metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.59D14, C7⋊Q16⋊5C4, Q8.6(C4×D7), (C4×Q8).7D7, C14.76(C4×D4), (Q8×C28).8C2, C4⋊C4.256D14, C7⋊4(Q16⋊C4), (C2×C28).260D4, C4.43(C4○D28), C28.63(C4○D4), C28.28(C22×C4), (C2×Q8).163D14, (C4×C28).101C22, (C2×C28).350C23, Dic14.17(C2×C4), (C4×Dic14).15C2, C14.Q16.10C2, Q8⋊Dic7.10C2, C4.Dic14.12C2, C42.D7.4C2, C2.4(D4.9D14), C2.4(C28.C23), C4⋊Dic7.333C22, (Q8×C14).198C22, C14.111(C8.C22), (C2×Dic14).267C22, C7⋊C8.4(C2×C4), C4.28(C2×C4×D7), C2.22(C4×C7⋊D4), (C7×Q8).13(C2×C4), (C2×C7⋊Q16).4C2, (C2×C14).481(C2×D4), (C2×C7⋊C8).103C22, C22.82(C2×C7⋊D4), (C2×C4).223(C7⋊D4), (C7×C4⋊C4).287C22, (C2×C4).450(C22×D7), SmallGroup(448,564)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.59D14
G = < a,b,c,d | a4=b4=1, c14=b2, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c13 >
Subgroups: 388 in 108 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C4×Q8, C2×Q16, C7⋊C8, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, Q16⋊C4, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7⋊Q16, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C42.D7, C4.Dic14, C14.Q16, Q8⋊Dic7, C4×Dic14, C2×C7⋊Q16, Q8×C28, C42.59D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8.C22, C4×D7, C7⋊D4, C22×D7, Q16⋊C4, C2×C4×D7, C4○D28, C2×C7⋊D4, C4×C7⋊D4, C28.C23, D4.9D14, C42.59D14
►Regular action on 448 points
Generators in S448
(1 265 337 378)(2 280 338 365)(3 267 339 380)(4 254 340 367)(5 269 341 382)(6 256 342 369)(7 271 343 384)(8 258 344 371)(9 273 345 386)(10 260 346 373)(11 275 347 388)(12 262 348 375)(13 277 349 390)(14 264 350 377)(15 279 351 392)(16 266 352 379)(17 253 353 366)(18 268 354 381)(19 255 355 368)(20 270 356 383)(21 257 357 370)(22 272 358 385)(23 259 359 372)(24 274 360 387)(25 261 361 374)(26 276 362 389)(27 263 363 376)(28 278 364 391)(29 129 239 304)(30 116 240 291)(31 131 241 306)(32 118 242 293)(33 133 243 308)(34 120 244 295)(35 135 245 282)(36 122 246 297)(37 137 247 284)(38 124 248 299)(39 139 249 286)(40 126 250 301)(41 113 251 288)(42 128 252 303)(43 115 225 290)(44 130 226 305)(45 117 227 292)(46 132 228 307)(47 119 229 294)(48 134 230 281)(49 121 231 296)(50 136 232 283)(51 123 233 298)(52 138 234 285)(53 125 235 300)(54 140 236 287)(55 127 237 302)(56 114 238 289)(57 85 309 422)(58 100 310 437)(59 87 311 424)(60 102 312 439)(61 89 313 426)(62 104 314 441)(63 91 315 428)(64 106 316 443)(65 93 317 430)(66 108 318 445)(67 95 319 432)(68 110 320 447)(69 97 321 434)(70 112 322 421)(71 99 323 436)(72 86 324 423)(73 101 325 438)(74 88 326 425)(75 103 327 440)(76 90 328 427)(77 105 329 442)(78 92 330 429)(79 107 331 444)(80 94 332 431)(81 109 333 446)(82 96 334 433)(83 111 335 448)(84 98 336 435)(141 209 401 182)(142 224 402 169)(143 211 403 184)(144 198 404 171)(145 213 405 186)(146 200 406 173)(147 215 407 188)(148 202 408 175)(149 217 409 190)(150 204 410 177)(151 219 411 192)(152 206 412 179)(153 221 413 194)(154 208 414 181)(155 223 415 196)(156 210 416 183)(157 197 417 170)(158 212 418 185)(159 199 419 172)(160 214 420 187)(161 201 393 174)(162 216 394 189)(163 203 395 176)(164 218 396 191)(165 205 397 178)(166 220 398 193)(167 207 399 180)(168 222 400 195)
