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G = C2×C6×Dic10order 480 = 25·3·5

Direct product of C2×C6 and Dic10

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

Aliases: C2×C6×Dic10, C30.69C24, C60.266C23, C306(C2×Q8), C101(C6×Q8), (C2×C30)⋊12Q8, C157(C22×Q8), C10.1(C23×C6), C6.69(C23×D5), C23.38(C6×D5), (C2×C12).437D10, (C22×C20).12C6, (C22×C60).19C2, C20.33(C22×C6), (C22×C12).18D5, (C2×C30).379C23, (C2×C60).515C22, C12.239(C22×D5), (C22×C6).135D10, Dic5.1(C22×C6), (C22×Dic5).8C6, (C3×Dic5).53C23, (C22×C30).164C22, (C6×Dic5).256C22, C51(Q8×C2×C6), C4.32(D5×C2×C6), (C2×C10)⋊6(C3×Q8), C2.3(D5×C22×C6), (C2×C4).87(C6×D5), C22.28(D5×C2×C6), (C2×C20).98(C2×C6), (C2×C6×Dic5).14C2, (C22×C4).10(C3×D5), (C2×C10).62(C22×C6), (C22×C10).51(C2×C6), (C2×Dic5).44(C2×C6), (C2×C6).375(C22×D5), SmallGroup(480,1135)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C6×Dic10
C1C5C10C30C3×Dic5C6×Dic5C2×C6×Dic5 — C2×C6×Dic10
C5C10 — C2×C6×Dic10
C1C22×C6C22×C12

Generators and relations for C2×C6×Dic10
 G = < a,b,c,d | a2=b6=c20=1, d2=c10, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 720 in 312 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×8], C22 [×7], C5, C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C10, C10 [×6], C12 [×4], C12 [×8], C2×C6 [×7], C15, C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic5 [×8], C20 [×4], C2×C10 [×7], C2×C12 [×6], C2×C12 [×12], C3×Q8 [×16], C22×C6, C30, C30 [×6], C22×Q8, Dic10 [×16], C2×Dic5 [×12], C2×C20 [×6], C22×C10, C22×C12, C22×C12 [×2], C6×Q8 [×12], C3×Dic5 [×8], C60 [×4], C2×C30 [×7], C2×Dic10 [×12], C22×Dic5 [×2], C22×C20, Q8×C2×C6, C3×Dic10 [×16], C6×Dic5 [×12], C2×C60 [×6], C22×C30, C22×Dic10, C6×Dic10 [×12], C2×C6×Dic5 [×2], C22×C60, C2×C6×Dic10
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], Q8 [×4], C23 [×15], D5, C2×C6 [×35], C2×Q8 [×6], C24, D10 [×7], C3×Q8 [×4], C22×C6 [×15], C3×D5, C22×Q8, Dic10 [×4], C22×D5 [×7], C6×Q8 [×6], C23×C6, C6×D5 [×7], C2×Dic10 [×6], C23×D5, Q8×C2×C6, C3×Dic10 [×4], D5×C2×C6 [×7], C22×Dic10, C6×Dic10 [×6], D5×C22×C6, C2×C6×Dic10

