Copied to
clipboard

## G = Dic30⋊8C4order 480 = 25·3·5

### 2nd semidirect product of Dic30 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic30⋊8C4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C60⋊5C4 — Dic30⋊8C4
 Lower central C15 — C30 — C60 — Dic30⋊8C4
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for Dic308C4
G = < a,b,c | a60=c4=1, b2=a30, bab-1=cac-1=a-1, cbc-1=a15b >

Subgroups: 500 in 84 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C2×C10, C24, Dic6, C2×Dic3, C2×C12, C30, Q8⋊C4, C40, Dic10, C2×Dic5, C2×C20, C4⋊Dic3, C2×C24, C2×Dic6, Dic15, C60, C2×C30, C4⋊Dic5, C2×C40, C2×Dic10, C2.Dic12, C120, Dic30, Dic30, C2×Dic15, C2×C60, C20.44D4, C605C4, C2×C120, C2×Dic30, Dic308C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, SD16, Q16, D10, C4×S3, D12, C3⋊D4, D15, Q8⋊C4, C4×D5, D20, C5⋊D4, C24⋊C2, Dic12, D6⋊C4, D30, C40⋊C2, Dic20, D10⋊C4, C2.Dic12, C4×D15, D60, C157D4, C20.44D4, C24⋊D5, Dic60, D303C4, Dic308C4