(1 46 15 32)(2 33 16 47)(3 48 17 34)(4 35 18 49)(5 50 19 36)(6 37 20 51)(7 52 21 38)(8 39 22 53)(9 54 23 40)(10 41 24 55)(11 56 25 42)(12 43 26 29)(13 30 27 44)(14 45 28 31)(57 403 71 417)(58 418 72 404)(59 405 73 419)(60 420 74 406)(61 407 75 393)(62 394 76 408)(63 409 77 395)(64 396 78 410)(65 411 79 397)(66 398 80 412)(67 413 81 399)(68 400 82 414)(69 415 83 401)(70 402 84 416)(85 184 99 170)(86 171 100 185)(87 186 101 172)(88 173 102 187)(89 188 103 174)(90 175 104 189)(91 190 105 176)(92 177 106 191)(93 192 107 178)(94 179 108 193)(95 194 109 180)(96 181 110 195)(97 196 111 182)(98 183 112 169)(113 274 127 260)(114 261 128 275)(115 276 129 262)(116 263 130 277)(117 278 131 264)(118 265 132 279)(119 280 133 266)(120 267 134 253)(121 254 135 268)(122 269 136 255)(123 256 137 270)(124 271 138 257)(125 258 139 272)(126 273 140 259)(141 321 155 335)(142 336 156 322)(143 323 157 309)(144 310 158 324)(145 325 159 311)(146 312 160 326)(147 327 161 313)(148 314 162 328)(149 329 163 315)(150 316 164 330)(151 331 165 317)(152 318 166 332)(153 333 167 319)(154 320 168 334)(197 422 211 436)(198 437 212 423)(199 424 213 438)(200 439 214 425)(201 426 215 440)(202 441 216 427)(203 428 217 442)(204 443 218 429)(205 430 219 444)(206 445 220 431)(207 432 221 446)(208 447 222 433)(209 434 223 448)(210 421 224 435)(225 362 239 348)(226 349 240 363)(227 364 241 350)(228 351 242 337)(229 338 243 352)(230 353 244 339)(231 340 245 354)(232 355 246 341)(233 342 247 356)(234 357 248 343)(235 344 249 358)(236 359 250 345)(237 346 251 360)(238 361 252 347)(281 366 295 380)(282 381 296 367)(283 368 297 382)(284 383 298 369)(285 370 299 384)(286 385 300 371)(287 372 301 386)(288 387 302 373)(289 374 303 388)(290 389 304 375)(291 376 305 390)(292 391 306 377)(293 378 307 392)(294 365 308 379)
(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)
(1 80 32 398 15 66 46 412)(2 397 47 79 16 411 33 65)(3 78 34 396 17 64 48 410)(4 395 49 77 18 409 35 63)(5 76 36 394 19 62 50 408)(6 393 51 75 20 407 37 61)(7 74 38 420 21 60 52 406)(8 419 53 73 22 405 39 59)(9 72 40 418 23 58 54 404)(10 417 55 71 24 403 41 57)(11 70 42 416 25 84 56 402)(12 415 29 69 26 401 43 83)(13 68 44 414 27 82 30 400)(14 413 31 67 28 399 45 81)(85 274 170 113 99 260 184 127)(86 140 185 273 100 126 171 259)(87 272 172 139 101 258 186 125)(88 138 187 271 102 124 173 257)(89 270 174 137 103 256 188 123)(90 136 189 269 104 122 175 255)(91 268 176 135 105 254 190 121)(92 134 191 267 106 120 177 253)(93 266 178 133 107 280 192 119)(94 132 193 265 108 118 179 279)(95 264 180 131 109 278 194 117)(96 130 195 263 110 116 181 277)(97 262 182 129 111 276 196 115)(98 128 169 261 112 114 183 275)(141 225 335 348 155 239 321 362)(142 347 322 252 156 361 336 238)(143 251 309 346 157 237 323 360)(144 345 324 250 158 359 310 236)(145 249 311 344 159 235 325 358)(146 343 326 248 160 357 312 234)(147 247 313 342 161 233 327 356)(148 341 328 246 162 355 314 232)(149 245 315 340 163 231 329 354)(150 339 330 244 164 353 316 230)(151 243 317 338 165 229 331 352)(152 337 332 242 166 351 318 