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G = C30.20D8order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.20D8, C30.8Q16, C60.11Q8, C20.4Dic6, C12.3Dic10, C153C812C4, C155(C2.D8), C4.8(C15⋊Q8), C20.43(C4×S3), C12.11(C4×D5), (C2×C20).61D6, (C2×C30).33D4, C53(C6.Q16), C30.29(C4⋊C4), C60.127(C2×C4), C6.14(D4⋊D5), (C2×C12).61D10, C32(C10.D8), C2.3(C15⋊D8), C4⋊Dic5.10S3, C4⋊Dic3.10D5, C6.6(C5⋊Q16), C10.14(D4⋊S3), C10.6(C3⋊Q16), C2.3(C15⋊Q16), C4.12(D30.C2), (C2×C60).186C22, C6.6(C10.D4), C2.4(Dic155C4), C10.12(Dic3⋊C4), C22.17(C15⋊D4), (C2×C4).191(S3×D5), (C5×C4⋊Dic3).9C2, (C3×C4⋊Dic5).9C2, (C2×C153C8).14C2, (C2×C6).49(C5⋊D4), (C2×C10).49(C3⋊D4), SmallGroup(480,65)

Series: Derived Chief Lower central Upper central

C1C60 — C30.20D8
C1C5C15C30C2×C30C2×C60C3×C4⋊Dic5 — C30.20D8
C15C30C60 — C30.20D8
C1C22C2×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 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3, C12 [×2], C12, C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5, C20 [×2], C20, C2×C10, C3⋊C8 [×2], C2×Dic3, C2×C12, C2×C12, C30 [×3], C2.D8, C52C8 [×2], C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, C60 [×2], C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C6.Q16, C153C8 [×2], C6×Dic5, C10×Dic3, C2×C60, C10.D8, C3×C4⋊Dic5, C5×C4⋊Dic3, C2×C153C8, C30.20D8
Quotients: C1, C2 [×3], C4 [×2], 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)
(1 423 174 380 352 455 310 203)(2 422 175 379 353 454 311 202)(3 421 176 378 354 453 312 201)(4 450 177 377 355 452 313 200)(5 449 178 376 356 451 314 199)(6 448 179 375 357 480 315 198)(7 447 180 374 358 479 316 197)(8 446 151 373 359 478 317 196)(9 445 152 372 360 477 318 195)(10 444 153 371 331 476 319 194)(11 443 154 370 332 475 320 193)(12 442 155 369 333 474 321 192)(13 441 156 368 334 473 322 191)(14 440 157 367 335 472 323 190)(15 439 158 366 336 471 324 189)(16 438 159 365 337 470 325 188)(17 437 160 364 338 469 326 187)(18 436 161 363 339 468 327 186)(19 435 162 362 340 467 328 185)(20 434 163 361 341 466 329 184)(21 433 164 390 342 465 330 183)(22 432 165 389 343 464 301 182)(23 431 166 388 344 463 302 181)(24 430 167 387 345 462 303 210)(25 429 168 386 346 461 304 209)(26 428 169 385 347 460 305 208)(27 427 170 384 348 459 306 207)(28 426 171 383 349 458 307 206)(29 425 172 382 350 457 308 205)(30 424 173 381 351 456 309 204)(31 96 269 139 391 231 300 65)(32 95 270 138 392 230 271 64)(33 94 241 137 393 229 272 63)(34 93 242 136 394 228 273 62)(35 92 243 135 395 227 274 61)(36 91 244 134 396 226 275 90)(37 120 245 133 397 225 276 89)(38 119 246 132 398 224 277 88)(39 118 247 131 399 223 278 87)(40 117 248 130 400 222 279 86)(41 116 249 129 401 221 280 85)(42 115 250 128 402 220 281 84)(43 114 251 127 403 219 282 83)(44 113 252 126 404 218 283 82)(45 112 253 125 405 217 284 81)(46 111 254 124 406 216 285 80)(47 110 255 123 407 215 286 79)(48 109 256 122 408 214 287 78)(49 108 257 121 409 213 288 77)(50 107 258 150 410 212 289 76)(51 106 259 149 411 211 290 75)(52 105 260 148 412 240 291 74)(53 104 261 147 413 239 292 73)(54 103 262 146 414 238 293 72)(55 102 263 145 415 237 294 71)(56 101 264 144 416 236 295 70)(57 100 265 143 417 235 296 69)(58 99 266 142 418 234 297 68)(59 98 267 141 419 233 298 67)(60 97 268 140 420 232 299 66)
