direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic5×C24, C40⋊7C12, C120⋊17C4, C30.32C42, C5⋊4(C4×C24), C15⋊15(C4×C8), C5⋊2C8⋊9C12, C6.18(C8×D5), C2.2(D5×C24), C30.50(C2×C8), C10.8(C4×C12), (C2×C40).11C6, C12.90(C4×D5), C4.20(D5×C12), (C2×C24).20D5, C20.46(C2×C12), C10.12(C2×C24), (C2×C120).27C2, C60.212(C2×C4), C6.15(C4×Dic5), C4.12(C6×Dic5), C2.2(C12×Dic5), C22.8(D5×C12), (C2×C12).441D10, (C4×Dic5).14C6, (C6×Dic5).31C4, C12.51(C2×Dic5), (C2×C60).541C22, (C2×Dic5).15C12, (C12×Dic5).29C2, (C3×C5⋊2C8)⋊19C4, (C2×C6).57(C4×D5), (C2×C8).10(C3×D5), (C2×C4).91(C6×D5), (C6×C5⋊2C8).28C2, (C2×C5⋊2C8).14C6, (C2×C20).107(C2×C6), (C2×C10).29(C2×C12), (C2×C30).143(C2×C4), SmallGroup(480,91)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — Dic5×C24 |
Generators and relations for Dic5×C24
G = < a,b,c | a24=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 176 in 88 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, C2×C8, C2×C8, Dic5, C20, C2×C10, C24, C24, C2×C12, C2×C12, C30, C30, C4×C8, C5⋊2C8, C40, C2×Dic5, C2×C20, C4×C12, C2×C24, C2×C24, C3×Dic5, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, C4×C24, C3×C5⋊2C8, C120, C6×Dic5, C2×C60, C8×Dic5, C6×C5⋊2C8, C12×Dic5, C2×C120, Dic5×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D5, C12, C2×C6, C42, C2×C8, Dic5, D10, C24, C2×C12, C3×D5, C4×C8, C4×D5, C2×Dic5, C4×C12, C2×C24, C3×Dic5, C6×D5, C8×D5, C4×Dic5, C4×C24, D5×C12, C6×Dic5, C8×Dic5, D5×C24, C12×Dic5, Dic5×C24
►Regular action on 480 points
Generators in S480
(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)
(1 200 87 65 40 324 157 366 277 107)(2 201 88 66 41 325 158 367 278 108)(3 202 89 67 42 326 159 368 279 109)(4 203 90 68 43 327 160 369 280 110)(5 204 91 69 44 328 161 370 281 111)(6 205 92 70 45 329 162 371 282 112)(7 206 93 71 46 330 163 372 283 113)(8 207 94 72 47 331 164 373 284 114)(9 208 95 49 48 332 165 374 285 115)(10 209 96 50 25 333 166 375 286 116)(11 210 73 51 26 334 167 376 287 117)(12 211 74 52 27 335 168 377 288 118)(13 212 75 53 28 336 145 378 265 119)(14 213 76 54 29 313 146 379 266 120)(15 214 77 55 30 314 147 380 267 97)(16 215 78 56 31 315 148 381 268 98)(17 216 79 57 32 316 149 382 269 99)(18 193 80 58 33 317 150 383 270 100)(19 194 81 59 34 318 151 384 271 101)(20 195 82 60 35 319 152 361 272 102)(21 196 83 61 36 320 153 362 273 103)(22 197 84 62 37 321 154 363 274 104)(23 198 85 63 38 322 155 364 275 105)(24 199 86 64 39 323 156 365 276 106)(121 395 261 360 461 185 430 443 222 311)(122 396 262 337 462 186 431 444 223 312)(123 397 263 338 463 187 432 445 224 289)(124 398 264 339 464 188 409 446 225 290)(125 399 241 340 465 189 410 447 226 291)(126 400 242 341 466 190 411 448 227 292)(127 401 243 342 467 191 412 449 228 293)(128 402 244 343 468 192 413 450 229 294)(129 403 245 344 469 169 414 451 230 295)(130 404 246 345 470 170 415 452 231 296)(131 405 247 346 471 171 416 453 232 297)(132 406 248 347 472 172 417 454 233 298)(133 407 249 348 473 173 418 455 234 299)(134 408 250 349 474 174 419 456 235 300)(135 385 251 350 475 175 420 433 236 301)(136 386 252 351 476 176 421 434 237 302)(137 387 253 352 477 177 422 435 238 303)(138 388 254 353 478 178 423 436 239 304)(139 389 255 354 479 179 424 437 240 305)(140 390 256 355 480 180 425 438 217 306)(141 391 257 356 457 181 426 439 218 307)(142 392 258 357 458 182 427 440 219 308)(143 393 259 358 459 183 428 441 220 309)(144 394 260 359 460 184 429 442 221 310)