228)(153 241 319 364 167 227 333 350)(154 363 334 240 168 349 320 226)(197 288 436 373 211 302 422 387)(198 372 423 287 212 386 437 301)(199 286 438 371 213 300 424 385)(200 370 425 285 214 384 439 299)(201 284 440 369 215 298 426 383)(202 368 427 283 216 382 441 297)(203 282 442 367 217 296 428 381)(204 366 429 281 218 380 443 295)(205 308 444 365 219 294 430 379)(206 392 431 307 220 378 445 293)(207 306 446 391 221 292 432 377)(208 390 433 305 222 376 447 291)(209 304 448 389 223 290 434 375)(210 388 435 303 224 374 421 289)
G:=sub<Sym(448)| (1,265,337,378)(2,280,338,365)(3,267,339,380)(4,254,340,367)(5,269,341,382)(6,256,342,369)(7,271,343,384)(8,258,344,371)(9,273,345,386)(10,260,346,373)(11,275,347,388)(12,262,348,375)(13,277,349,390)(14,264,350,377)(15,279,351,392)(16,266,352,379)(17,253,353,366)(18,268,354,381)(19,255,355,368)(20,270,356,383)(21,257,357,370)(22,272,358,385)(23,259,359,372)(24,274,360,387)(25,261,361,374)(26,276,362,389)(27,263,363,376)(28,278,364,391)(29,129,239,304)(30,116,240,291)(31,131,241,306)(32,118,242,293)(33,133,243,308)(34,120,244,295)(35,135,245,282)(36,122,246,297)(37,137,247,284)(38,124,248,299)(39,139,249,286)(40,126,250,301)(41,113,251,288)(42,128,252,303)(43,115,225,290)(44,130,226,305)(45,117,227,292)(46,132,228,307)(47,119,229,294)(48,134,230,281)(49,121,231,296)(50,136,232,283)(51,123,233,298)(52,138,234,285)(53,125,235,300)(54,140,236,287)(55,127,237,302)(56,114,238,289)(57,85,309,422)(58,100,310,437)(59,87,311,424)(60,102,312,439)(61,89,313,426)(62,104,314,441)(63,91,315,428)(64,106,316,443)(65,93,317,430)(66,108,318,445)(67,95,319,432)(68,110,320,447)(69,97,321,434)(70,112,322,421)(71,99,323,436)(72,86,324,423)(73,101,325,438)(74,88,326,425)(75,103,327,440)(76,90,328,427)(77,105,329,442)(78,92,330,429)(79,107,331,444)(80,94,332,431)(81,109,333,446)(82,96,334,433)(83,111,335,448)(84,98,336,435)(141,209,401,182)(142,224,402,169)(143,211,403,184)(144,198,404,171)(145,213,405,186)(146,200,406,173)(147,215,407,188)(148,202,408,175)(149,217,409,190)(150,204,410,177)(151,219,411,192)(152,206,412,179)(153,221,413,194)(154,208,414,181)(155,223,415,196)(156,210,416,183)(157,197,417,170)(158,212,418,185)(159,199,419,172)(160,214,420,187)(161,201,393,174)(162,216,394,189)(163,203,395,176)(164,218,396,191)(165,205,397,178)(166,220,398,193)(167,207,399,180)(168,222,400,195), (1,46,15,32)(2,33,16,47)(3,48,17,34)(4,35,18,49)(5,50,19,36)(6,37,20,51)(7,52,21,38)(8,39,22,53)(9,54,23,40)(10,41,24,55)(11,56,25,42)(12,43,26,29)(13,30,27,44)(14,45,28,31)(57,403,71,417)(58,418,72,404)(59,405,73,419)(60,420,74,406)(61,407,75,393)(62,394,76,408)(63,409,77,395)(64,396,78,410)(65,411,79,397)(66,398,80,412)(67,413,81,399)(68,400,82,414)(69,415,83,401)(70,402,84,416)(85,184,99,170)(86,171,100,185)(87,186,101,172)(88,173,102,187)(89,188,103,174)(90,175,104,189)(91,190,105,176)(92,177,106,191)(93,192,107,178)(94,179,108,193)(95,194,109,180)(96,181,110,195)(97,196,111,182)(98,183,112,169)(113,274,127,260)(114,261,128,275)(115,276,129,262)(116,263,130,277)(117,278,131,264)(118,265,132,279)(119,280,133,266)(120,267,134,253)(121,254,135,268)(122,269,136