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G = Dic309C4order 480 = 25·3·5

3rd semidirect product of Dic30 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.2D4, C4.10D60, Dic309C4, C12.10D20, C20.10D12, C30.17Q16, C30.26SD16, C4⋊C4.3D15, C4.2(C4×D15), C20.48(C4×S3), C60.78(C2×C4), (C2×C20).67D6, (C2×C4).35D30, C12.16(C4×D5), (C2×C12).68D10, (C2×C30).138D4, C53(C6.SD16), C32(C10.Q16), C6.8(D4.D5), C6.8(C5⋊Q16), C10.30(D6⋊C4), C1511(Q8⋊C4), C2.2(D4.D15), (C2×C60).53C22, C10.8(D4.S3), (C2×Dic30).6C2, C2.2(C157Q16), C10.8(C3⋊Q16), C2.6(D303C4), C30.72(C22⋊C4), C6.15(D10⋊C4), C22.15(C157D4), (C5×C4⋊C4).3S3, (C3×C4⋊C4).3D5, (C15×C4⋊C4).3C2, (C2×C153C8).3C2, (C2×C6).70(C5⋊D4), (C2×C10).70(C3⋊D4), SmallGroup(480,170)

Series: Derived Chief Lower central Upper central

C1C60 — Dic309C4
C1C5C15C30C2×C30C2×C60C2×Dic30 — Dic309C4
C15C30C60 — Dic309C4
C1C22C2×C4C4⋊C4

Generators and relations for Dic309C4
 G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a31, cbc-1=a45b >

Subgroups: 420 in 84 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], Dic3 [×2], C12 [×2], C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5 [×2], C20 [×2], C20, C2×C10, C3⋊C8, Dic6 [×3], C2×Dic3, C2×C12, C2×C12, C30 [×3], Q8⋊C4, C52C8, Dic10 [×3], C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, Dic15 [×2], C60 [×2], C60, C2×C30, C2×C52C8, C5×C4⋊C4, C2×Dic10, C6.SD16, C153C8, Dic30 [×2], Dic30, C2×Dic15, C2×C60, C2×C60, C10.Q16, C2×C153C8, C15×C4⋊C4, C2×Dic30, Dic309C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, SD16, Q16, D10, C4×S3, D12, C3⋊D4, D15, Q8⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, D4.S3, C3⋊Q16, D30, D10⋊C4, D4.D5, C5⋊Q16, C6.SD16, C4×D15, D60, C157D4, C10.Q16, D303C4, D4.D15, C157Q16, Dic309C4

