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G = Dic1510Q8order 480 = 25·3·5

The semidirect product of Dic15 and Q8 acting through Inn(Dic15)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1510Q8, Dic3011C4, C1518(C4×Q8), C4⋊C4.7D15, C4.4(C4×D15), C6.38(Q8×D5), C2.1(Q8×D15), C20.59(C4×S3), C60.84(C2×C4), C12.27(C4×D5), (C2×C4).28D30, C30.91(C2×Q8), C10.38(S3×Q8), (C2×C20).208D6, (C2×C12).206D10, C34(Dic53Q8), C55(Dic6⋊C4), (C2×Dic30).8C2, C30.4Q8.5C2, C30.218(C4○D4), C6.95(D42D5), C2.3(D42D15), C30.159(C22×C4), (C2×C60).176C22, (C2×C30).284C23, Dic15.30(C2×C4), (C4×Dic15).10C2, C10.95(D42S3), C22.15(C22×D15), (C2×Dic15).160C22, C6.64(C2×C4×D5), (C5×C4⋊C4).4S3, C10.96(S3×C2×C4), (C3×C4⋊C4).4D5, C2.10(C2×C4×D15), (C15×C4⋊C4).4C2, (C2×C6).280(C22×D5), (C2×C10).279(C22×S3), SmallGroup(480,852)

Series: Derived Chief Lower central Upper central

C1C30 — Dic1510Q8
C1C5C15C30C2×C30C2×Dic15C4×Dic15 — Dic1510Q8
C15C30 — Dic1510Q8
C1C22C4⋊C4

Generators and relations for Dic1510Q8
 G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a31, bc=cb >

Subgroups: 612 in 140 conjugacy classes, 65 normal (33 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10 [×3], Dic3 [×7], C12 [×2], C12 [×2], C2×C6, C15, C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×Q8, Dic5 [×7], C20 [×2], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C4×Dic3 [×3], Dic3⋊C4 [×2], C3×C4⋊C4, C2×Dic6, Dic15 [×6], Dic15, C60 [×2], C60 [×2], C2×C30, C4×Dic5 [×3], C10.D4 [×2], C5×C4⋊C4, C2×Dic10, Dic6⋊C4, Dic30 [×4], C2×Dic15 [×2], C2×Dic15 [×2], C2×C60, C2×C60 [×2], Dic53Q8, C4×Dic15, C4×Dic15 [×2], C30.4Q8 [×2], C15×C4⋊C4, C2×Dic30, Dic1510Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D5, D6 [×3], C22×C4, C2×Q8, C4○D4, D10 [×3], C4×S3 [×2], C22×S3, D15, C4×Q8, C4×D5 [×2], C22×D5, S3×C2×C4, D42S3, S3×Q8, D30 [×3], C2×C4×D5, D42D5, Q8×D5, Dic6⋊C4, C4×D15 [×2], C22×D15, Dic53Q8, C2×C4×D15, D42D15, Q8×D15, Dic1510Q8