,255)(123,256,137,270)(124,271,138,257)(125,258,139,272)(126,273,140,259)(141,321,155,335)(142,336,156,322)(143,323,157,309)(144,310,158,324)(145,325,159,311)(146,312,160,326)(147,327,161,313)(148,314,162,328)(149,329,163,315)(150,316,164,330)(151,331,165,317)(152,318,166,332)(153,333,167,319)(154,320,168,334)(197,422,211,436)(198,437,212,423)(199,424,213,438)(200,439,214,425)(201,426,215,440)(202,441,216,427)(203,428,217,442)(204,443,218,429)(205,430,219,444)(206,445,220,431)(207,432,221,446)(208,447,222,433)(209,434,223,448)(210,421,224,435)(225,362,239,348)(226,349,240,363)(227,364,241,350)(228,351,242,337)(229,338,243,352)(230,353,244,339)(231,340,245,354)(232,355,246,341)(233,342,247,356)(234,357,248,343)(235,344,249,358)(236,359,250,345)(237,346,251,360)(238,361,252,347)(281,366,295,380)(282,381,296,367)(283,368,297,382)(284,383,298,369)(285,370,299,384)(286,385,300,371)(287,372,301,386)(288,387,302,373)(289,374,303,388)(290,389,304,375)(291,376,305,390)(292,391,306,377)(293,378,307,392)(294,365,308,379), (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), (1,80,32,398,15,66,46,412)(2,397,47,79,16,411,33,65)(3,78,34,396,17,64,48,410)(4,395,49,77,18,409,35,63)(5,76,36,394,19,62,50,408)(6,393,51,75,20,407,37,61)(7,74,38,420,21,60,52,406)(8,419,53,73,22,405,39,59)(9,72,40,418,23,58,54,404)(10,417,55,71,24,403,41,57)(11,70,42,416,25,84,56,402)(12,415,29,69,26,401,43,83)(13,68,44,414,27,82,30,400)(14,413,31,67,28,399,45,81)(85,274,170,113,99,260,184,127)(86,140,185,273,100,126,171,259)(87,272,172,139,101,258,186,125)(88,138,187,271,102,124,173,257)(89,270,174,137,103,256,188,123)(90,136,189,269,104,122,175,255)(91,268,176,135,105,254,190,121)(92,134,191,267,106,120,177,253)(93,266,178,133,107,280,192,119)(94,132,193,265,108,118,179,279)(95,264,180,131,109,278,194,117)(96,130,195,263,110,116,181,277)(97,262,182,129,111,276,196,115)(98,128,169,261,112,114,183,275)(141,225,335,348,155,239,321,362)(142,347,322,252,156,361,336,238)(143,251,309,346,157,237,323,360)(144,345,324,250,158,359,310,236)(145,249,311,344,159,235,325,358)(146,343,326,248,160,357,312,234)(147,247,313,342,161,233,327,356)(148,341,328,246,162,355,314,232)(149,245,315,340,163,231,329,354)(150,339,330,244,164,353,316,230)(151,243,317,338,165,229,331,352)(152,337,332,242,166,351,318,228)(153,241,319,364,167,227,333,350)(154,363,334,240,168,349,320,226)(197,288,436,373,211,302,422,387)(198,372,423,287,212,386,437,301)(199,286,438,371,213,300,424,385)(200,370,425,285,214,384,439,299)(201,284,440,369,215,298,426,383)(202,368,427,283,216,382,441,297)(203,282,442,367,217,296,428,381)(204,366,429,281,218,380,443,295)(205,308,444,365,219,294,430,379)(206,392,431,307,220,378,445,293)(207,306,446,391,221,292,432,377)(208,390,433,305,222,376,447,291)(209,304,448,389,223,290,434,375)(210,388,435,303,224,374,421,289)>;