(1 360 324 222)(2 337 325 223)(3 338 326 224)(4 339 327 225)(5 340 328 226)(6 341 329 227)(7 342 330 228)(8 343 331 229)(9 344 332 230)(10 345 333 231)(11 346 334 232)(12 347 335 233)(13 348 336 234)(14 349 313 235)(15 350 314 236)(16 351 315 237)(17 352 316 238)(18 353 317 239)(19 354 318 240)(20 355 319 217)(21 356 320 218)(22 357 321 219)(23 358 322 220)(24 359 323 221)(25 296 116 470)(26 297 117 471)(27 298 118 472)(28 299 119 473)(29 300 120 474)(30 301 97 475)(31 302 98 476)(32 303 99 477)(33 304 100 478)(34 305 101 479)(35 306 102 480)(36 307 103 457)(37 308 104 458)(38 309 105 459)(39 310 106 460)(40 311 107 461)(41 312 108 462)(42 289 109 463)(43 290 110 464)(44 291 111 465)(45 292 112 466)(46 293 113 467)(47 294 114 468)(48 295 115 469)(49 129 285 169)(50 130 286 170)(51 131 287 171)(52 132 288 172)(53 133 265 173)(54 134 266 174)(55 135 267 175)(56 136 268 176)(57 137 269 177)(58 138 270 178)(59 139 271 179)(60 140 272 180)(61 141 273 181)(62 142 274 182)(63 143 275 183)(64 144 276 184)(65 121 277 185)(66 122 278 186)(67 123 279 187)(68 124 280 188)(69 125 281 189)(70 126 282 190)(71 127 283 191)(72 128 284 192)(73 405 376 416)(74 406 377 417)(75 407 378 418)(76 408 379 419)(77 385 380 420)(78 386 381 421)(79 387 382 422)(80 388 383 423)(81 389 384 424)(82 390 361 425)(83 391 362 426)(84 392 363 427)(85 393 364 428)(86 394 365 429)(87 395 366 430)(88 396 367 431)(89 397 368 432)(90 398 369 409)(91 399 370 410)(92 400 371 411)(93 401 372 412)(94 402 373 413)(95 403 374 414)(96 404 375 415)(145 455 212 249)(146 456 213 250)(147 433 214 251)(148 434 215 252)(149 435 216 253)(150 436 193 254)(151 437 194 255)(152 438 195 256)(153 439 196 257)(154 440 197 258)(155 441 198 259)(156 442 199 260)(157 443 200 261)(158 444 201 262)(159 445 202 263)(160 446 203 264)(161 447 204 241)(162 448 205 242)(163 449 206 243)(164 450 207 244)(165 451 208 245)(166 452 209 246)(167 453 210 247)(168 454 211 248)
G:=sub<Sym(480)| (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), (1,200,87,65,40,324,157,366,277,107)(2,201,88,66,41,325,158,367,278,108)(3,202,89,67,42,326,159,368,279,109)(4,203,90,68,43,327,160,369,280,110)(5,204,91,69,44,328,161,370,281,111)(6,205,92,70,45,329,162,371,282,112)(7,206,93,71,46,330,163,372,283,113)(8,207,94,72,47,331,164,373,284,114)(9,208,95,49,48,332,165,374,285,115)(10,209,96,50,25,333,166,375,286,116)(11,210,73,51,26,334,167,376,287,117)(12,211,74,52,27,335,168,377,288,118)(13,212,75,53,28,336,145,378,265,119)(14,213,76,54,29,313,146,379,266,120)(15,214,77,55,30,314,147,380,267,97)(16,215,78,56,31,315,148,381,268,98)(17,216,79,57,32,316,149,382,269,99)(18,193,80,58,33,317,150,383,270,100)(19,194,81,59,34,318,151,384,271,101)(20,195,82,60,35,319,152,361,272,102)(21,196,83,61,36,320,153,362,273,103)(22,