Smallest permutation representation of Dic1510Q8
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)
(1 218 31 188)(2 217 32 187)(3 216 33 186)(4 215 34 185)(5 214 35 184)(6 213 36 183)(7 212 37 182)(8 211 38 181)(9 210 39 240)(10 209 40 239)(11 208 41 238)(12 207 42 237)(13 206 43 236)(14 205 44 235)(15 204 45 234)(16 203 46 233)(17 202 47 232)(18 201 48 231)(19 200 49 230)(20 199 50 229)(21 198 51 228)(22 197 52 227)(23 196 53 226)(24 195 54 225)(25 194 55 224)(26 193 56 223)(27 192 57 222)(28 191 58 221)(29 190 59 220)(30 189 60 219)(61 164 91 134)(62 163 92 133)(63 162 93 132)(64 161 94 131)(65 160 95 130)(66 159 96 129)(67 158 97 128)(68 157 98 127)(69 156 99 126)(70 155 100 125)(71 154 101 124)(72 153 102 123)(73 152 103 122)(74 151 104 121)(75 150 105 180)(76 149 106 179)(77 148 107 178)(78 147 108 177)(79 146 109 176)(80 145 110 175)(81 144 111 174)(82 143 112 173)(83 142 113 172)(84 141 114 171)(85 140 115 170)(86 139 116 169)(87 138 117 168)(88 137 118 167)(89 136 119 166)(90 135 120 165)(241 399 271 369)(242 398 272 368)(243 397 273 367)(244 396 274 366)(245 395 275 365)(246 394 276 364)(247 393 277 363)(248 392 278 362)(249 391 279 361)(250 390 280 420)(251 389 281 419)(252 388 282 418)(253 387 283 417)(254 386 284 416)(255 385 285 415)(256 384 286 414)(257 383 287 413)(258 382 288 412)(259 381 289 411)(260 380 290 410)(261 379 291 409)(262 378 292 408)(263 377 293 407)(264 376 294 406)(265 375 295 405)(266 374 296 404)(267 373 297 403)(268 372 298 402)(269 371 299 401)(270 370 300 400)(301 453 331 423)(302 452 332 422)(303 451 333 421)(304 450 334 480)(305 449 335 479)(306 448 336 478)(307 447 337 477)(308 446 338 476)(309 445 339 475)(310 444 340 474)(311 443 341 473)(312 442 342 472)(313 441 343 471)(314 440 344 470)(315 439 345 469)(316 438 346 468)(317 437 347 467)(318 436 348 466)(319 435 349 465)(320 434 350 464)(321 433 351 463)(322 432 352 462)(323 431 353 461)(324 430 354 460)(325 429 355 459)(326 428 356 458)(327 427 357 457)(328 426 358 456)(329 425 359 455)(330 424 360 454)
(1 154 365 448)(2 125 366 479)(3 156 367 450)(4 127 368 421)(5 158 369 452)(6 129 370 423)(7 160 371 454)(8 131 372 425)(9 162 373 456)(10 133 374 427)(11 164 375 458)(12 135 376 429)(13 166 377 460)(14 137 378 431)(15 168 379 462)(16 139 380 433)(17 170 381 464)(18 141 382 435)(19 172 383 466)(20 143 384 437)(21 174 385 468)(22 145 386 439)(23 176 387 470)(24 147 388 441)(25 178 389 472)(26 149 390 443)(27 180 391 474)(28 151 392 445)(29 122 393 476)(30 153 394 447)(31 124 395 478)(32 155 396 449)(33 126 397 480)(34 157 398 451)(35 128 399 422)(36 159 400 453)(37 130 401 424)(38 161 402 455)(39 132 403 426)(40 163 404 457)(41 134 405 428)(42 165 406 459)(43 136 407 430)(44 167 408 461)(45 138 409 432)(46 169 410 463)(47 140 411 434)(48 171 412 465)(49 142 413 436)(50 173 414 467)(51 144 415 438)(52 175 416 469)(53 146 417 440)(54 177 418 471)(55 148 419 442)(56 179 420 473)(57 150 361 444)(58 121 362 475)(59 152 363 446)(60 123 364 477)(61 265 356 238)(62 296 357 209)(63 267 358 240)(64 298 359 211)(65 269 360 182)(66 300 301 213)(67 271 302 184)(68 242 303 215)(69 273 304 186)(70 244 305 217)(71 275 306 188)(72 246 307 219)(73 277 308 190)(74 248 309 221)(75 279 310 192)(76 250 311 223)(77 281 312 194)(78 252 313 225)(79 283 314 196)(80 254 315 227)(81 285 316 198)(82 256 317 229)(83 287 318 200)(84 258 319 231)(85 289 320 202)(86 260 321 233)(87 291 322 204)(88 262 323 235)(89 293 324 206)(90 264 325 237)(91 295 326 208)(92 266 327 239)(93 297 328 210)(94 268 329 181)(95 299 330 212)(96 270 331 183)(97 241 332 214)(98 272 333 185)(99 243 334 216)(100 274 335 187)(101 245 336 218)(102 276 337 189)(103 247 338 220)(104 278 339 191)(105 249 340 222)(106 280 341 193)(107 251 342 224)(108 282 343 195)(109 253 344 226)(110 284 345 197)(111 255 346 228)(112 286 347 199)(113 257 348 230)(114 288 349 201)(115 259 350 232)(116 290 351 203)(117 261 352 234)(118 292 353 205)(119 263 354 236)(120 294 355 207)

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

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

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

90 conjugacy classes

class 1 2A2B2C 3 4A···4F4G4H4I4J4K···4P5A5B6A6B6C10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order122234···444444···45566610···1012···121515151520···2030···3060···60
size111122···21515151530···30222222···24···422224···42···24···4

90 irreducible representations

dim11111122222222222444444
type++++++-+++++------
imageC1C2C2C2C2C4S3Q8D5D6C4○D4D10C4×S3D15C4×D5D30C4×D15D42S3S3×Q8D42D5Q8×D5D42D15Q8×D15
kernelDic1510Q8C4×Dic15C30.4Q8C15×C4⋊C4C2×Dic30Dic30C5×C4⋊C4Dic15C3×C4⋊C4C2×C20C30C2×C12C20C4⋊C4C12C2×C4C4C10C10C6C6C2C2
# reps1321181223264481216112244

Matrix representation of Dic1510Q8 in GL5(𝔽61)

10000
0283700
0244700
0002732
0004234
,
10000
014400
006000
0006021
000581
,
110000
01000
00100
0001113
0003350

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,28,24,0,0,0,37,47,0,0,0,0,0,27,42,0,0,0,32,34],[1,0,0,0,0,0,1,0,0,0,0,44,60,0,0,0,0,0,60,58,0,0,0,21,1],[11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,33,0,0,0,13,50] >;

Dic1510Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_{10}Q_8
% in TeX

G:=Group("Dic15:10Q8");
// GroupNames label

G:=SmallGroup(480,852);
// by ID

G=gap.SmallGroup(480,852);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,135,142,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=c^4=1,b^2=a^30,b*a*b^-1=a^-1,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations

׿
×
𝔽