).348D10, C6.10(C5⋊Q16), (C2×C60).271C22, C6.15(C10.D4), C52(C3×C2.D8), (C3×C52C8)⋊7C4, (C5×C4⋊C4).1C6, C10.9(C3×C4⋊C4), C2.1(C3×D4⋊D5), (C3×C4⋊C4).8D5, C4⋊C4.1(C3×D5), (C15×C4⋊C4).8C2, (C2×C52C8).1C6, (C2×C20).7(C2×C6), (C2×C4).27(C6×D5), C2.1(C3×C5⋊Q16), (C6×C52C8).13C2, (C2×C10).29(C3×D4), (C2×C6).84(C5⋊D4), (C3×C4⋊Dic5).22C2, C2.3(C3×C10.D4), C22.12(C3×C5⋊D4), SmallGroup(480,85)

Series: Derived Chief Lower central Upper central

C1C20 — C3×C10.D8
C1C5C10C20C2×C20C2×C60C6×C52C8 — C3×C10.D8
C5C10C20 — C3×C10.D8
C1C2×C6C2×C12C3×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 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], C12 [×2], C12 [×2], C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, Dic5, C20 [×2], C20, C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C30 [×3], C2.D8, C52C8 [×2], C2×Dic5, C2×C20, C2×C20, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×Dic5, C60 [×2], C60, C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C3×C2.D8, C3×C52C8 [×2], C6×Dic5, C2×C60, C2×C60, C10.D8, C6×C52C8, C3×C4⋊Dic5, C15×C4⋊C4, C3×C10.D8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4, Q8, D5, C12 [×2], 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
(1 104 64)(2 105 65)(3 106 66)(4 107 67)(5 108 68)(6 109 69)(7 110 70)(8 101 61)(9 102 62)(10 103 63)(11 91 51)(12 92 52)(13 93 53)(14 94 54)(15 95 55)(16 96 56)(17 97 57)(18 98 58)(19 99 59)(20 100 60)(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 37 251 20 265)(2 289 46 272 38 260 11 264)(3 288 47 271 39 259 12 263)(4 287 48 280 40 258 13 262)(5 286 49 279 31 257 14 261)(6 285 50 278 32 256 15 270)(7 284 41 277 33 255 16 269)(8 283 42 276 34 254 17 268)(9 282 43 275 35 253 18 267)(10 281 44 274 36 252 19 266)(21 250 469 230 455 216 477 233)(22 249 470 229 456 215 478 232)(23 248 461 228 457 214 479 231)(24 247 462 227 458 213 480 240)(25 246 463 226 459 212 471 239)(26 245 464 225 460 211 472 238)(27 244 465 224 451 220 473 237)(28 243 466 223 452 219 474 236)(29 242 467 222 453 218 475 235)(30 241 468 221 454 217 476 234)(51 304 65 329 86 312 78 300)(52 303 66 328 87 311 79 299)(53 302 67 327 88 320 80 298)(54 301 68 326 89 319 71 297)(55 310 69 325 90 318 72 296)(56 309 70 324 81 317 73 295)(57 308 61 323 82 316 74 294)(58 307 62 322 83 315 75 293)(59 306 63 321 84 314 76 292)(60 305 64 330 85 313 77 291)(91 344 105 369 126 352 118 340)(92 343 106 368 127 351 119 339)(93 342 107 367 128 360 120 338)(94 341 108 366 129 359 111 337)(95 350 109 365 130 358 112 336)(96 349 110 364 121 357 113 335)(97 348 101 363 122 356 114 334)(98 347 102 362 123 355 115 333)(99 346 103 361 124 354 116 332)(100 345 104 370 125 353 117 331)(131 392 158 409 166 384 145 380)(132 391 159 408 167 383 146 379)(133 400 160 407 168 382 147 378)(134 399 151 406 169 381 148 377)(135 398 152 405 170 390 149 376)(136 397 153 404 161 389 150 375)(137 396 154 403 162 388 141 374)(138 395 155 402 163 387 142 373)(139 394 156 401 164 386 143 372)(140 393 157 410 165 385 144 371)(171 432 198 449 206 424 185 420)(172 431 199 448 207 423 186 419)(173 440 200 447 208 422 187 418)(174 439 191 446 209 421 188 417)(175 438 192 445 210 430 189 416)(176 437 193 444 201 429 190 415)(177 436 194 443 202 428 181 414)(178 435 195 442 203 427 182 413)(179 434 196 441 204 426 183 412)(180 433 197 450 205 425 184 411)