197,84,62,37,321,154,363,274,104)(23,198,85,63,38,322,155,364,275,105)(24,199,86,64,39,323,156,365,276,106)(121,395,261,360,461,185,430,443,222,311)(122,396,262,337,462,186,431,444,223,312)(123,397,263,338,463,187,432,445,224,289)(124,398,264,339,464,188,409,446,225,290)(125,399,241,340,465,189,410,447,226,291)(126,400,242,341,466,190,411,448,227,292)(127,401,243,342,467,191,412,449,228,293)(128,402,244,343,468,192,413,450,229,294)(129,403,245,344,469,169,414,451,230,295)(130,404,246,345,470,170,415,452,231,296)(131,405,247,346,471,171,416,453,232,297)(132,406,248,347,472,172,417,454,233,298)(133,407,249,348,473,173,418,455,234,299)(134,408,250,349,474,174,419,456,235,300)(135,385,251,350,475,175,420,433,236,301)(136,386,252,351,476,176,421,434,237,302)(137,387,253,352,477,177,422,435,238,303)(138,388,254,353,478,178,423,436,239,304)(139,389,255,354,479,179,424,437,240,305)(140,390,256,355,480,180,425,438,217,306)(141,391,257,356,457,181,426,439,218,307)(142,392,258,357,458,182,427,440,219,308)(143,393,259,358,459,183,428,441,220,309)(144,394,260,359,460,184,429,442,221,310), (1,360,324,222)(2,337,325,223)(3,338,326,224)(4,339,327,225)(5,340,328,226)(6,341,329,227)(7,342,330,228)(8,343,331,229)(9,344,332,230)(10,345,333,231)(11,346,334,232)(12,347,335,233)(13,348,336,234)(14,349,313,235)(15,350,314,236)(16,351,315,237)(17,352,316,238)(18,353,317,239)(19,354,318,240)(20,355,319,217)(21,356,320,218)(22,357,321,219)(23,358,322,220)(24,359,323,221)(25,296,116,470)(26,297,117,471)(27,298,118,472)(28,299,119,473)(29,300,120,474)(30,301,97,475)(31,302,98,476)(32,303,99,477)(33,304,100,478)(34,305,101,479)(35,306,102,480)(36,307,103,457)(37,308,104,458)(38,309,105,459)(39,310,106,460)(40,311,107,461)(41,312,108,462)(42,289,109,463)(43,290,110,464)(44,291,111,465)(45,292,112,466)(46,293,113,467)(47,294,114,468)(48,295,115,469)(49,129,285,169)(50,130,286,170)(51,131,287,171)(52,132,288,172)(53,133,265,173)(54,134,266,174)(55,135,267,175)(56,136,268,176)(57,137,269,177)(58,138,270,178)(59,139,271,179)(60,140,272,180)(61,141,273,181)(62,142,274,182)(63,143,275,183)(64,144,276,184)(65,121,277,185)(66,122,278,186)(67,123,279,187)(68,124,280,188)(69,125,281,189)(70,126,282,190)(71,127,283,191)(72,128,284,192)(73,405,376,416)(74,406,377,417)(75,407,378,418)(76,408,379,419)(77,385,380,420)(78,386,381,421)(79,387,382,422)(80,388,383,423)(81,389,384,424)(82,390,361,425)(83,391,362,426)(84,392,363,427)(85,393,364,428)(86,394,365,429)(87,395,366,430)(88,396,367,431)(89,397,368,432)(90,398,369,409)(91,399,370,410)(92,400,371,411)(93,401,372,412)(94,402,373,413)(95,403,374,414)(96,404,375,415)(145,455,212,249)(146,456,213,250)(147,433,214,251)(148,434,215,252)(149,435,216,253)(150,436,193,254)(151,437,194,255)(152,438,195,256)(153,439,196,257)(154,440,197,258)(155,441,198,259)(156,442,199,260)(157,443,200,261)(158,444,201,262)(159,445,202,263)(160,446,203,264)(161,447,204,241)(162,448,205,242)(163,449,206,243)(164,450,207,244)(165,451,208,245)(166,452,209,246)(167,453,210,247)(168,454,211,248)>;
