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G = C10×Dic12order 480 = 25·3·5

Direct product of C10 and Dic12

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

Aliases: C10×Dic12, C306Q16, C40.79D6, C20.46D12, C60.184D4, C120.97C22, C60.270C23, C61(C5×Q16), C31(C10×Q16), C1512(C2×Q16), C4.8(C5×D12), C8.16(S3×C10), (C2×C40).14S3, (C2×C24).6C10, C6.12(D4×C10), C12.31(C5×D4), (C2×C120).21C2, C24.18(C2×C10), C30.299(C2×D4), C10.83(C2×D12), C2.14(C10×D12), (C2×C10).55D12, (C2×C30).126D4, (C2×C20).435D6, Dic6.7(C2×C10), (C2×Dic6).4C10, C22.14(C5×D12), C12.31(C22×C10), (C2×C60).529C22, C20.234(C22×S3), (C10×Dic6).14C2, (C5×Dic6).49C22, C4.31(S3×C2×C10), (C2×C8).4(C5×S3), (C2×C6).19(C5×D4), (C2×C4).83(S3×C10), (C2×C12).95(C2×C10), SmallGroup(480,784)

Series: Derived Chief Lower central Upper central

C1C12 — C10×Dic12
C1C3C6C12C60C5×Dic6C10×Dic6 — C10×Dic12
C3C6C12 — C10×Dic12
C1C2×C10C2×C20C2×C40

Generators and relations for C10×Dic12
 G = < a,b,c | a10=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 260 in 120 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], C20 [×2], C20 [×4], C2×C10, C24 [×2], Dic6 [×4], Dic6 [×2], C2×Dic3 [×2], C2×C12, C30, C30 [×2], C2×Q16, C40 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×6], Dic12 [×4], C2×C24, C2×Dic6 [×2], C5×Dic3 [×4], C60 [×2], C2×C30, C2×C40, C5×Q16 [×4], Q8×C10 [×2], C2×Dic12, C120 [×2], C5×Dic6 [×4], C5×Dic6 [×2], C10×Dic3 [×2], C2×C60, C10×Q16, C5×Dic12 [×4], C2×C120, C10×Dic6 [×2], C10×Dic12
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], Q16 [×2], C2×D4, C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C2×Q16, C5×D4 [×2], C22×C10, Dic12 [×2], C2×D12, S3×C10 [×3], C5×Q16 [×2], D4×C10, C2×Dic12, C5×D12 [×2], S3×C2×C10, C10×Q16, C5×Dic12 [×2], C10×D12, C10×Dic12