(1 390 6 385)(2 389 7 384)(3 388 8 383)(4 387 9 382)(5 386 10 381)(11 375 16 380)(12 374 17 379)(13 373 18 378)(14 372 19 377)(15 371 20 376)(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)(51 415 56 420)(52 414 57 419)(53 413 58 418)(54 412 59 417)(55 411 60 416)(61 423 66 428)(62 422 67 427)(63 421 68 426)(64 430 69 425)(65 429 70 424)(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)(91 455 96 460)(92 454 97 459)(93 453 98 458)(94 452 99 457)(95 451 100 456)(101 463 106 468)(102 462 107 467)(103 461 108 466)(104 470 109 465)(105 469 110 464)(111 474 116 479)(112 473 117 478)(113 472 118 477)(114 471 119 476)(115 480 120 475)(131 260 136 255)(132 259 137 254)(133 258 138 253)(134 257 139 252)(135 256 140 251)(141 268 146 263)(142 267 147 262)(143 266 148 261)(144 265 149 270)(145 264 150 269)(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)(171 300 176 295)(172 299 177 294)(173 298 178 293)(174 297 179 292)(175 296 180 291)(181 308 186 303)(182 307 187 302)(183 306 188 301)(184 305 189 310)(185 304 190 309)(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)(211 340 216 335)(212 339 217 334)(213 338 218 333)(214 337 219 332)(215 336 220 331)(221 348 226 343)(222 347 227 342)(223 346 228 341)(224 345 229 350)(225 344 230 349)(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)| (1,104,64)(2,105,65)(3,106,66)(4,107,67)(5,108,68)(6,109,69)(7,110,70)(8,101,61)(9,102,62)(10,103,63)(11,91,51)(12,92,52)(13,93,53)(14,94,54)(15,95,55)(16,96,56)(17,97,57)(18,98,58)(19,99,59)(20,100,60)(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), (1,290,45,273,37,251,20,265)(2,289,46,272,38,260,11,264)(3,288,47,271,39,259,12,263)(4,287,48,280,40,258,13,262)(5,286,49,279,31,257,14,261)(6,285,50,278,32,256,15,270)(7,284,41,277,33,255,16,269)(8,283,42,276,34,254,17,268)(9,282,43,275,35,253,18,267)(10,281,44,274,36,252,19,266)(21,250,469,230,455,216,477,233)(22,249,470,229,456,215,478,232)(23,248,461,228,457,214,479,231)(24,247,462,227,458,213,480,240)(25,246,463,226,459,212,471,239)(26,245,464,225,460,211,472,238)(27,244,465,224,451,220,473,237)(28,243,466,223,452,219,474,236)(29,242,467,222,453,218,475,235)(30,241,468,221,454,217,476,234)(51,304,65,329,86,312,78,300)(52,303,66,328,87,311,79,299)(53,302,67,327,88,320,80,298)(54,301,68,326,89,319,71,297)(55,310,69,325,90,318,72,296)(56,309,70,324,81,317,73,295)(57,308,61,323,82,316,74,294)(58,307,62,322,83,315,75,293)(59,306,63,321,84,314,76,292)(60,305,64,330,85,313,77,291)(91,344,105,369,126,352,118,340)(92,343,106,368,127,351,119,339)(93,342,107,367,128,360,120,338)(94,341,108,366,129,359,111,337)(95,350,109,365,130,358,112,336)(96,349,110,364,121,357,113,335)(97,348,101,363,122,356,114,334)(98,347,102,362,123,355,115,333)(99,346,103,361,124,354,116,332)(100,345,104,370,125,353,117,331)(131,392,158,409,166,384,145,380)(132,391,159,408,167,383,146,379)(133,400,160,407,168,382,147,378)(134,399,151,406,169,381,148,377)(135,398,152,405,170,390,149,376)(136,397,153,404,161,389,150,375)(137,396,154,403,162,388,141,374)(138,395,155,402,163,387,142,373)(139,394,156,401,164,386,143,372)(140,393,157,410,165,385,144,371)(171,432,198,449,206,424,185,420)(172,431,199,448,207,423,186,419)(173