G:=Group( (1,265,337,378)(2,280,338,365)(3,267,339,380)(4,254,340,367)(5,269,341,382)(6,256,342,369)(7,271,343,384)(8,258,344,371)(9,273,345,386)(10,260,346,373)(11,275,347,388)(12,262,348,375)(13,277,349,390)(14,264,350,377)(15,279,351,392)(16,266,352,379)(17,253,353,366)(18,268,354,381)(19,255,355,368)(20,270,356,383)(21,257,357,370)(22,272,358,385)(23,259,359,372)(24,274,360,387)(25,261,361,374)(26,276,362,389)(27,263,363,376)(28,278,364,391)(29,129,239,304)(30,116,240,291)(31,131,241,306)(32,118,242,293)(33,133,243,308)(34,120,244,295)(35,135,245,282)(36,122,246,297)(37,137,247,284)(38,124,248,299)(39,139,249,286)(40,126,250,301)(41,113,251,288)(42,128,252,303)(43,115,225,290)(44,130,226,305)(45,117,227,292)(46,132,228,307)(47,119,229,294)(48,134,230,281)(49,121,231,296)(50,136,232,283)(51,123,233,298)(52,138,234,285)(53,125,235,300)(54,140,236,287)(55,127,237,302)(56,114,238,289)(57,85,309,422)(58,100,310,437)(59,87,311,424)(60,102,312,439)(61,89,313,426)(62,104,314,441)(63,91,315,428)(64,106,316,443)(65,93,317,430)(66,108,318,445)(67,95,319,432)(68,110,320,447)(69,97,321,434)(70,112,322,421)(71,99,323,436)(72,86,324,423)(73,101,325,438)(74,88,326,425)(75,103,327,440)(76,90,328,427)(77,105,329,442)(78,92,330,429)(79,107,331,444)(80,94,332,431)(81,109,333,446)(82,96,334,433)(83,111,335,448)(84,98,336,435)(141,209,401,182)(142,224,402,169)(143,211,403,184)(144,198,404,171)(145,213,405,186)(146,200,406,173)(147,215,407,188)(148,202,408,175)(149,217,409,190)(150,204,410,177)(151,219,411,192)(152,206,412,179)(153,221,413,194)(154,208,414,181)(155,223,415,196)(156,210,416,183)(157,197,417,170)(158,212,418,185)(159,199,419,172)(160,214,420,187)(161,201,393,174)(162,216,394,189)(163,203,395,176)(164,218,396,191)(165,205,397,178)(166,220,398,193)(167,207,399,180)(168,222,400,195), (1,46,15,32)(2,33,16,47)(3,48,17,34)(4,35,18,49)(5,50,19,36)(6,37,20,51)(7,52,21,38)(8,39,22,53)(9,54,23,40)(10,41,24,55)(11,56,25,42)(12,43,26,29)(13,30,27,44)(14,45,28,31)(57,403,71,417)(58,418,72,404)(59,405,73,419)(60,420,74,406)(61,407,75,393)(62,394,76,408)(63,409,77,395)(64,396,78,410)(65,411,79,397)(66,398,80,412)(67,413,81,399)(68,400,82,414)(69,415,83,401)(70,402,84,416)(85,184,99,170)(86,171,100,185)(87,186,101,172)(88,173,102,187)(89,188,103,174)(90,175,104,189)(91,190,105,176)(92,177,106,191)(93,192,107,178)(94,179,108,193)(95,194,109,180)(96,181,110,195)(97,196,111,182)(98,183,112,169)(113,274,127,260)(114,261,128,275)(115,276,129,262)(116,263,130,277)(117,278,131,264)(118,265,132,279)(119,280,133,266)(120,267,134,253)(121,254,135,268)(122,269,136,255)(123,256,137,270)(124,271,138,257)(125,258,139,272)(126,273,140,259)(141,321,155,335)(142,336,156,322)(143,323,157,309)(144,310,158,324)(145,325,159,311)(146,312,160,326)(147,327,161,313)(148,314,162,328)(149,329,163,315)(150,316,164,330)(151,331,165,317)(152,318,166,332)(153,333,167,319)(154,320,168,334)(197,422,211,436)(198,437,212,423)(199,424,213,438)(200,439,214,425)(201,426,215,440)(202,441,216,427)(203,428,217,442)(204,443,218,429)(205,430,219,444)(206,445,220,431)(207,432,221,446)(208,447,222,433)(209,434,223,448)(210,421,224,435)(225,362,239,348)(226,349,240,363)(227,364,241,350)(228,351,242,337)(229,338,243,352)(230,353,244,339)(231,340,245,354)(232,355,246,341)(233,342,247,356)(234,357,248,343)(235,344,249,358)(236,359,250,345)(237,346,251,360)(238,361,252,347)(281,366,295,380)(282,381,296,367)(283,368