Smallest permutation representation of C10×Dic12
Regular action on 480 points
Generators in S480
(1 117 382 427 85 288 212 136 224 400)(2 118 383 428 86 265 213 137 225 401)(3 119 384 429 87 266 214 138 226 402)(4 120 361 430 88 267 215 139 227 403)(5 97 362 431 89 268 216 140 228 404)(6 98 363 432 90 269 193 141 229 405)(7 99 364 409 91 270 194 142 230 406)(8 100 365 410 92 271 195 143 231 407)(9 101 366 411 93 272 196 144 232 408)(10 102 367 412 94 273 197 121 233 385)(11 103 368 413 95 274 198 122 234 386)(12 104 369 414 96 275 199 123 235 387)(13 105 370 415 73 276 200 124 236 388)(14 106 371 416 74 277 201 125 237 389)(15 107 372 417 75 278 202 126 238 390)(16 108 373 418 76 279 203 127 239 391)(17 109 374 419 77 280 204 128 240 392)(18 110 375 420 78 281 205 129 217 393)(19 111 376 421 79 282 206 130 218 394)(20 112 377 422 80 283 207 131 219 395)(21 113 378 423 81 284 208 132 220 396)(22 114 379 424 82 285 209 133 221 397)(23 115 380 425 83 286 210 134 222 398)(24 116 381 426 84 287 211 135 223 399)(25 55 159 192 303 359 470 439 330 253)(26 56 160 169 304 360 471 440 331 254)(27 57 161 170 305 337 472 441 332 255)(28 58 162 171 306 338 473 442 333 256)(29 59 163 172 307 339 474 443 334 257)(30 60 164 173 308 340 475 444 335 258)(31 61 165 174 309 341 476 445 336 259)(32 62 166 175 310 342 477 446 313 260)(33 63 167 176 311 343 478 447 314 261)(34 64 168 177 312 344 479 448 315 262)(35 65 145 178 289 345 480 449 316 263)(36 66 146 179 290 346 457 450 317 264)(37 67 147 180 291 347 458 451 318 241)(38 68 148 181 292 348 459 452 319 242)(39 69 149 182 293 349 460 453 320 243)(40 70 150 183 294 350 461 454 321 244)(41 71 151 184 295 351 462 455 322 245)(42 72 152 185 296 352 463 456 323 246)(43 49 153 186 297 353 464 433 324 247)(44 50 154 187 298 354 465 434 325 248)(45 51 155 188 299 355 466 435 326 249)(46 52 156 189 300 356 467 436 327 250)(47 53 157 190 301 357 468 437 328 251)(48 54 158 191 302 358 469 438 329 252)
(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)
(1 341 13 353)(2 340 14 352)(3 339 15 351)(4 338 16 350)(5 337 17 349)(6 360 18 348)(7 359 19 347)(8 358 20 346)(9 357 21 345)(10 356 22 344)(11 355 23 343)(12 354 24 342)(25 282 37 270)(26 281 38 269)(27 280 39 268)(28 279 40 267)(29 278 41 266)(30 277 42 265)(31 276 43 288)(32 275 44 287)(33 274 45 286)(34 273 46 285)(35 272 47 284)(36 271 48 283)(49 212 61 200)(50 211 62 199)(51 210 63 198)(52 209 64 197)(53 208 65 196)(54 207 66 195)(55 206 67 194)(56 205 68 193)(57 204 69 216)(58 203 70 215)(59 202 71 214)(60 201 72 213)(73 247 85 259)(74 246 86 258)(75 245 87 257)(76 244 88 256)(77 243 89 255)(78 242 90 254)(79 241 91 253)(80 264 92 252)(81 263 93 251)(82 262 94 250)(83 261 95 249)(84 260 96 248)(97 472 109 460)(98 471 110 459)(99 470 111 458)(100 469 112 457)(101 468 113 480)(102 467 114 479)(103 466 115 478)(104 465 116 477)(105 464 117 476)(106 463 118 475)(107 462 119 474)(108 461 120 473)(121 156 133 168)(122 155 134 167)(123 154 135 166)(124 153 136 165)(125 152 137 164)(126 151 138 163)(127 150 139 162)(128 149 140 161)(129 148 141 160)(130 147 142 159)(131 146 143 158)(132 145 144 157)(169 217 181 229)(170 240 182 228)(171 239 183 227)(172 238 184 226)(173 237 185 225)(174 236 186 224)(175 235 187 223)(176 234 188 222)(177 233 189 221)(178 232 190 220)(179 231 191 219)(180 230 192 218)(289 408 301 396)(290 407 302 395)(291 406 303 394)(292 405 304 393)(293 404 305 392)(294 403 306 391)(295 402 307 390)(296 401 308 389)(297 400 309 388)(298 399 310 387)(299 398 311 386)(300 397 312 385)(313 414 325 426)(314 413 326 425)(315 412 327 424)(316 411 328 423)(317 410 329 422)(318 409 330 421)(319 432 331 420)(320 431 332 419)(321 430 333 418)(322 429 334 417)(323 428 335 416)(324 427 336 415)(361 442 373 454)(362 441 374 453)(363 440 375 452)(364 439 376 451)(365 438 377 450)(366 437 378 449)(367 436 379 448)(368 435 380 447)(369 434 381 446)(370 433 382 445)(371 456 383 444)(372 455 384 443)

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

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

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

150 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B5C5D6A6B6C8A8B8C8D10A···10L12A12B12C12D15A15B15C15D20A···20H20I···20X24A···24H30A···30L40A···40P60A···60P120A···120AF
order122234444445555666888810···10121212121515151520···2020···2024···2430···3040···4060···60120···120
size111122212121212111122222221···1222222222···212···122···22···22···22···22···2

150 irreducible representations

dim11111111222222222222222222
type+++++++++-++-
imageC1C2C2C2C5C10C10C10S3D4D4D6D6Q16D12D12C5×S3C5×D4C5×D4Dic12S3×C10S3×C10C5×Q16C5×D12C5×D12C5×Dic12
kernelC10×Dic12C5×Dic12C2×C120C10×Dic6C2×Dic12Dic12C2×C24C2×Dic6C2×C40C60C2×C30C40C2×C20C30C20C2×C10C2×C8C12C2×C6C10C8C2×C4C6C4C22C2
# reps14124164811121422444884168832

Matrix representation of C10×Dic12 in GL3(𝔽241) generated by

24000
0360
0036
,
24000
0232105
0136127
,
100
020131
0111221
G:=sub<GL(3,GF(241))| [240,0,0,0,36,0,0,0,36],[240,0,0,0,232,136,0,105,127],[1,0,0,0,20,111,0,131,221] >;

C10×Dic12 in GAP, Magma, Sage, TeX

C_{10}\times {\rm Dic}_{12}
% in TeX

G:=Group("C10xDic12");
// GroupNames label

G:=SmallGroup(480,784);
// by ID

G=gap.SmallGroup(480,784);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,926,646,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c|a^10=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