,440,200,447,208,422,187,418)(174,439,191,446,209,421,188,417)(175,438,192,445,210,430,189,416)(176,437,193,444,201,429,190,415)(177,436,194,443,202,428,181,414)(178,435,195,442,203,427,182,413)(179,434,196,441,204,426,183,412)(180,433,197,450,205,425,184,411), (1,390,6,385)(2,389,7,384)(3,388,8,383)(4,387,9,382)(5,386,10,381)(11,375,16,380)(12,374,17,379)(13,373,18,378)(14,372,19,377)(15,371,20,376)(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)(51,415,56,420)(52,414,57,419)(53,413,58,418)(54,412,59,417)(55,411,60,416)(61,423,66,428)(62,422,67,427)(63,421,68,426)(64,430,69,425)(65,429,70,424)(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)(91,455,96,460)(92,454,97,459)(93,453,98,458)(94,452,99,457)(95,451,100,456)(101,463,106,468)(102,462,107,467)(103,461,108,466)(104,470,109,465)(105,469,110,464)(111,474,116,479)(112,473,117,478)(113,472,118,477)(114,471,119,476)(115,480,120,475)(131,260,136,255)(132,259,137,254)(133,258,138,253)(134,257,139,252)(135,256,140,251)(141,268,146,263)(142,267,147,262)(143,266,148,261)(144,265,149,270)(145,264,150,269)(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)(171,300,176,295)(172,299,177,294)(173,298,178,293)(174,297,179,292)(175,296,180,291)(181,308,186,303)(182,307,187,302)(183,306,188,301)(184,305,189,310)(185,304,190,309)(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)(211,340,216,335)(212,339,217,334)(213,338,218,333)(214,337,219,332)(215,336,220,331)(221,348,226,343)(222,347,227,342)(223,346,228,341)(224,345,229,350)(225,344,230,349)(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( (1,104,64)(2,105,65)(3,106,66)(4,107,67)(5,108,68)(6,109,69)(7,110,70)(8,101,61)(9,102,62)(10,103,63)(11,91,51)(12,92,52)(13,93,53)(14,94,54)(15,95,55)(16,96,56)(17,97,57)(18,98,58)(19,99,59)(20,100,60)(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), (1,290,45,273,37,251,20,265)(2,289,46,272,38,260,11,264)(3,288,47,271,39,259,12,263)(4,287,48,280,40,258,13,262)(5,286,49,279,31,257,14,261)(6,285,50,278,32,256,15,270)(7,284,41,277,33,255,16,269)(8,283,42,276,34,254,17,268)(9,282,43,275,35,253,18,267)(10,281,44,274,36,252,19,266)(21,250,469,230,455,216,477,233)(22,249,470,229,456,215,478,232)(23,248,461,228,457,214,479,231)(24,247,462,227,458,213,480,240)(25,246,463,226,459,212,471,239)(26,245,464,225,460,211,472,238)(27,244,465,224,451,220,473,237)(28,243,466,223,452,219,474,236)(29,242,467,222,453,218,475,235)(30,241,468,221,454,217,476,234)(51,304,65,329,86,312,78,300)(52,303,66,328,87,311,79,299)(53,302,67,327,88,320,80,298)(54,301,68,326,89,319,71,297)(55,310,69,325,90,318,72,296)(56,309,70,324,81,317,73,295)(57,308,61,323,82,316,74,294)(58,307,62,322,83,315,75,293)(59,306,63,321,84,314,76,292)(60,305,64,330,85,313,77,291)(91,344,105,369,126,352,118,340)(92,343,106,368,127,351,119,339)(93,342,107,367,128,360,120,338)(94,341,108,366,129,359,111,337)(95,350,109,365,130,358,112,336)(96,349,110,364,121,357,113,335)(97,348,101,363,122,356,114,334)(98,347,102,362,123,355,115,333)(99,346,103,361,124,354,116,332)(100,345,104,370,125,353,117,331)(131,392,158,409,166,384,145,380)(132,391,159,408,167,383,146,379)(133,400,160,407,168,382,147,378)(134,399,151,406,169,381,148,377)(135,398,152,405,170,390,149,376)(136,397,153,404,161,389,150,375)(137