Copied to
clipboard

## G = C30.20D8order 480 = 25·3·5

### 20th non-split extension by C30 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C30.20D8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C3×C4⋊Dic5 — C30.20D8
 Lower central C15 — C30 — C60 — C30.20D8
 Upper central C1 — C22 — C2×C4

Generators and relations for C30.20D8
G = < a,b,c | a30=b8=1, c2=a15, bab-1=a-1, cac-1=a19, cbc-1=b-1 >

Subgroups: 268 in 72 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C30, C2.D8, C52C8, C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, C60, C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C6.Q16, C153C8, C6×Dic5, C10×Dic3, C2×C60, C10.D8, C3×C4⋊Dic5, C5×C4⋊Dic3, C2×C153C8, C30.20D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D8, Q16, D10, Dic6, C4×S3, C3⋊D4, C2.D8, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, D4⋊S3, C3⋊Q16, S3×D5, C10.D4, D4⋊D5, C5⋊Q16, C6.Q16, D30.C2, C15⋊D4, C15⋊Q8, C10.D8, C15⋊D8, C15⋊Q16, Dic155C4, C30.20D8

Smallest permutation representation of C30.20D8
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)
(1 120 393 227 167 370 456 75)(2 119 394 226 168 369 457 74)(3 118 395 225 169 368 458 73)(4 117 396 224 170 367 459 72)(5 116 397 223 171 366 460 71)(6 115 398 222 172 365 461 70)(7 114 399 221 173 364 462 69)(8 113 400 220 174 363 463 68)(9 112 401 219 175 362 464 67)(10 111 402 218 176 361 465 66)(11 110 403 217 177 390 466 65)(12 109 404 216 178 389 467 64)(13 108 405 215 179 388 468 63)(14 107 406 214 180 387 469 62)(15 106 407 213 151 386 470 61)(16 105 408 212 152 385 471 90)(17 104 409 211 153 384 472 89)(18 103 410 240 154 383 473 88)(19 102 411 239 155 382 474 87)(20 101 412 238 156 381 475 86)(21 100 413 237 157 380 476 85)(22 99 414 236 158 379 477 84)(23 98 415 235 159 378 478 83)(24 97 416 234 160 377 479 82)(25 96 417 233 161 376 480 81)(26 95 418 232 162 375 451 80)(27 94 419 231 163 374 452 79)(28 93 420 230 164 373 453 78)(29 92 391 229 165 372 454 77)(30 91 392 228 166 371 455 76)(31 311 434 266 359 185 290 133)(32 310 435 265 360 184 291 132)(33 309 436 264 331 183 292 131)(34 308 437 263 332 182 293 130)(35 307 438 262 333 181 294 129)(36 306 439 261 334 210 295 128)(37 305 440 260 335 209 296 127)(38 304 441 259 336 208 297 126)(39 303 442 258 337 207 298 125)(40 302 443 257 338 206 299 124)(41 301 444 256 339 205 300 123)(42 330 445 255 340 204 271 122)(43 329 446 254 341 203 272 121)(44 328 447 253 342 202 273 150)(45 327 448 252 343 201 274 149)(46 326 449 251 344 200 275 148)(47 325 450 250 345 199 276 147)(48 324 421 249 346 198 277 146)(49 323 422 248 347 197 278 145)(50 322 423 247 348 196 279 144)(51 321 424 246 349 195 280 143)(52 320 425 245 350 194 281 142)(53 319 426 244 351 193 282 141)(54 318 427 243 352 192 283 140)(55 317 428 242 353 191 284 139)(56 316 429 241 354 190 285 138)(57 315 430 270 355 189 286 137)(58 314 431 269 356 188 287 136)(59 313 432 268 357 187 288 135)(60 312 433 267 358 186 289 134)
(1 184 16 199)(2 203 17 188)(3 192 18 207)(4 181 19 196)(5 200 20 185)(6 189 21 204)(7 208 22 193)(8 197 23 182)(9 186 24 201)(10 205 25 190)(11 194 26 209)(12 183 27 198)(13 202 28 187)(14 191 29 206)(15 210 30 195)(31 366 46 381)(32 385 47 370)(33 374 48 389)(34 363 49 378)(35 382 50 367)(36 371 51 386)(37 390 52 375)(38 379 53 364)(39 368 54 383)(40 387 55 372)(41 376 56 361)(42 365 57 380)(43 384 58 369)(44 373 59 388)(45 362 60 377)(61 295 76 280)(62 284 77 299)(63 273 78 288)(64 292 79 277)(65 281 80 296)(66 300 81 285)(67 289 82 274)(68 278 83 293)(69 297 84 282)(70 286 85 271)(71 275 86 290)(72 294 87 279)(73 283 88 298)(74 272 89 287)(75 291 90 276)(91 349 106 334)(92 338 107 353)(93 357 108 342)(94 346 109 331)(95 335 110 350)(96 354 111 339)(97 343 112 358)(98 332 113 347)(99 351 114 336)(100 340 115 355)(101 359 116 344)(102 348 117 333)(103 337 118 352)(104 356 119 341)(105 345 120 360)(121 472 136 457)(122 461 137 476)(123 480 138 465)(124 469 139 454)(125 458 140 473)(126 477 141 462)(127 466 142 451)(128 455 143 470)(129 474 144 459)(130 463 145 478)(131 452 146 467)(132 471 147 456)(133 460 148 475)(134 479 149 464)(135 468 150 453)(151 306 166 321)(152 325 167 310)(153 314 168 329)(154 303 169 318)(155 322 170 307)(156 311 171 326)(157 330 172 315)(158 319 173 304)(159 308 174 323)(160 327 175 312)(161 316 176 301)(162 305 177 320)(163 324 178 309)(164 313 179 328)(165 302 180 317)(211 431 226 446)(212 450 227 435)(213 439 228 424)(214 428 229 443)(215 447 230 432)(216 436 231 421)(217 425 232 440)(218 444 233 429)(219 433 234 448)(220 422 235 437)(221 441 236 426)(222 430 237 445)(223 449 238 434)(224 438 239 423)(225 427 240 442)(241 402 256 417)(242 391 257 406)(243 410 258 395)(244 399 259 414)(245 418 260 403)(246 407 261 392)(247 396 262 411)(248 415 263 400)(249 404 264 419)(250 393 265 408)(251 412 266 397)(252 401 267 416)(253 420 268 405)(254 409 269 394)(255 398 270 413)```