Smallest permutation representation of C2×C6×Dic10
Regular action on 480 points
Generators in S480
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 161)(13 162)(14 163)(15 164)(16 165)(17 166)(18 167)(19 168)(20 169)(21 343)(22 344)(23 345)(24 346)(25 347)(26 348)(27 349)(28 350)(29 351)(30 352)(31 353)(32 354)(33 355)(34 356)(35 357)(36 358)(37 359)(38 360)(39 341)(40 342)(41 190)(42 191)(43 192)(44 193)(45 194)(46 195)(47 196)(48 197)(49 198)(50 199)(51 200)(52 181)(53 182)(54 183)(55 184)(56 185)(57 186)(58 187)(59 188)(60 189)(61 323)(62 324)(63 325)(64 326)(65 327)(66 328)(67 329)(68 330)(69 331)(70 332)(71 333)(72 334)(73 335)(74 336)(75 337)(76 338)(77 339)(78 340)(79 321)(80 322)(81 298)(82 299)(83 300)(84 281)(85 282)(86 283)(87 284)(88 285)(89 286)(90 287)(91 288)(92 289)(93 290)(94 291)(95 292)(96 293)(97 294)(98 295)(99 296)(100 297)(101 431)(102 432)(103 433)(104 434)(105 435)(106 436)(107 437)(108 438)(109 439)(110 440)(111 421)(112 422)(113 423)(114 424)(115 425)(116 426)(117 427)(118 428)(119 429)(120 430)(121 157)(122 158)(123 159)(124 160)(125 141)(126 142)(127 143)(128 144)(129 145)(130 146)(131 147)(132 148)(133 149)(134 150)(135 151)(136 152)(137 153)(138 154)(139 155)(140 156)(201 248)(202 249)(203 250)(204 251)(205 252)(206 253)(207 254)(208 255)(209 256)(210 257)(211 258)(212 259)(213 260)(214 241)(215 242)(216 243)(217 244)(218 245)(219 246)(220 247)(221 478)(222 479)(223 480)(224 461)(225 462)(226 463)(227 464)(228 465)(229 466)(230 467)(231 468)(232 469)(233 470)(234 471)(235 472)(236 473)(237 474)(238 475)(239 476)(240 477)(261 308)(262 309)(263 310)(264 311)(265 312)(266 313)(267 314)(268 315)(269 316)(270 317)(271 318)(272 319)(273 320)(274 301)(275 302)(276 303)(277 304)(278 305)(279 306)(280 307)(361 392)(362 393)(363 394)(364 395)(365 396)(366 397)(367 398)(368 399)(369 400)(370 381)(371 382)(372 383)(373 384)(374 385)(375 386)(376 387)(377 388)(378 389)(379 390)(380 391)(401 451)(402 452)(403 453)(404 454)(405 455)(406 456)(407 457)(408 458)(409 459)(410 460)(411 441)(412 442)(413 443)(414 444)(415 445)(416 446)(417 447)(418 448)(419 449)(420 450)
(1 402 95 462 213 324)(2 403 96 463 214 325)(3 404 97 464 215 326)(4 405 98 465 216 327)(5 406 99 466 217 328)(6 407 100 467 218 329)(7 408 81 468 219 330)(8 409 82 469 220 331)(9 410 83 470 201 332)(10 411 84 471 202 333)(11 412 85 472 203 334)(12 413 86 473 204 335)(13 414 87 474 205 336)(14 415 88 475 206 337)(15 416 89 476 207 338)(16 417 90 477 208 339)(17 418 91 478 209 340)(18 419 92 479 210 321)(19 420 93 480 211 322)(20 401 94 461 212 323)(21 160 112 370 269 182)(22 141 113 371 270 183)(23 142 114 372 271 184)(24 143 115 373 272 185)(25 144 116 374 273 186)(26 145 117 375 274 187)(27 146 118 376 275 188)(28 147 119 377 276 189)(29 148 120 378 277 190)(30 149 101 379 278 191)(31 150 102 380 279 192)(32 151 103 361 280 193)(33 152 104 362 261 194)(34 153 105 363 262 195)(35 154 106 364 263 196)(36 155 107 365 264 197)(37 156 108 366 265 198)(38 157 109 367 266 199)(39 158 110 368 267 200)(40 159 111 369 268 181)(41 351 132 430 389 304)(42 352 133 431 390 305)(43 353 134 432 391 306)(44 354 135 433 392 307)(45 355 136 434 393 308)(46 356 137 435 394 309)(47 357 138 436 395 310)(48 358 139 437 396 311)(49 359 140 438 397 312)(50 360 121 439 398 313)(51 341 122 440 399 314)(52 342 123 421 400 315)(53 343 124 422 381 316)(54 344 125 423 382 317)(55 345 126 424 383 318)(56 346 127 425 384 319)(57 347 128 426 385 320)(58 348 129 427 386 301)(59 349 130 428 387 302)(60 350 131 429 388 303)(61 169 451 291 224 259)(62 170 452 292 225 260)(63 171 453 293 226 241)(64 172 454 294 227 242)(65 173 455 295 228 243)(66 174 456 296 229 244)(67 175 457 297 230 245)(68 176 458 298 231 246)(69 177 459 299 232 247)(70 178 460 300 233 248)(71 179 441 281 234 249)(72 180 442 282 235 250)(73 161 443 283 236 251)(74 162 444 284 237 252)(75 163 445 285 238 253)(76 164 446 286 239 254)(77 165 447 287 240 255)(78 166 448 288 221 256)(79 167 449 289 222 257)(80 168 450 290 223 258)
(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)
(1 353 11 343)(2 352 12 342)(3 351 13 341)(4 350 14 360)(5 349 15 359)(6 348 16 358)(7 347 17 357)(8 346 18 356)(9 345 19 355)(10 344 20 354)(21 170 31 180)(22 169 32 179)(23 168 33 178)(24 167 34 177)(25 166 35 176)(26 165 36 175)(27 164 37 174)(28 163 38 173)(29 162 39 172)(30 161 40 171)(41 336 51 326)(42 335 52 325)(43 334 53 324)(44 333 54 323)(45 332 55 322)(46 331 56 321)(47 330 57 340)(48 329 58 339)(49 328 59 338)(50 327 60 337)(61 193 71 183)(62 192 72 182)(63 191 73 181)(64 190 74 200)(65 189 75 199)(66 188 76 198)(67 187 77 197)(68 186 78 196)(69 185 79 195)(70 184 80 194)(81 426 91 436)(82 425 92 435)(83 424 93 434)(84 423 94 433)(85 422 95 432)(86 421 96 431)(87 440 97 430)(88 439 98 429)(89 438 99 428)(90 437 100 427)(101 283 111 293)(102 282 112 292)(103 281 113 291)(104 300 114 290)(105 299 115 289)(106 298 116 288)(107 297 117 287)(108 296 118 286)(109 295 119 285)(110 294 120 284)(121 405 131 415)(122 404 132 414)(123 403 133 413)(124 402 134 412)(125 401 135 411)(126 420 136 410)(127 419 137 409)(128 418 138 408)(129 417 139 407)(130 416 140 406)(141 451 151 441)(142 450 152 460)(143 449 153 459)(144 448 154 458)(145 447 155 457)(146 446 156 456)(147 445 157 455)(148 444 158 454)(149 443 159 453)(150 442 160 452)(201 318 211 308)(202 317 212 307)(203 316 213 306)(204 315 214 305)(205 314 215 304)(206 313 216 303)(207 312 217 302)(208 311 218 301)(209 310 219 320)(210 309 220 319)(221 364 231 374)(222 363 232 373)(223 362 233 372)(224 361 234 371)(225 380 235 370)(226 379 236 369)(227 378 237 368)(228 377 238 367)(229 376 239 366)(230 375 240 365)(241 278 251 268)(242 277 252 267)(243 276 253 266)(244 275 254 265)(245 274 255 264)(246 273 256 263)(247 272 257 262)(248 271 258 261)(249 270 259 280)(250 269 260 279)(381 462 391 472)(382 461 392 471)(383 480 393 470)(384 479 394 469)(385 478 395 468)(386 477 396 467)(387 476 397 466)(388 475 398 465)(389 474 399 464)(390 473 400 463)