,396,154,403,162,388,141,374)(138,395,155,402,163,387,142,373)(139,394,156,401,164,386,143,372)(140,393,157,410,165,385,144,371)(171,432,198,449,206,424,185,420)(172,431,199,448,207,423,186,419)(173,440,200,447,208,422,187,418)(174,439,191,446,209,421,188,417)(175,438,192,445,210,430,189,416)(176,437,193,444,201,429,190,415)(177,436,194,443,202,428,181,414)(178,435,195,442,203,427,182,413)(179,434,196,441,204,426,183,412)(180,433,197,450,205,425,184,411), (1,390,6,385)(2,389,7,384)(3,388,8,383)(4,387,9,382)(5,386,10,381)(11,375,16,380)(12,374,17,379)(13,373,18,378)(14,372,19,377)(15,371,20,376)(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)(51,415,56,420)(52,414,57,419)(53,413,58,418)(54,412,59,417)(55,411,60,416)(61,423,66,428)(62,422,67,427)(63,421,68,426)(64,430,69,425)(65,429,70,424)(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)(91,455,96,460)(92,454,97,459)(93,453,98,458)(94,452,99,457)(95,451,100,456)(101,463,106,468)(102,462,107,467)(103,461,108,466)(104,470,109,465)(105,469,110,464)(111,474,116,479)(112,473,117,478)(113,472,118,477)(114,471,119,476)(115,480,120,475)(131,260,136,255)(132,259,137,254)(133,258,138,253)(134,257,139,252)(135,256,140,251)(141,268,146,263)(142,267,147,262)(143,266,148,261)(144,265,149,270)(145,264,150,269)(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)(171,300,176,295)(172,299,177,294)(173,298,178,293)(174,297,179,292)(175,296,180,291)(181,308,186,303)(182,307,187,302)(183,306,188,301)(184,305,189,310)(185,304,190,309)(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)(211,340,216,335)(212,339,217,334)(213,338,218,333)(214,337,219,332)(215,336,220,331)(221,348,226,343)(222,347,227,342)(223,346,228,341)(224,345,229,350)(225,344,230,349)(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([(1,104,64),(2,105,65),(3,106,66),(4,107,67),(5,108,68),(6,109,69),(7,110,70),(8,101,61),(9,102,62),(10,103,63),(11,91,51),(12,92,52),(13,93,53),(14,94,54),(15,95,55),(16,96,56),(17,97,57),(18,98,58),(19,99,59),(20,100,60),(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)], [(1,290,45,273,37,251,20,265),(2,289,46,272,38,260,11,264),(3,288,47,271,39,259,12,263),(4,287,48,280,40,258,13,262),(5,286,49,279,31,257,14,261),(6,285,50,278,32,256,15,270),(7,284,41,277,33,255,16,269),(8,283,42,276,34,254,17,268),(9,282,43,275,35,253,18,267),(10,281,44,274,36,252,19,266),(21,250,469,230,455,216,477,233),(22,249,470,229,456,215,478,232),(23,248,461,228,457,214,479,231),(24,247,462,227,458,213,480,240),(25,246,463,226,459,212,471,239),(26,245,464,225,460,211,472,238),(27,244,465,224,451,220,473,237),(28,243,466,223,452,219,474,236),(29,242,467,222,453,218,475,235),(30,241,468,221,454,217,476,234),(51,304,65,329,86,312,78,300),(52,303,66,328,87,311,79,299),(53,302,67,327,88,320,80,298),(54,301,68,326,89,319,71,297),(55,310,69,325,90,318,72,296),(56,309,70,324,81,317,73,295),(57,308,61,323,82,316,74,294),(58,307,62,322,83,315,75,293),(59,306,63,321,84,314,76,292),(60,305,64,330,85,313,77,291),(91,344,105,369,126,352,118,340),(92,343,106,368,127,351,119,339),(93,342,107,367,128,360,120,338),(94,341,108,366,129,359,111,337),(95,350,109,365,130,358,112,336),(96,349,110,364,121,357,113,335),(97,348,101,363,122,356,114,334),(98,347,102,362,123,355,115,333),(99,346,103