G:=Group( (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), (1,200,87,65,40,324,157,366,277,107)(2,201,88,66,41,325,158,367,278,108)(3,202,89,67,42,326,159,368,279,109)(4,203,90,68,43,327,160,369,280,110)(5,204,91,69,44,328,161,370,281,111)(6,205,92,70,45,329,162,371,282,112)(7,206,93,71,46,330,163,372,283,113)(8,207,94,72,47,331,164,373,284,114)(9,208,95,49,48,332,165,374,285,115)(10,209,96,50,25,333,166,375,286,116)(11,210,73,51,26,334,167,376,287,117)(12,211,74,52,27,335,168,377,288,118)(13,212,75,53,28,336,145,378,265,119)(14,213,76,54,29,313,146,379,266,120)(15,214,77,55,30,314,147,380,267,97)(16,215,78,56,31,315,148,381,268,98)(17,216,79,57,32,316,149,382,269,99)(18,193,80,58,33,317,150,383,270,100)(19,194,81,59,34,318,151,384,271,101)(20,195,82,60,35,319,152,361,272,102)(21,196,83,61,36,320,153,362,273,103)(22,197,84,62,37,321,154,363,274,104)(23,198,85,63,38,322,155,364,275,105)(24,199,86,64,39,323,156,365,276,106)(121,395,261,360,461,185,430,443,222,311)(122,396,262,337,462,186,431,444,223,312)(123,397,263,338,463,187,432,445,224,289)(124,398,264,339,464,188,409,446,225,290)(125,399,241,340,465,189,410,447,226,291)(126,400,242,341,466,190,411,448,227,292)(127,401,243,342,467,191,412,449,228,293)(128,402,244,343,468,192,413,450,229,294)(129,403,245,344,469,169,414,451,230,295)(130,404,246,345,470,170,415,452,231,296)(131,405,247,346,471,171,416,453,232,297)(132,406,248,347,472,172,417,454,233,298)(133,407,249,348,473,173,418,455,234,299)(134,408,250,349,474,174,419,456,235,300)(135,385,251,350,475,175,420,433,236,301)(136,386,252,351,476,176,421,434,237,302)(137,387,253,352,477,177,422,435,238,303)(138,388,254,353,478,178,423,436,239,304)(139,389,255,354,479,179,424,437,240,305)(140,390,256,355,480,180,425,438,217,306)(141,391,257,356,457,181,426,439,218,307)(142,392,258,357,458,182,427,440,219,308)(143,393,259,358,459,183,428,441,220,309)(144,394,260,359,460,184,429,442,221,310), (1,360,324,222)(2,337,325,223)(3,338,326,224)(4,339,327,225)(5,340,328,226)(6,341,329,227)(7,342,330,228)(8,343,331,229)(9,344,332,230)(10,345,333,231)(11,346,334,232)(12,347,335,233)(13,348,336,234)(14,349,313,235)(15,350,314,236)(16,351,315,237)(17,352,316,238)(18,353,317,239)(19,354,318,240)(20,355,319,217)(21,356,320,218)(22,357,321,219)(23,358,322,220)(24,359,323,221)(25,296,116,470)(26,297,117,471)(27,298,118,472)(28,299,119,473)(29,300,120,474)(30,301,97,475)(31,302,98,476)(32,303,99,477)(33,304,100,478)(34,305,101,479)(35,306,102,480)(36,307,103,457)(37,308,104,458)(38,309,105,459)(39,310,106,460)(40,311,107,461)(41,312,108,462)(42,289,109,463)(43,290,110,464)(44,291,111,465)(45,292,112,466)(46,293,113,467)(47,294,114,468)(48,295,115,469)(49,129,285,169)(50,130,286,170)(51,131,287,171)(52,132,288,172)(53,133,265,173)(54,134,266,174)(55,135,267,175)(56,136,268,176)(57,137,269,177)(58,138,270,178)(59,139,271,179)(60,140,272,180)(61,141,273,181)(62,142,274,182)(63,143,275,183)(64,144,276,184)(65,121,277,185)(66,122,278,186)(67,123,279,187)(68,124,280,188)(69,125,281,189)(70,126,282,190)(71,127,283,191)(72,128,284,192)(73,405,376,416)(74,406,377,417)(75,407,378,418)(76,408,379,419)(77,385,380,420)(78,386,381,421)(79,387,