,297,382)(284,383,298,369)(285,370,299,384)(286,385,300,371)(287,372,301,386)(288,387,302,373)(289,374,303,388)(290,389,304,375)(291,376,305,390)(292,391,306,377)(293,378,307,392)(294,365,308,379), (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), (1,80,32,398,15,66,46,412)(2,397,47,79,16,411,33,65)(3,78,34,396,17,64,48,410)(4,395,49,77,18,409,35,63)(5,76,36,394,19,62,50,408)(6,393,51,75,20,407,37,61)(7,74,38,420,21,60,52,406)(8,419,53,73,22,405,39,59)(9,72,40,418,23,58,54,404)(10,417,55,71,24,403,41,57)(11,70,42,416,25,84,56,402)(12,415,29,69,26,401,43,83)(13,68,44,414,27,82,30,400)(14,413,31,67,28,399,45,81)(85,274,170,113,99,260,184,127)(86,140,185,273,100,126,171,259)(87,272,172,139,101,258,186,125)(88,138,187,271,102,124,173,257)(89,270,174,137,103,256,188,123)(90,136,189,269,104,122,175,255)(91,268,176,135,105,254,190,121)(92,134,191,267,106,120,177,253)(93,266,178,133,107,280,192,119)(94,132,193,265,108,118,179,279)(95,264,180,131,109,278,194,117)(96,130,195,263,110,116,181,277)(97,262,182,129,111,276,196,115)(98,128,169,261,112,114,183,275)(141,225,335,348,155,239,321,362)(142,347,322,252,156,361,336,238)(143,251,309,346,157,237,323,360)(144,345,324,250,158,359,310,236)(145,249,311,344,159,235,325,358)(146,343,326,248,160,357,312,234)(147,247,313,342,161,233,327,356)(148,341,328,246,162,355,314,232)(149,245,315,340,163,231,329,354)(150,339,330,244,164,353,316,230)(151,243,317,338,165,229,331,352)(152,337,332,242,166,351,318,228)(153,241,319,364,167,227,333,350)(154,363,334,240,168,349,320,226)(197,288,436,373,211,302,422,387)(198,372,423,287,212,386,437,301)(199,286,438,371,213,300,424,385)(200,370,425,285,214,384,439,299)(201,284,440,369,215,298,426,383)(202,368,427,283,216,382,441,297)(203,282,442,367,217,296,428,381)(204,366,429,281,218,380,443,295)(205,308,444,365,219,294,430,379)(206,392,431,307,220,378,445,293)(207,306,446,391,221,292,432,377)(208,390,433,305,222,376,447,291)(209,304,448,389,223,290,434,375)(210,388,435,303,224,374,421,289) );
G=PermutationGroup([[(1,265,337,378),(2,280,338,365),(3,267,339,380),(4,254,340,367),(5,269,341,382),(6,256,342,369),(7,271,343,384),(8,258,344,371),(9,273,345,386),(10,260,346,373),(11,275,347,388),(12,262,348,375),(13,277,349,390),(14,264,350,377),(15,279,351,392),(16,266,352,379),(17,253,353,366),(18,268,354,381),(19,255,355,368),(20,270,356,383),(21,257,357,370),(22,272,358,385),(23,259,359,372),(24,274,360,387),(25,261,361,374),(26,276,362,389),(27,263,363,376),(28,278,364,391),(29,129,239,304),(30,116,240,291),(31,131,241,306),(32,118,242,293),(33,133,243,308),(34,120,244,295),(35,135,245,282),(36,122,246,297),(37,137,247,284),(38,124,248,299),(39,139,249,286),(40,126,250,301),(41,113,251,288),(42,128,252,303),(43,115,225,290),(44,130,226,305),(45,117,227,292),(46,132,228,307),(47,119,229,294),(48,134,230,281),(49,121,231,296),(50,136,232,283),(51,123,233,298),(52,138,234,285),(53,125,235,300),(54,140,236,287),(55,127,237,302),(56,114,238,289),(57,85,309,422),(58,100,310,437),(59,87,311,424),(60,102,312,439),(61,89,313,426),(62,104,314,441),(63,91,315,428),(64,106,316,443),(65,93,317,430),(66,108