Smallest permutation representation of Dic308C4
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)
(1 112 31 82)(2 111 32 81)(3 110 33 80)(4 109 34 79)(5 108 35 78)(6 107 36 77)(7 106 37 76)(8 105 38 75)(9 104 39 74)(10 103 40 73)(11 102 41 72)(12 101 42 71)(13 100 43 70)(14 99 44 69)(15 98 45 68)(16 97 46 67)(17 96 47 66)(18 95 48 65)(19 94 49 64)(20 93 50 63)(21 92 51 62)(22 91 52 61)(23 90 53 120)(24 89 54 119)(25 88 55 118)(26 87 56 117)(27 86 57 116)(28 85 58 115)(29 84 59 114)(30 83 60 113)(121 475 151 445)(122 474 152 444)(123 473 153 443)(124 472 154 442)(125 471 155 441)(126 470 156 440)(127 469 157 439)(128 468 158 438)(129 467 159 437)(130 466 160 436)(131 465 161 435)(132 464 162 434)(133 463 163 433)(134 462 164 432)(135 461 165 431)(136 460 166 430)(137 459 167 429)(138 458 168 428)(139 457 169 427)(140 456 170 426)(141 455 171 425)(142 454 172 424)(143 453 173 423)(144 452 174 422)(145 451 175 421)(146 450 176 480)(147 449 177 479)(148 448 178 478)(149 447 179 477)(150 446 180 476)(181 382 211 412)(182 381 212 411)(183 380 213 410)(184 379 214 409)(185 378 215 408)(186 377 216 407)(187 376 217 406)(188 375 218 405)(189 374 219 404)(190 373 220 403)(191 372 221 402)(192 371 222 401)(193 370 223 400)(194 369 224 399)(195 368 225 398)(196 367 226 397)(197 366 227 396)(198 365 228 395)(199 364 229 394)(200 363 230 393)(201 362 231 392)(202 361 232 391)(203 420 233 390)(204 419 234 389)(205 418 235 388)(206 417 236 387)(207 416 237 386)(208 415 238 385)(209 414 239 384)(210 413 240 383)(241 302 271 332)(242 301 272 331)(243 360 273 330)(244 359 274 329)(245 358 275 328)(246 357 276 327)(247 356 277 326)(248 355 278 325)(249 354 279 324)(250 353 280 323)(251 352 281 322)(252 351 282 321)(253 350 283 320)(254 349 284 319)(255 348 285 318)(256 347 286 317)(257 346 287 316)(258 345 288 315)(259 344 289 314)(260 343 290 313)(261 342 291 312)(262 341 292 311)(263 340 293 310)(264 339 294 309)(265 338 295 308)(266 337 296 307)(267 336 297 306)(268 335 298 305)(269 334 299 304)(270 333 300 303)
(1 419 281 131)(2 418 282 130)(3 417 283 129)(4 416 284 128)(5 415 285 127)(6 414 286 126)(7 413 287 125)(8 412 288 124)(9 411 289 123)(10 410 290 122)(11 409 291 121)(12 408 292 180)(13 407 293 179)(14 406 294 178)(15 405 295 177)(16 404 296 176)(17 403 297 175)(18 402 298 174)(19 401 299 173)(20 400 300 172)(21 399 241 171)(22 398 242 170)(23 397 243 169)(24 396 244 168)(25 395 245 167)(26 394 246 166)(27 393 247 165)(28 392 248 164)(29 391 249 163)(30 390 250 162)(31 389 251 161)(32 388 252 160)(33 387 253 159)(34 386 254 158)(35 385 255 157)(36 384 256 156)(37 383 257 155)(38 382 258 154)(39 381 259 153)(40 380 260 152)(41 379 261 151)(42 378 262 150)(43 377 263 149)(44 376 264 148)(45 375 265 147)(46 374 266 146)(47 373 267 145)(48 372 268 144)(49 371 269 143)(50 370 270 142)(51 369 271 141)(52 368 272 140)(53 367 273 139)(54 366 274 138)(55 365 275 137)(56 364 276 136)(57 363 277 135)(58 362 278 134)(59 361 279 133)(60 420 280 132)(61 210 331 441)(62 209 332 440)(63 208 333 439)(64 207 334 438)(65 206 335 437)(66 205 336 436)(67 204 337 435)(68 203 338 434)(69 202 339 433)(70 201 340 432)(71 200 341 431)(72 199 342 430)(73 198 343 429)(74 197 344 428)(75 196 345 427)(76 195 346 426)(77 194 347 425)(78 193 348 424)(79 192 349 423)(80 191 350 422)(81 190 351 421)(82 189 352 480)(83 188 353 479)(84 187 354 478)(85 186 355 477)(86 185 356 476)(87 184 357 475)(88 183 358 474)(89 182 359 473)(90 181 360 472)(91 240 301 471)(92 239 302 470)(93 238 303 469)(94 237 304 468)(95 236 305 467)(96 235 306 466)(97 234 307 465)(98 233 308 464)(99 232 309 463)(100 231 310 462)(101 230 311 461)(102 229 312 460)(103 228 313 459)(104 227 314 458)(105 226 315 457)(106 225 316 456)(107 224 317 455)(108 223 318 454)(109 222 319 453)(110 221 320 452)(111 220 321 451)(112 219 322 450)(113 218 323 449)(114 217 324 448)(115 216 325 447)(116 215 326 446)(117 214 327 445)(118 213 328 444)(119 212 329 443)(120 211 330 442)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 2 2 2 60 60 60 60 2 2 2 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + - + + + + - + - + - image C1 C2 C2 C2 C4 S3 D4 D4 D5 D6 SD16 Q16 D10 C4×S3 C3⋊D4 D12 D15 C4×D5 C5⋊D4 D20 C24⋊C2 Dic12 D30 C40⋊C2 Dic20 C4×D15 C15⋊7D4 D60 C24⋊D5 Dic60 kernel Dic30⋊8C4 C60⋊5C4 C2×C120 C2×Dic30 Dic30 C2×C40 C60 C2×C30 C2×C24 C2×C20 C30 C30 C2×C12 C20 C20 C2×C10 C2×C8 C12 C12 C2×C6 C10 C10 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 2 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 8 8 16 16

Matrix representation of Dic308C4 in GL3(𝔽241) generated by

 1 0 0 0 78 128 0 113 170
,
 1 0 0 0 156 238 0 79 85
,
 177 0 0 0 84 35 0 5 157
G:=sub<GL(3,GF(241))| [1,0,0,0,78,113,0,128,170],[1,0,0,0,156,79,0,238,85],[177,0,0,0,84,5,0,35,157] >;

Dic308C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{30}\rtimes_8C_4
% in TeX

G:=Group("Dic30:8C4");
// GroupNames label

G:=SmallGroup(480,176);
// by ID

G=gap.SmallGroup(480,176);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,85,92,422,100,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=c^4=1,b^2=a^30,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^15*b>;
// generators/relations

׿
×
𝔽