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

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

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

156 conjugacy classes

class 1 2A···2G3A3B4A4B4C4D4E···4L5A5B6A···6N10A···10N12A···12H12I···12X15A15B15C15D20A···20P30A···30AB60A···60AF
order12···23344444···4556···610···1012···1212···121515151520···2030···3060···60
size11···111222210···10221···12···22···210···1022222···22···22···2

156 irreducible representations

dim111111112222222222
type++++-+++-
imageC1C2C2C2C3C6C6C6Q8D5D10D10C3×Q8C3×D5Dic10C6×D5C6×D5C3×Dic10
kernelC2×C6×Dic10C6×Dic10C2×C6×Dic5C22×C60C22×Dic10C2×Dic10C22×Dic5C22×C20C2×C30C22×C12C2×C12C22×C6C2×C10C22×C4C2×C6C2×C4C23C22
# reps112212244242122841624432

Matrix representation of C2×C6×Dic10 in GL5(𝔽61)

600000
060000
006000
000600
000060
,
600000
01000
00100
000480
000048
,
10000
00100
0601700
000236
0002729
,
600000
006000
060000
0005358
000428

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,48,0,0,0,0,0,48],[1,0,0,0,0,0,0,60,0,0,0,1,17,0,0,0,0,0,2,27,0,0,0,36,29],[60,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,53,42,0,0,0,58,8] >;

C2×C6×Dic10 in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm Dic}_{10}
% in TeX

G:=Group("C2xC6xDic10");
// GroupNames label

G:=SmallGroup(480,1135);
// by ID

G=gap.SmallGroup(480,1135);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,336,1571,192,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^20=1,d^2=c^10,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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