,361,124,354,116,332),(100,345,104,370,125,353,117,331),(131,392,158,409,166,384,145,380),(132,391,159,408,167,383,146,379),(133,400,160,407,168,382,147,378),(134,399,151,406,169,381,148,377),(135,398,152,405,170,390,149,376),(136,397,153,404,161,389,150,375),(137,396,154,403,162,388,141,374),(138,395,155,402,163,387,142,373),(139,394,156,401,164,386,143,372),(140,393,157,410,165,385,144,371),(171,432,198,449,206,424,185,420),(172,431,199,448,207,423,186,419),(173,440,200,447,208,422,187,418),(174,439,191,446,209,421,188,417),(175,438,192,445,210,430,189,416),(176,437,193,444,201,429,190,415),(177,436,194,443,202,428,181,414),(178,435,195,442,203,427,182,413),(179,434,196,441,204,426,183,412),(180,433,197,450,205,425,184,411)], [(1,390,6,385),(2,389,7,384),(3,388,8,383),(4,387,9,382),(5,386,10,381),(11,375,16,380),(12,374,17,379),(13,373,18,378),(14,372,19,377),(15,371,20,376),(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),(51,415,56,420),(52,414,57,419),(53,413,58,418),(54,412,59,417),(55,411,60,416),(61,423,66,428),(62,422,67,427),(63,421,68,426),(64,430,69,425),(65,429,70,424),(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),(91,455,96,460),(92,454,97,459),(93,453,98,458),(94,452,99,457),(95,451,100,456),(101,463,106,468),(102,462,107,467),(103,461,108,466),(104,470,109,465),(105,469,110,464),(111,474,116,479),(112,473,117,478),(113,472,118,477),(114,471,119,476),(115,480,120,475),(131,260,136,255),(132,259,137,254),(133,258,138,253),(134,257,139,252),(135,256,140,251),(141,268,146,263),(142,267,147,262),(143,266,148,261),(144,265,149,270),(145,264,150,269),(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),(171,300,176,295),(172,299,177,294),(173,298,178,293),(174,297,179,292),(175,296,180,291),(181,308,186,303),(182,307,187,302),(183,306,188,301),(184,305,189,310),(185,304,190,309),(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),(211,340,216,335),(212,339,217,334),(213,338,218,333),(214,337,219,332),(215,336,220,331),(221,348,226,343),(222,347,227,342),(223,346,228,341),(224,345,229,350),(225,344,230,349),(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 2A2B2C3A3B4A4B4C4D4E4F5A5B6A···6F8A8B8C8D10A···10F12A12B12C12D12E12F12G12H12I12J12K12L15A15B15C15D20A···20L24A···24H30A···30L60A···60X
order122233444444556···6888810···101212121212121212121212121515151520···2024···2430···3060···60
size11111122442020221···1101010102···2222244442020202022224···410···102···24···4

102 irreducible representations

dim11111111112222222222222222224444
type++++-+++-+-+-
imageC1C2C2C2C3C4C6C6C6C12Q8D4D5D8Q16D10C3×Q8C3×D4C3×D5Dic10C4×D5C5⋊D4C3×D8C3×Q16C6×D5C3×Dic10D5×C12C3×C5⋊D4D4⋊D5C5⋊Q16C3×D4⋊D5C3×C5⋊Q16
kernelC3×C10.D8C6×C52C8C3×C4⋊Dic5C15×C4⋊C4C10.D8C3×C52C8C2×C52C8C4⋊Dic5C5×C4⋊C4C52C8C60C2×C30C3×C4⋊C4C30C30C2×C12C20C2×C10C4⋊C4C12C12C2×C6C10C10C2×C4C4C4C22C6C6C2C2
# reps11112422281122222244444448882244

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

100000
010000
00225000
00022500
000010
000001
,
100000
010000
00240000
00024000
000051189
0000510
,
11110000
230110000
002083500
001413300
0000133185
00006108
,
1641960000
196770000
00818200
0020816000
0000133185
00006108

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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