(1 149 16 134)(2 138 17 123)(3 127 18 142)(4 146 19 131)(5 135 20 150)(6 124 21 139)(7 143 22 128)(8 132 23 147)(9 121 24 136)(10 140 25 125)(11 129 26 144)(12 148 27 133)(13 137 28 122)(14 126 29 141)(15 145 30 130)(31 375 46 390)(32 364 47 379)(33 383 48 368)(34 372 49 387)(35 361 50 376)(36 380 51 365)(37 369 52 384)(38 388 53 373)(39 377 54 362)(40 366 55 381)(41 385 56 370)(42 374 57 389)(43 363 58 378)(44 382 59 367)(45 371 60 386)(61 341 76 356)(62 360 77 345)(63 349 78 334)(64 338 79 353)(65 357 80 342)(66 346 81 331)(67 335 82 350)(68 354 83 339)(69 343 84 358)(70 332 85 347)(71 351 86 336)(72 340 87 355)(73 359 88 344)(74 348 89 333)(75 337 90 352)(91 174 106 159)(92 163 107 178)(93 152 108 167)(94 171 109 156)(95 160 110 175)(96 179 111 164)(97 168 112 153)(98 157 113 172)(99 176 114 161)(100 165 115 180)(101 154 116 169)(102 173 117 158)(103 162 118 177)(104 151 119 166)(105 170 120 155)(181 413 196 398)(182 402 197 417)(183 391 198 406)(184 410 199 395)(185 399 200 414)(186 418 201 403)(187 407 202 392)(188 396 203 411)(189 415 204 400)(190 404 205 419)(191 393 206 408)(192 412 207 397)(193 401 208 416)(194 420 209 405)(195 409 210 394)(211 325 226 310)(212 314 227 329)(213 303 228 318)(214 322 229 307)(215 311 230 326)(216 330 231 315)(217 319 232 304)(218 308 233 323)(219 327 234 312)(220 316 235 301)(221 305 236 320)(222 324 237 309)(223 313 238 328)(224 302 239 317)(225 321 240 306)(241 426 256 441)(242 445 257 430)(243 434 258 449)(244 423 259 438)(245 442 260 427)(246 431 261 446)(247 450 262 435)(248 439 263 424)(249 428 264 443)(250 447 265 432)(251 436 266 421)(252 425 267 440)(253 444 268 429)(254 433 269 448)(255 422 270 437)(271 469 286 454)(272 458 287 473)(273 477 288 462)(274 466 289 451)(275 455 290 470)(276 474 291 459)(277 463 292 478)(278 452 293 467)(279 471 294 456)(280 460 295 475)(281 479 296 464)(282 468 297 453)(283 457 298 472)(284 476 299 461)(285 465 300 480)

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444455666888810···1012121212121215152020202020···2030···3060···60
size11112221212202022222303030302···2442020202044444412···124···44···4

60 irreducible representations

dim11111222222222222224444444444
type+++++-++++-+--+-++-+--
imageC1C2C2C2C4S3Q8D4D5D6D8Q16D10Dic6C4×S3C3⋊D4Dic10C4×D5C5⋊D4D4⋊S3C3⋊Q16S3×D5D4⋊D5C5⋊Q16D30.C2C15⋊Q8C15⋊D4C15⋊D8C15⋊Q16
kernelC30.20D8C3×C4⋊Dic5C5×C4⋊Dic3C2×C153C8C153C8C4⋊Dic5C60C2×C30C4⋊Dic3C2×C20C30C30C2×C12C20C20C2×C10C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111212222224441122222244

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

110000
24000000
0051100
00240000
000010
000001
,
1011930000
921400000
0011316200
0018312800
00000219
000011219
,
6400000
0640000
001031500
006413800
00001973
00007844

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

׿
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𝔽