Smallest permutation representation of Dic309C4
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)
(1 326 31 356)(2 325 32 355)(3 324 33 354)(4 323 34 353)(5 322 35 352)(6 321 36 351)(7 320 37 350)(8 319 38 349)(9 318 39 348)(10 317 40 347)(11 316 41 346)(12 315 42 345)(13 314 43 344)(14 313 44 343)(15 312 45 342)(16 311 46 341)(17 310 47 340)(18 309 48 339)(19 308 49 338)(20 307 50 337)(21 306 51 336)(22 305 52 335)(23 304 53 334)(24 303 54 333)(25 302 55 332)(26 301 56 331)(27 360 57 330)(28 359 58 329)(29 358 59 328)(30 357 60 327)(61 249 91 279)(62 248 92 278)(63 247 93 277)(64 246 94 276)(65 245 95 275)(66 244 96 274)(67 243 97 273)(68 242 98 272)(69 241 99 271)(70 300 100 270)(71 299 101 269)(72 298 102 268)(73 297 103 267)(74 296 104 266)(75 295 105 265)(76 294 106 264)(77 293 107 263)(78 292 108 262)(79 291 109 261)(80 290 110 260)(81 289 111 259)(82 288 112 258)(83 287 113 257)(84 286 114 256)(85 285 115 255)(86 284 116 254)(87 283 117 253)(88 282 118 252)(89 281 119 251)(90 280 120 250)(121 463 151 433)(122 462 152 432)(123 461 153 431)(124 460 154 430)(125 459 155 429)(126 458 156 428)(127 457 157 427)(128 456 158 426)(129 455 159 425)(130 454 160 424)(131 453 161 423)(132 452 162 422)(133 451 163 421)(134 450 164 480)(135 449 165 479)(136 448 166 478)(137 447 167 477)(138 446 168 476)(139 445 169 475)(140 444 170 474)(141 443 171 473)(142 442 172 472)(143 441 173 471)(144 440 174 470)(145 439 175 469)(146 438 176 468)(147 437 177 467)(148 436 178 466)(149 435 179 465)(150 434 180 464)(181 405 211 375)(182 404 212 374)(183 403 213 373)(184 402 214 372)(185 401 215 371)(186 400 216 370)(187 399 217 369)(188 398 218 368)(189 397 219 367)(190 396 220 366)(191 395 221 365)(192 394 222 364)(193 393 223 363)(194 392 224 362)(195 391 225 361)(196 390 226 420)(197 389 227 419)(198 388 228 418)(199 387 229 417)(200 386 230 416)(201 385 231 415)(202 384 232 414)(203 383 233 413)(204 382 234 412)(205 381 235 411)(206 380 236 410)(207 379 237 409)(208 378 238 408)(209 377 239 407)(210 376 240 406)
(1 240 89 143)(2 211 90 174)(3 182 91 145)(4 213 92 176)(5 184 93 147)(6 215 94 178)(7 186 95 149)(8 217 96 180)(9 188 97 151)(10 219 98 122)(11 190 99 153)(12 221 100 124)(13 192 101 155)(14 223 102 126)(15 194 103 157)(16 225 104 128)(17 196 105 159)(18 227 106 130)(19 198 107 161)(20 229 108 132)(21 200 109 163)(22 231 110 134)(23 202 111 165)(24 233 112 136)(25 204 113 167)(26 235 114 138)(27 206 115 169)(28 237 116 140)(29 208 117 171)(30 239 118 142)(31 210 119 173)(32 181 120 144)(33 212 61 175)(34 183 62 146)(35 214 63 177)(36 185 64 148)(37 216 65 179)(38 187 66 150)(39 218 67 121)(40 189 68 152)(41 220 69 123)(42 191 70 154)(43 222 71 125)(44 193 72 156)(45 224 73 127)(46 195 74 158)(47 226 75 129)(48 197 76 160)(49 228 77 131)(50 199 78 162)(51 230 79 133)(52 201 80 164)(53 232 81 135)(54 203 82 166)(55 234 83 137)(56 205 84 168)(57 236 85 139)(58 207 86 170)(59 238 87 141)(60 209 88 172)(241 476 346 381)(242 447 347 412)(243 478 348 383)(244 449 349 414)(245 480 350 385)(246 451 351 416)(247 422 352 387)(248 453 353 418)(249 424 354 389)(250 455 355 420)(251 426 356 391)(252 457 357 362)(253 428 358 393)(254 459 359 364)(255 430 360 395)(256 461 301 366)(257 432 302 397)(258 463 303 368)(259 434 304 399)(260 465 305 370)(261 436 306 401)(262 467 307 372)(263 438 308 403)(264 469 309 374)(265 440 310 405)(266 471 311 376)(267 442 312 407)(268 473 313 378)(269 444 314 409)(270 475 315 380)(271 446 316 411)(272 477 317 382)(273 448 318 413)(274 479 319 384)(275 450 320 415)(276 421 321 386)(277 452 322 417)(278 423 323 388)(279 454 324 419)(280 425 325 390)(281 456 326 361)(282 427 327 392)(283 458 328 363)(284 429 329 394)(285 460 330 365)(286 431 331 396)(287 462 332 367)(288 433 333 398)(289 464 334 369)(290 435 335 400)(291 466 336 371)(292 437 337 402)(293 468 338 373)(294 439 339 404)(295 470 340 375)(296 441 341 406)(297 472 342 377)(298 443 343 408)(299 474 344 379)(300 445 345 410)

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222344444455666888810···1012···121515151520···2030···3060···60
size111122244606022222303030302···24···422224···42···24···4

84 irreducible representations

dim111112222222222222222222444444
type+++++++++-++++++------
imageC1C2C2C2C4S3D4D4D5D6SD16Q16D10C4×S3D12C3⋊D4D15C4×D5D20C5⋊D4D30C4×D15D60C157D4D4.S3C3⋊Q16D4.D5C5⋊Q16D4.D15C157Q16
kernelDic309C4C2×C153C8C15×C4⋊C4C2×Dic30Dic30C5×C4⋊C4C60C2×C30C3×C4⋊C4C2×C20C30C30C2×C12C20C20C2×C10C4⋊C4C12C12C2×C6C2×C4C4C4C22C10C10C6C6C2C2
# reps111141112122222244444888112244

Matrix representation of Dic309C4 in GL4(𝔽241) generated by

8414700
9411000
001035
0047138
,
1018200
21023100
00243
00140239
,
64000
06400
0065240
00127176
G:=sub<GL(4,GF(241))| [84,94,0,0,147,110,0,0,0,0,103,47,0,0,5,138],[10,210,0,0,182,231,0,0,0,0,2,140,0,0,43,239],[64,0,0,0,0,64,0,0,0,0,65,127,0,0,240,176] >;

Dic309C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{30}\rtimes_9C_4
% in TeX

G:=Group("Dic30:9C4");
// GroupNames label

G:=SmallGroup(480,170);
// by ID

G=gap.SmallGroup(480,170);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,36,675,346,80,2693,18822]);
// 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,c*b*c^-1=a^45*b>;
// generators/relations

׿
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