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

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

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

60 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 12E 12F 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H 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 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 12 12 20 20 2 2 2 2 2 30 30 30 30 2 ··· 2 4 4 20 20 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - + + + + - + - - + - + + - + - - image C1 C2 C2 C2 C4 S3 Q8 D4 D5 D6 D8 Q16 D10 Dic6 C4×S3 C3⋊D4 Dic10 C4×D5 C5⋊D4 D4⋊S3 C3⋊Q16 S3×D5 D4⋊D5 C5⋊Q16 D30.C2 C15⋊Q8 C15⋊D4 C15⋊D8 C15⋊Q16 kernel C30.20D8 C3×C4⋊Dic5 C5×C4⋊Dic3 C2×C15⋊3C8 C15⋊3C8 C4⋊Dic5 C60 C2×C30 C4⋊Dic3 C2×C20 C30 C30 C2×C12 C20 C20 C2×C10 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 1 1 2 2 2 2 2 2 4 4

Matrix representation of C30.20D8 in GL6(𝔽241)

 1 1 0 0 0 0 240 0 0 0 0 0 0 0 51 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 101 193 0 0 0 0 92 140 0 0 0 0 0 0 113 162 0 0 0 0 183 128 0 0 0 0 0 0 0 219 0 0 0 0 11 219
,
 64 0 0 0 0 0 0 64 0 0 0 0 0 0 103 15 0 0 0 0 64 138 0 0 0 0 0 0 197 3 0 0 0 0 78 44

`G:=sub<GL(6,GF(241))| [1,240,0,0,0,0,1,0,0,0,0,0,0,0,51,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[101,92,0,0,0,0,193,140,0,0,0,0,0,0,113,183,0,0,0,0,162,128,0,0,0,0,0,0,0,11,0,0,0,0,219,219],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,103,64,0,0,0,0,15,138,0,0,0,0,0,0,197,78,0,0,0,0,3,44] >;`

C30.20D8 in GAP, Magma, Sage, TeX

`C_{30}._{20}D_8`
`% in TeX`

`G:=Group("C30.20D8");`
`// GroupNames label`

`G:=SmallGroup(480,65);`
`// by ID`

`G=gap.SmallGroup(480,65);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,36,675,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=b^8=1,c^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=a^19,c*b*c^-1=b^-1>;`
`// generators/relations`

׿
×
𝔽