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G = C30.Q16order 480 = 25·3·5

1st non-split extension by C30 of Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.8D20, C60.78D4, C30.1Q16, C30.7SD16, Dic106Dic3, C12.4(C4×D5), (C2×C30).17D4, (C2×C20).50D6, C4⋊Dic3.8D5, C4.7(D5×Dic3), C60.122(C2×C4), C153(Q8⋊C4), (C2×C12).51D10, C53(Q82Dic3), C33(C10.Q16), C6.5(D4.D5), C6.4(C5⋊Q16), (C3×Dic10)⋊10C4, C20.12(C3⋊D4), C4.22(C3⋊D20), (C6×Dic10).6C2, (C2×Dic10).7S3, C20.23(C2×Dic3), C10.4(C3⋊Q16), C2.1(C15⋊Q16), C30.53(C22⋊C4), (C2×C60).183C22, C10.5(Q82S3), C2.2(C20.D6), C6.28(D10⋊C4), C2.7(D10⋊Dic3), C22.14(C15⋊D4), C10.17(C6.D4), (C2×C4).186(S3×D5), (C5×C4⋊Dic3).7C2, (C2×C153C8).11C2, (C2×C6).45(C5⋊D4), (C2×C10).45(C3⋊D4), SmallGroup(480,46)

Series: Derived Chief Lower central Upper central

C1C60 — C30.Q16
C1C5C15C30C2×C30C2×C60C6×Dic10 — C30.Q16
C15C30C60 — C30.Q16
C1C22C2×C4

Generators and relations for C30.Q16
 G = < a,b,c | a30=b8=1, c2=a15b4, bab-1=a-1, cac-1=a11, cbc-1=a15b-1 >

Subgroups: 316 in 84 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], Dic3, C12 [×2], C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5 [×2], C20 [×2], C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8 [×3], C30 [×3], Q8⋊C4, C52C8, Dic10 [×2], Dic10, C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C5×Dic3, C3×Dic5 [×2], C60 [×2], C2×C30, C2×C52C8, C5×C4⋊C4, C2×Dic10, Q82Dic3, C153C8, C3×Dic10 [×2], C3×Dic10, C6×Dic5, C10×Dic3, C2×C60, C10.Q16, C5×C4⋊Dic3, C2×C153C8, C6×Dic10, C30.Q16
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, SD16, Q16, D10, C2×Dic3, C3⋊D4 [×2], Q8⋊C4, C4×D5, D20, C5⋊D4, Q82S3, C3⋊Q16, C6.D4, S3×D5, D10⋊C4, D4.D5, C5⋊Q16, Q82Dic3, D5×Dic3, C15⋊D4, C3⋊D20, C10.Q16, C20.D6, C15⋊Q16, D10⋊Dic3, C30.Q16

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

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

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

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

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++++++++-+-+++-+---+-
imageC1C2C2C2C4S3D4D4D5Dic3D6SD16Q16D10C3⋊D4C3⋊D4C4×D5D20C5⋊D4Q82S3C3⋊Q16S3×D5D4.D5C5⋊Q16D5×Dic3C3⋊D20C15⋊D4C20.D6C15⋊Q16
kernelC30.Q16C5×C4⋊Dic3C2×C153C8C6×Dic10C3×Dic10C2×Dic10C60C2×C30C4⋊Dic3Dic10C2×C20C30C30C2×C12C20C2×C10C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111221222224441122222244

Matrix representation of C30.Q16 in GL6(𝔽241)

02400000
1520000
0024024000
001000
00002400
00000240
,
2081600000
189330000
0017214600
002156900
00000184
00009338
,
6400000
0640000
0017214600
002156900
000014107
000068227

G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,52,0,0,0,0,0,0,240,1,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[208,189,0,0,0,0,160,33,0,0,0,0,0,0,172,215,0,0,0,0,146,69,0,0,0,0,0,0,0,93,0,0,0,0,184,38],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,172,215,0,0,0,0,146,69,0,0,0,0,0,0,14,68,0,0,0,0,107,227] >;

C30.Q16 in GAP, Magma, Sage, TeX

C_{30}.Q_{16}
% in TeX

G:=Group("C30.Q16");
// GroupNames label

G:=SmallGroup(480,46);
// by ID

G=gap.SmallGroup(480,46);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,36,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^30=b^8=1,c^2=a^15*b^4,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b^-1>;
// generators/relations

׿
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