382,422)(80,388,383,423)(81,389,384,424)(82,390,361,425)(83,391,362,426)(84,392,363,427)(85,393,364,428)(86,394,365,429)(87,395,366,430)(88,396,367,431)(89,397,368,432)(90,398,369,409)(91,399,370,410)(92,400,371,411)(93,401,372,412)(94,402,373,413)(95,403,374,414)(96,404,375,415)(145,455,212,249)(146,456,213,250)(147,433,214,251)(148,434,215,252)(149,435,216,253)(150,436,193,254)(151,437,194,255)(152,438,195,256)(153,439,196,257)(154,440,197,258)(155,441,198,259)(156,442,199,260)(157,443,200,261)(158,444,201,262)(159,445,202,263)(160,446,203,264)(161,447,204,241)(162,448,205,242)(163,449,206,243)(164,450,207,244)(165,451,208,245)(166,452,209,246)(167,453,210,247)(168,454,211,248) );
G=PermutationGroup([[(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)], [(1,200,87,65,40,324,157,366,277,107),(2,201,88,66,41,325,158,367,278,108),(3,202,89,67,42,326,159,368,279,109),(4,203,90,68,43,327,160,369,280,110),(5,204,91,69,44,328,161,370,281,111),(6,205,92,70,45,329,162,371,282,112),(7,206,93,71,46,330,163,372,283,113),(8,207,94,72,47,331,164,373,284,114),(9,208,95,49,48,332,165,374,285,115),(10,209,96,50,25,333,166,375,286,116),(11,210,73,51,26,334,167,376,287,117),(12,211,74,52,27,335,168,377,288,118),(13,212,75,53,28,336,145,378,265,119),(14,213,76,54,29,313,146,379,266,120),(15,214,77,55,30,314,147,380,267,97),(16,215,78,56,31,315,148,381,268,98),(17,216,79,57,32,316,149,382,269,99),(18,193,80,58,33,317,150,383,270,100),(19,194,81,59,34,318,151,384,271,101),(20,195,82,60,35,319,152,361,272,102),(21,196,83,61,36,320,153,362,273,103),(22,197,84,62,37,321,154,363,274,104),(23,198,85,63,38,322,155,364,275,105),(24,199,86,64,39,323,156,365,276,106),(121,395,261,360,461,185,430,443,222,311),(122,396,262,337,462,186,431,444,223,312),(123,397,263,338,463,187,432,445,224,289),(124,398,264,339,464,188,409,446,225,290),(125,399,241,340,465,189,410,447,226,291),(126,400,242,341,466,190,411,448,227,292),(127,401,243,342,467,191,412,449,228,293),(128,402,244,343,468,192,413,450,229,294),(129,403,245,344,469,169,414,451,230,295),(130,404,246,345,470,170,415,452,231,296),(131,405,247,346,471,171,416,453,232,297),(132,406,248,347,472,172,417,454,233,298),(133,407,249,348,473,173,418,455,234,299),(134,408,250,349,474,174,419,456,235,300),(135,385,251,350,475,175,420,433,236,301),(136,386,252,351,476,176,421,434,237,302),(137,387,253,352,477,177,422,435,238,303),(138,388,254,353,478,178,423,436,239,304),(139,389,255,354,479,179,424,437,240,305),(140,390,256,355,480,180,425,438,217,306),(141,391,257,356,457,181,426,439,218,307),(142,392,258,357,458,182,427,440,219,308),(143,393,259,358,459,183,428,441,220,309),(144,394,260,359,460,184,429,442,221,310)], [(1,360,324,222),(2,337,325,223),(3,338,326,224),(4,339,327,225),(5,340,328,226),(6,341,329,227),(7,342,330,228),(8,343,331,229),(9,344,332,230),(10,345,333,231),(11,346,334,232),(12,347,335,233),(13,348,336,234),(14,349,313,235),(15,350,314,236),(16,351,315,237),(17,352,316,238),(18,353,317,239),(19,354,318,240),(20,355,319,217),(21,356,320,218),(22,357,321,219),(23,358,322,220),(24,359,323,221),(25,296,116,470),(