,318,445),(67,95,319,432),(68,110,320,447),(69,97,321,434),(70,112,322,421),(71,99,323,436),(72,86,324,423),(73,101,325,438),(74,88,326,425),(75,103,327,440),(76,90,328,427),(77,105,329,442),(78,92,330,429),(79,107,331,444),(80,94,332,431),(81,109,333,446),(82,96,334,433),(83,111,335,448),(84,98,336,435),(141,209,401,182),(142,224,402,169),(143,211,403,184),(144,198,404,171),(145,213,405,186),(146,200,406,173),(147,215,407,188),(148,202,408,175),(149,217,409,190),(150,204,410,177),(151,219,411,192),(152,206,412,179),(153,221,413,194),(154,208,414,181),(155,223,415,196),(156,210,416,183),(157,197,417,170),(158,212,418,185),(159,199,419,172),(160,214,420,187),(161,201,393,174),(162,216,394,189),(163,203,395,176),(164,218,396,191),(165,205,397,178),(166,220,398,193),(167,207,399,180),(168,222,400,195)], [(1,46,15,32),(2,33,16,47),(3,48,17,34),(4,35,18,49),(5,50,19,36),(6,37,20,51),(7,52,21,38),(8,39,22,53),(9,54,23,40),(10,41,24,55),(11,56,25,42),(12,43,26,29),(13,30,27,44),(14,45,28,31),(57,403,71,417),(58,418,72,404),(59,405,73,419),(60,420,74,406),(61,407,75,393),(62,394,76,408),(63,409,77,395),(64,396,78,410),(65,411,79,397),(66,398,80,412),(67,413,81,399),(68,400,82,414),(69,415,83,401),(70,402,84,416),(85,184,99,170),(86,171,100,185),(87,186,101,172),(88,173,102,187),(89,188,103,174),(90,175,104,189),(91,190,105,176),(92,177,106,191),(93,192,107,178),(94,179,108,193),(95,194,109,180),(96,181,110,195),(97,196,111,182),(98,183,112,169),(113,274,127,260),(114,261,128,275),(115,276,129,262),(116,263,130,277),(117,278,131,264),(118,265,132,279),(119,280,133,266),(120,267,134,253),(121,254,135,268),(122,269,136,255),(123,256,137,270),(124,271,138,257),(125,258,139,272),(126,273,140,259),(141,321,155,335),(142,336,156,322),(143,323,157,309),(144,310,158,324),(145,325,159,311),(146,312,160,326),(147,327,161,313),(148,314,162,328),(149,329,163,315),(150,316,164,330),(151,331,165,317),(152,318,166,332),(153,333,167,319),(154,320,168,334),(197,422,211,436),(198,437,212,423),(199,424,213,438),(200,439,214,425),(201,426,215,440),(202,441,216,427),(203,428,217,442),(204,443,218,429),(205,430,219,444),(206,445,220,431),(207,432,221,446),(208,447,222,433),(209,434,223,448),(210,421,224,435),(225,362,239,348),(226,349,240,363),(227,364,241,350),(228,351,242,337),(229,338,243,352),(230,353,244,339),(231,340,245,354),(232,355,246,341),(233,342,247,356),(234,357,248,343),(235,344,249,358),(236,359,250,345),(237,346,251,360),(238,361,252,347),(281,366,295,380),(282,381,296,367),(283,368,297,382),(284,383,298,369),(285,370,299,384),(286,385,300,371),(287,372,301,386),(288,387,302,373),(289,374,303,388),(290,389,304,375),(291,376,305,390),(292,391,306,377),(293,378,307,392),(294,365,308,379)], [(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)], [(1,80,32,398,15,66,46,412),(2,397,47,79,16,411,33,65),(3,78,34,396,17,64,48,410),(4,395,49,77,18,409,35,63),(5,76,36,394,19,62,50,408),(6,393,51,75,20,407,37,61),(7,74,38,420,21,60,52,406),(8,419,53,73,22,405,39,59),(9,72,40,418,23,58,54,404),(10,417,55,71,24,403,41,57),(11,70,42,416,25,84,56,402),(12,415,29,69,26,401,43,83),(13,68,44,414,27,82,30,400),(14,413,31,67,28,399,45,81),(85,274,170,113,99,260,184,127),(86,140,185,273,100,126,171,259),(87,272,172,139,101,258,186,125),(88,138,187,271,102,124,173,257),(89,270,174,137,103,256,188,123),(90,136,189,269,104,122,175,255),(91,268,176,135,105,254,190,121),(92,134,191,267,106,120,177,253),(93,266,178,133,107,280,192,119),(94,132,193,265,108,118,179,279),(95,264,180,131,109,278,194,117),(96,130,195,263,110,116,181,277),(97,262,182,129,111,276,196,115),(98,128,169,261,112,114,183,275),(141,225,335,348,155,239,321,362),(142,347,322,252,156,361,336,238),(143,251,309,346,157,237,323,360),(144,345,324,250,158,359,310,236),(145,249,311,344,159,235,325,358),(146,343,326,248,160,357,312,234),(147,247,313,342,161,233,327,356),(148,341,328,246,162,355,314,232),(149,245,315,340,163,231,329,354),(150,339,330,244,164,353,316,230),(151,243,317,338,165,229,331,352),(152,337,332,242,166,351,318,228),(153,241,319,364,167,227,333,350),(154,363,334,240,168,349,320,226),(197,288,436,373,211,302,422,387),(198,372,423,287,212,386,437,301),(199,286,438,371,213,300,424,385),(200,370,425,285,214,384,439,299),(201,284,440,369,215,298,426,383),(202,368,427,283,216,382,441,297),(203,282,442,367,217,296,428,381),(204,366,429,281,218,380,443,295),(205,308,444,365,219,294,430,379),(206,392,431,307,220,378,445,293),(207,306,446,391,221,292,432,377),(208,390,433,305,222,376,447,291),(209,304,448,389,223,290,434,375),(210,388,435,303,224,374,421,289)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | C4×D7 | C4○D28 | C8.C22 | C28.C23 | D4.9D14 |
| kernel | C42.59D14 | C42.D7 | C4.Dic14 | C14.Q16 | Q8⋊Dic7 | C4×Dic14 | C2×C7⋊Q16 | Q8×C28 | C7⋊Q16 | C2×C28 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C2×C4 | Q8 | C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 2 | 6 | 6 |
Matrix representation of C42.59D14 ►in GL6(𝔽113)
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 8 |
| 0 | 0 | 0 | 0 | 105 | 96 |
| 0 | 0 | 96 | 105 | 0 | 0 |
| 0 | 0 | 8 | 17 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 6 | 54 | 0 | 0 | 0 | 0 |
| 24 | 84 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 81 | 99 | 17 |
| 0 | 0 | 32 | 33 | 96 | 55 |
| 0 | 0 | 99 | 17 | 103 | 32 |
| 0 | 0 | 96 | 55 | 81 | 80 |
| 11 | 17 | 0 | 0 | 0 | 0 |
| 46 | 102 | 0 | 0 | 0 | 0 |
| 0 | 0 | 85 | 35 | 1 | 9 |
| 0 | 0 | 41 | 28 | 33 | 112 |
| 0 | 0 | 112 | 104 | 85 | 35 |
| 0 | 0 | 80 | 1 | 41 | 28 |
G:=sub<GL(6,GF(113))| [98,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,96,8,0,0,0,0,105,17,0,0,17,105,0,0,0,0,8,96,0,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,1,0,0,0,0,0,0,1,0,0],[6,24,0,0,0,0,54,84,0,0,0,0,0,0,10,32,99,96,0,0,81,33,17,55,0,0,99,96,103,81,0,0,17,55,32,80],[11,46,0,0,0,0,17,102,0,0,0,0,0,0,85,41,112,80,0,0,35,28,104,1,0,0,1,33,85,41,0,0,9,112,35,28] >;
C42.59D14 in GAP, Magma, Sage, TeX
C_4^2._{59}D_{14} % in TeX
G:=Group("C4^2.59D14"); // GroupNames label
G:=SmallGroup(448,564);
// by ID
G=gap.SmallGroup(448,564);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,232,387,58,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^13>;
// generators/relations