26,297,117,471),(27,298,118,472),(28,299,119,473),(29,300,120,474),(30,301,97,475),(31,302,98,476),(32,303,99,477),(33,304,100,478),(34,305,101,479),(35,306,102,480),(36,307,103,457),(37,308,104,458),(38,309,105,459),(39,310,106,460),(40,311,107,461),(41,312,108,462),(42,289,109,463),(43,290,110,464),(44,291,111,465),(45,292,112,466),(46,293,113,467),(47,294,114,468),(48,295,115,469),(49,129,285,169),(50,130,286,170),(51,131,287,171),(52,132,288,172),(53,133,265,173),(54,134,266,174),(55,135,267,175),(56,136,268,176),(57,137,269,177),(58,138,270,178),(59,139,271,179),(60,140,272,180),(61,141,273,181),(62,142,274,182),(63,143,275,183),(64,144,276,184),(65,121,277,185),(66,122,278,186),(67,123,279,187),(68,124,280,188),(69,125,281,189),(70,126,282,190),(71,127,283,191),(72,128,284,192),(73,405,376,416),(74,406,377,417),(75,407,378,418),(76,408,379,419),(77,385,380,420),(78,386,381,421),(79,387,382,422),(80,388,383,423),(81,389,384,424),(82,390,361,425),(83,391,362,426),(84,392,363,427),(85,393,364,428),(86,394,365,429),(87,395,366,430),(88,396,367,431),(89,397,368,432),(90,398,369,409),(91,399,370,410),(92,400,371,411),(93,401,372,412),(94,402,373,413),(95,403,374,414),(96,404,375,415),(145,455,212,249),(146,456,213,250),(147,433,214,251),(148,434,215,252),(149,435,216,253),(150,436,193,254),(151,437,194,255),(152,438,195,256),(153,439,196,257),(154,440,197,258),(155,441,198,259),(156,442,199,260),(157,443,200,261),(158,444,201,262),(159,445,202,263),(160,446,203,264),(161,447,204,241),(162,448,205,242),(163,449,206,243),(164,450,207,244),(165,451,208,245),(166,452,209,246),(167,453,210,247),(168,454,211,248)]])
192 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 6A | ··· | 6F | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 12A | ··· | 12H | 12I | ··· | 12X | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24P | 24Q | ··· | 24AF | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
192 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C12 | C24 | D5 | Dic5 | D10 | C3×D5 | C4×D5 | C4×D5 | C3×Dic5 | C6×D5 | C8×D5 | D5×C12 | D5×C12 | D5×C24 |
| kernel | Dic5×C24 | C6×C5⋊2C8 | C12×Dic5 | C2×C120 | C8×Dic5 | C3×C5⋊2C8 | C120 | C6×Dic5 | C2×C5⋊2C8 | C4×Dic5 | C2×C40 | C3×Dic5 | C5⋊2C8 | C40 | C2×Dic5 | Dic5 | C2×C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 16 | 8 | 8 | 8 | 32 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of Dic5×C24 ►in GL4(𝔽241) generated by
| 226 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 190 |
| 240 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 182 | 190 |
| 0 | 0 | 73 | 59 |
G:=sub<GL(4,GF(241))| [226,0,0,0,0,177,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,240,0,0,0,0,0,1,0,0,240,190],[240,0,0,0,0,177,0,0,0,0,182,73,0,0,190,59] >;
Dic5×C24 in GAP, Magma, Sage, TeX
{\rm Dic}_5\times C_{24} % in TeX
G:=Group("Dic5xC24"); // GroupNames label
G:=SmallGroup(480,91);
// by ID
G=gap.SmallGroup(480,91);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,176,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^24=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations