metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.76D14, C7⋊C8⋊4Q8, C7⋊5(C8⋊Q8), C4⋊Q8.6D7, C4.36(Q8×D7), C4⋊C4.81D14, C28.37(C2×Q8), (C2×C28).293D4, C14.32(C4⋊Q8), C28.6Q8.8C2, C14.97(C8⋊C22), (C2×C28).398C23, (C4×C28).127C22, C28.Q8.16C2, C4.Dic14.17C2, C42.D7.7C2, C14.93(C8.C22), C4⋊Dic7.158C22, C2.12(Dic7⋊Q8), C2.18(D4.D14), C2.14(C28.C23), (C7×C4⋊Q8).6C2, (C2×C14).529(C2×D4), (C2×C4).71(C7⋊D4), (C2×C7⋊C8).133C22, (C7×C4⋊C4).128C22, (C2×C4).495(C22×D7), C22.201(C2×C7⋊D4), SmallGroup(448,614)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.76D14
G = < a,b,c,d | a4=b4=1, c14=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc13 >
Subgroups: 332 in 90 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×Q8, C8⋊Q8, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, Q8×C14, C42.D7, C28.Q8, C4.Dic14, C28.6Q8, C7×C4⋊Q8, C42.76D14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, C8⋊C22, C8.C22, C7⋊D4, C22×D7, C8⋊Q8, Q8×D7, C2×C7⋊D4, D4.D14, C28.C23, Dic7⋊Q8, C42.76D14
►Regular action on 448 points
Generators in S448
(1 376 15 390)(2 391 16 377)(3 378 17 392)(4 365 18 379)(5 380 19 366)(6 367 20 381)(7 382 21 368)(8 369 22 383)(9 384 23 370)(10 371 24 385)(11 386 25 372)(12 373 26 387)(13 388 27 374)(14 375 28 389)(29 394 43 408)(30 409 44 395)(31 396 45 410)(32 411 46 397)(33 398 47 412)(34 413 48 399)(35 400 49 414)(36 415 50 401)(37 402 51 416)(38 417 52 403)(39 404 53 418)(40 419 54 405)(41 406 55 420)(42 393 56 407)(57 224 71 210)(58 211 72 197)(59 198 73 212)(60 213 74 199)(61 200 75 214)(62 215 76 201)(63 202 77 216)(64 217 78 203)(65 204 79 218)(66 219 80 205)(67 206 81 220)(68 221 82 207)(69 208 83 222)(70 223 84 209)(85 183 99 169)(86 170 100 184)(87 185 101 171)(88 172 102 186)(89 187 103 173)(90 174 104 188)(91 189 105 175)(92 176 106 190)(93 191 107 177)(94 178 108 192)(95 193 109 179)(96 180 110 194)(97 195 111 181)(98 182 112 196)(113 154 127 168)(114 141 128 155)(115 156 129 142)(116 143 130 157)(117 158 131 144)(118 145 132 159)(119 160 133 146)(120 147 134 161)(121 162 135 148)(122 149 136 163)(123 164 137 150)(124 151 138 165)(125 166 139 152)(126 153 140 167)(225 260 239 274)(226 275 240 261)(227 262 241 276)(228 277 242 263)(229 264 243 278)(230 279 244 265)(231 266 245 280)(232 253 246 267)(233 268 247 254)(234 255 248 269)(235 270 249 256)(236 257 250 271)(237 272 251 258)(238 259 252 273)(281 435 295 421)(282 422 296 436)(283 437 297 423)(284 424 298 438)(285 439 299 425)(286 426 300 440)(287 441 301 427)(288 428 302 442)(289 443 303 429)(290 430 304 444)(291 445 305 431)(292 432 306 446)(293 447 307 433)(294 434 308 448)(309 354 323 340)(310 341 324 355)(311 356 325 342)(312 343 326 357)(313 358 327 344)(314 345 328 359)(315 360 329 346)(316 347 330 361)(317 362 331 348)(318 349 332 363)(319 364 333 350)(320 351 334 337)(321 338 335 352)(322 353 336 339)
(1 60 35 131)(2 132 36 61)(3 62 37 133)(4 134 38 63)(5 64 39 135)(6 136 40 65)(7 66 41 137)(8 138 42 67)(9 68 43 139)(10 140 44 69)(11 70 45 113)(12 114 46 71)(13 72 47 115)(14 116 48 73)(15 74 49 117)(16 118 50 75)(17 76 51 119)(18 120 52 77)(19 78 53 121)(20 122 54 79)(21 80 55 123)(22 124 56 81)(23 82 29 125)(24 126 30 83)(25 84 31 127)(26 128 32 57)(27 58 33 129)(28 130 34 59)(85 245 290 340)(86 341 291 246)(87 247 292 342)(88 343 293 248)(89 249 294 344)(90 345 295 250)(91 251 296 346)(92 347 297 252)(93 225 298 348)(94 349 299 226)(95 227 300 350)(96 351 301 228)(97 229 302 352)(98 353 303 230)(99 231 304 354)(100 355 305 232)(101 233 306 356)(102 357 307 234)(103 235 308 358)(104 359 281 236)(105 237 282 360)(106 361 283 238)(107 239 284 362)(108 363 285 240)(109 241 286 364)(110 337 287 242)(111 243 288 338)(112 339 289 244)(141 397 210 373)(142 374 211 398)(143 399 212 375)(144 376 213 400)(145 401 214 377)(146 378 215 402)(147 403 216 379)(148 380 217 404)(149 405 218 381)(150 382 219 406)(151 407 220 383)(152 384 221 408)(153 409 222 385)(154 386 223 410)(155 411 224 387)(156 388 197 412)(157 413 198 389)(158 390 199 414)(159 415 200 391)(160 392 201 416)(161 417 202 365)(162 366 203 418)(163 419 204 367)(164 368 205 420)(165 393 206 369)(166 370 207 394)(167 395 208 371)(168 372 209 396)(169 266 444 323)(170 324 445 267)(171 268 446 325)(172 326 447 269)(173 270 448 327)(174 328 421 271)(175 272 422 329)(176 330 423 273)(177 274 424 331)(178 332 425 275)(179 276 426 333)(180 334 427 277)(181 278 428 335)(182 336 429 279)(183 280 430 309)(184 310 431 253)(185 254 432 311)(186 312 433 255)(187 256 434 313)(188 314 435 257)(189 258 436 315)(190 316 437 259)(191 260 438 317)(192 318 439 261)(193 262 440 319)(194 320 441 263)(195 264 442 321)(196 322 443 265)
(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)
(1 276 15 262)(2 439 16 425)(3 274 17 260)(4 437 18 423)(5 272 19 258)(6 435 20 421)(7 270 21 256)(8 433 22 447)(9 268 23 254)(10 431 24 445)(11 266 25 280)(12 429 26 443)(13 264 27 278)(14 427 28 441)(29 311 43 325)(30 170 44 184)(31 309 45 323)(32 196 46 182)(33 335 47 321)(34 194 48 180)(35 333 49 319)(36 192 50 178)(37 331 51 317)(38 190 52 176)(39 329 53 315)(40 188 54 174)(41 327 55 313)(42 186 56 172)(57 265 71 279)(58 181 72 195)(59 263 73 277)(60 179 74 193)(61 261 75 275)(62 177 76 191)(63 259 77 273)(64 175 78 189)(65 257 79 271)(66 173 80 187)(67 255 81 269)(68 171 82 185)(69 253 83 267)(70 169 84 183)(85 154 99 168)(86 371 100 385)(87 152 101 166)(88 369 102 383)(89 150 103 164)(90 367 104 381)(91 148 105 162)(92 365 106 379)(93 146 107 160)(94 391 108 377)(95 144 109 158)(96 389 110 375)(97 142 111 156)(98 387 112 373)(113 444 127 430)(114 336 128 322)(115 442 129 428)(116 334 130 320)(117 440 131 426)(118 332 132 318)(119 438 133 424)(120 330 134 316)(121 436 135 422)(122 328 136 314)(123 434 137 448)(124 326 138 312)(125 432 139 446)(126 324 140 310)(141 230 155 244)(143 228 157 242)(145 226 159 240)(147 252 161 238)(149 250 163 236)(151 248 165 234)(153 246 167 232)(197 302 211 288)(198 337 212 351)(199 300 213 286)(200 363 214 349)(201 298 215 284)(202 361 216 347)(203 296 217 282)(204 359 218 345)(205 294 219 308)(206 357 220 343)(207 292 221 306)(208 355 222 341)(209 290 223 304)(210 353 224 339)(225 402 239 416)(227 400 241 414)(229 398 243 412)(231 396 245 410)(233 394 247 408)(235 420 249 406)(237 418 251 404)(281 405 295 419)(283 403 297 417)(285 401 299 415)(287 399 301 413)(289 397 303 411)(291 395 305 409)(293 393 307 407)(338 388 352 374)(340 386 354 372)(342 384 356 370)(344 382 358 368)(346 380 360 366)(348 378 362 392)(350 376 364 390)
G:=sub<Sym(448)| (1,376,15,390)(2,391,16,377)(3,378,17,392)(4,365,18,379)(5,380,19,366)(6,367,20,381)(7,382,21,368)(8,369,22,383)(9,384,23,370)(10,371,24,385)(11,386,25,372)(12,373,26,387)(13,388,27,374)(14,375,28,389)(29,394,43,408)(30,409,44,395)(31,396,45,410)(32,411,46,397)(33,398,47,412)(34,413,48,399)(35,400,49,414)(36,415,50,401)(37,402,51,416)(38,417,52,403)(39,404,53,418)(40,419,54,405)(41,406,55,420)(42,393,56,407)(57,224,71,210)(58,211,72,197)(59,198,73,212)(60,213,74,199)(61,200,75,214)(62,215,76,201)(63,202,77,216)(64,217,78,203)(65,204,79,218)(66,219,80,205)(67,206,81,220)(68,221,82,207)(69,208,83,222)(70,223,84,209)(85,183,99,169)(86,170,100,184)(87,185,101,171)(88,172,102,186)(89,187,103,173)(90,174,104,188)(91,189,105,175)(92,176,106,190)(93,191,107,177)(94,178,108,192)(95,193,109,179)(96,180,110,194)(97,195,111,181)(98,182,112,196)(113,154,127,168)(114,141,128,155)(115,156,129,142)(116,143,130,157)(117,158,131,144)(118,145,132,159)(119,160,133,146)(120,147,134,161)(121,162,135,148)(122,149,136,163)(123,164,137,150)(124,151,138,165)(125,166,139,152)(126,153,140,167)(225,260,239,274)(226,275,240,261)(227,262,241,276)(228,277,242,263)(229,264,243,278)(230,279,244,265)(231,266,245,280)(232,253,246,267)(233,268,247,254)(234,255,248,269)(235,270,249,256)(236,257,250,271)(237,272,251,258)(238,259,252,273)(281,435,295,421)(282,422,296,436)(283,437,297,423)(284,424,298,438)(285,439,299,425)(286,426,300,440)(287,441,301,427)(288,428,302,442)(289,443,303,429)(290,430,304,444)(291,445,305,431)(292,432,306,446)(293,447,307,433)(294,434,308,448)(309,354,323,340)(310,341,324,355)(311,356,325,342)(312,343,326,357)(313,358,327,344)(314,345,328,359)(315,360,329,346)(316,347,330,361)(317,362,331,348)(318,349,332,363)(319,364,333,350)(320,351,334,337)(321,338,335,352)(322,353,336,339), (1,60,35,131)(2,132,36,61)(3,62,37,133)(4,134,38,63)(5,64,39,135)(6,136,40,65)(7,66,41,137)(8,138,42,67)(9,68,43,139)(10,140,44,69)(11,70,45,113)(12,114,46,71)(13,72,47,115)(14,116,48,73)(15,74,49,117)(16,118,50,75)(17,76,51,119)(18,120,52,77)(19,78,53,121)(20,122,54,79)(21,80,55,123)(22,124,56,81)(23,82,29,125)(24,126,30,83)(25,84,31,127)(26,128,32,57)(27,58,33,129)(28,130,34,59)(85,245,290,340)(86,341,291,246)(87,247,292,342)(88,343,293,248)(89,249,294,344)(90,345,295,250)(91,251,296,346)(92,347,297,252)(93,225,298,348)(94,349,299,226)(95,227,300,350)(96,351,301,228)(97,229,302,352)(98,353,303,230)(99,231,304,354)(100,355,305,232)(101,233,306,356)(102,357,307,234)(103,235,308,358)(104,359,281,236)(105,237,282,360)(106,361,283,238)(107,239,284,362)(108,363,285,240)(109,241,286,364)(110,337,287,242)(111,243,288,338)(112,339,289,244)(141,397,210,373)(142,374,211,398)(143,399,212,375)(144,376,213,400)(145,401,214,377)(146,378,215,402)(147,403,216,379)(148,380,217,404)(149,405,218,381)(150,382,219,406)(151,407,220,383)(152,384,221,408)(153,409,222,385)(154,386,223,410)(155,411,224,387)(156,388,197,412)(157,413,198,389)(158,390,199,414)(159,415,200,391)(160,392,201,416)(161,417,202,365)(162,366,203,418)(163,419,204,367)(164,368,205,420)(165,393,206,369)(166,370,207,394)(167,395,208,371)(168,372,209,396)(169,266,444,323)(170,324,445,267)(171,268,446,325)(172,326,447,269)(173,270,448,327)(174,328,421,271)(175,272,422,329)(176,330,423,273)(177,274,424,331)(178,332,425,275)(179,276,426,333)(180,334,427,277)(181,278,428,335)(182,336,429,279)(183,280,430,309)(184,310,431,253)(185,254,432,311)(186,312,433,255)(187,256,434,313)(188,314,435,257)(189,258,436,315)(190,316,437,259)(191,260,438,317)(192,318,439,261)(193,262,440,319)(194,320,441,263)(195,264,442,321)(196,322,443,265), (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), (1,276,15,262)(2,439,16,425)(3,274,17,260)(4,437,18,423)(5,272,19,258)(6,435,20,421)(7,270,21,256)(8,433,22,447)(9,268,23,254)(10,431,24,445)(11,266,25,280)(12,429,26,443)(13,264,27,278)(14,427,28,441)(29,311,43,325)(30,170,44,184)(31,309,45,323)(32,196,46,182)(33,335,47,321)(34,194,48,180)(35,333,49,319)(36,192,50,178)(37,331,51,317)(38,190,52,176)(39,329,53,315)(40,188,54,174)(41,327,55,313)(42,186,56,172)(57,265,71,279)(58,181,72,195)(59,263,73,277)(60,179,74,193)(61,261,75,275)(62,177,76,191)(63,259,77,273)(64,175,78,189)(65,257,79,271)(66,173,80,187)(67,255,81,269)(68,171,82,185)(69,253,83,267)(70,169,84,183)(85,154,99,168)(86,371,100,385)(87,152,101,166)(88,369,102,383)(89,150,103,164)(90,367,104,381)(91,148,105,162)(92,365,106,379)(93,146,107,160)(94,391,108,377)(95,144,109,158)(96,389,110,375)(97,142,111,156)(98,387,112,373)(113,444,127,430)(114,336,128,322)(115,442,129,428)(116,334,130,320)(117,440,131,426)(118,332,132,318)(119,438,133,424)(120,330,134,316)(121,436,135,422)(122,328,136,314)(123,434,137,448)(124,326,138,312)(125,432,139,446)(126,324,140,310)(141,230,155,244)(143,228,157,242)(145,226,159,240)(147,252,161,238)(149,250,163,236)(151,248,165,234)(153,246,167,232)(197,302,211,288)(198,337,212,351)(199,300,213,286)(200,363,214,349)(201,298,215,284)(202,361,216,347)(203,296,217,282)(204,359,218,345)(205,294,219,308)(206,357,220,343)(207,292,221,306)(208,355,222,341)(209,290,223,304)(210,353,224,339)(225,402,239,416)(227,400,241,414)(229,398,243,412)(231,396,245,410)(233,394,247,408)(235,420,249,406)(237,418,251,404)(281,405,295,419)(283,403,297,417)(285,401,299,415)(287,399,301,413)(289,397,303,411)(291,395,305,409)(293,393,307,407)(338,388,352,374)(340,386,354,372)(342,384,356,370)(344,382,358,368)(346,380,360,366)(348,378,362,392)(350,376,364,390)>;
G:=Group( (1,376,15,390)(2,391,16,377)(3,378,17,392)(4,365,18,379)(5,380,19,366)(6,367,20,381)(7,382,21,368)(8,369,22,383)(9,384,23,370)(10,371,24,385)(11,386,25,372)(12,373,26,387)(13,388,27,374)(14,375,28,389)(29,394,43,408)(30,409,44,395)(31,396,45,410)(32,411,46,397)(33,398,47,412)(34,413,48,399)(35,400,49,414)(36,415,50,401)(37,402,51,416)(38,417,52,403)(39,404,53,418)(40,419,54,405)(41,406,55,420)(42,393,56,407)(57,224,71,210)(58,211,72,197)(59,198,73,212)(60,213,74,199)(61,200,75,214)(62,215,76,201)(63,202,77,216)(64,217,78,203)(65,204,79,218)(66,219,80,205)(67,206,81,220)(68,221,82,207)(69,208,83,222)(70,223,84,209)(85,183,99,169)(86,170,100,184)(87,185,101,171)(88,172,102,186)(89,187,103,173)(90,174,104,188)(91,189,105,175)(92,176,106,190)(93,191,107,177)(94,178,108,192)(95,193,109,179)(96,180,110,194)(97,195,111,181)(98,182,112,196)(113,154,127,168)(114,141,128,155)(115,156,129,142)(116,143,130,157)(117,158,131,144)(118,145,132,159)(119,160,133,146)(120,147,134,161)(121,162,135,148)(122,149,136,163)(123,164,137,150)(124,151,138,165)(125,166,139,152)(126,153,140,167)(225,260,239,274)(226,275,240,261)(227,262,241,276)(228,277,242,263)(229,264,243,278)(230,279,244,265)(231,266,245,280)(232,253,246,267)(233,268,247,254)(234,255,248,269)(235,270,249,256)(236,257,250,271)(237,272,251,258)(238,259,252,273)(281,435,295,421)(282,422,296,436)(283,437,297,423)(284,424,298,438)(285,439,299,425)(286,426,300,440)(287,441,301,427)(288,428,302,442)(289,443,303,429)(290,430,304,444)(291,445,305,431)(292,432,306,446)(293,447,307,433)(294,434,308,448)(309,354,323,340)(310,341,324,355)(311,356,325,342)(312,343,326,357)(313,358,327,344)(314,345,328,359)(315,360,329,346)(316,347,330,361)(317,362,331,348)(318,349,332,363)(319,364,333,350)(320,351,334,337)(321,338,335,352)(322,353,336,339), (1,60,35,131)(2,132,36,61)(3,62,37,133)(4,134,38,63)(5,64,39,135)(6,136,40,65)(7,66,41,137)(8,138,42,67)(9,68,43,139)(10,140,44,69)(11,70,45,113)(12,114,46,71)(13,72,47,115)(14,116,48,73)(15,74,49,117)(16,118,50,75)(17,76,51,119)(18,120,52,77)(19,78,53,121)(20,122,54,79)(21,80,55,123)(22,124,56,81)(23,82,29,125)(24,126,30,83)(25,84,31,127)(26,128,32,57)(27,58,33,129)(28,130,34,59)(85,245,290,340)(86,341,291,246)(87,247,292,342)(88,343,293,248)(89,249,294,344)(90,345,295,250)(91,251,296,346)(92,347,297,252)(93,225,298,348)(94,349,299,226)(95,227,300,350)(96,351,301,228)(97,229,302,352)(98,353,303,230)(99,231,304,354)(100,355,305,232)(101,233,306,356)(102,357,307,234)(103,235,308,358)(104,359,281,236)(105,237,282,360)(106,361,283,238)(107,239,284,362)(108,363,285,240)(109,241,286,364)(110,337,287,242)(111,243,288,338)(112,339,289,244)(141,397,210,373)(142,374,211,398)(143,399,212,375)(144,376,213,400)(145,401,214,377)(146,378,215,402)(147,403,216,379)(148,380,217,404)(149,405,218,381)(150,382,219,406)(151,407,220,383)(152,384,221,408)(153,409,222,385)(154,386,223,410)(155,411,224,387)(156,388,197,412)(157,413,198,389)(158,390,199,414)(159,415,200,391)(160,392,201,416)(161,417,202,365)(162,366,203,418)(163,419,204,367)(164,368,205,420)(165,393,206,369)(166,370,207,394)(167,395,208,371)(168,372,209,396)(169,266,444,323)(170,324,445,267)(171,268,446,325)(172,326,447,269)(173,270,448,327)(174,328,421,271)(175,272,422,329)(176,330,423,273)(177,274,424,331)(178,332,425,275)(179,276,426,333)(180,334,427,277)(181,278,428,335)(182,336,429,279)(183,280,430,309)(184,310,431,253)(185,254,432,311)(186,312,433,255)(187,256,434,313)(188,314,435,257)(189,258,436,315)(190,316,437,259)(191,260,438,317)(192,318,439,261)(193,262,440,319)(194,320,441,263)(195,264,442,321)(196,322,443,265), (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), (1,276,15,262)(2,439,16,425)(3,274,17,260)(4,437,18,423)(5,272,19,258)(6,435,20,421)(7,270,21,256)(8,433,22,447)(9,268,23,254)(10,431,24,445)(11,266,25,280)(12,429,26,443)(13,264,27,278)(14,427,28,441)(29,311,43,325)(30,170,44,184)(31,309,45,323)(32,196,46,182)(33,335,47,321)(34,194,48,180)(35,333,49,319)(36,192,50,178)(37,331,51,317)(38,190,52,176)(39,329,53,315)(40,188,54,174)(41,327,55,313)(42,186,56,172)(57,265,71,279)(58,181,72,195)(59,263,73,277)(60,179,74,193)(61,261,75,275)(62,177,76,191)(63,259,77,273)(64,175,78,189)(65,257,79,271)(66,173,80,187)(67,255,81,269)(68,171,82,185)(69,253,83,267)(70,169,84,183)(85,154,99,168)(86,371,100,385)(87,152,101,166)(88,369,102,383)(89,150,103,164)(90,367,104,381)(91,148,105,162)(92,365,106,379)(93,146,107,160)(94,391,108,377)(95,144,109,158)(96,389,110,375)(97,142,111,156)(98,387,112,373)(113,444,127,430)(114,336,128,322)(115,442,129,428)(116,334,130,320)(117,440,131,426)(118,332,132,318)(119,438,133,424)(120,330,134,316)(121,436,135,422)(122,328,136,314)(123,434,137,448)(124,326,138,312)(125,432,139,446)(126,324,140,310)(141,230,155,244)(143,228,157,242)(145,226,159,240)(147,252,161,238)(149,250,163,236)(151,248,165,234)(153,246,167,232)(197,302,211,288)(198,337,212,351)(199,300,213,286)(200,363,214,349)(201,298,215,284)(202,361,216,347)(203,296,217,282)(204,359,218,345)(205,294,219,308)(206,357,220,343)(207,292,221,306)(208,355,222,341)(209,290,223,304)(210,353,224,339)(225,402,239,416)(227,400,241,414)(229,398,243,412)(231,396,245,410)(233,394,247,408)(235,420,249,406)(237,418,251,404)(281,405,295,419)(283,403,297,417)(285,401,299,415)(287,399,301,413)(289,397,303,411)(291,395,305,409)(293,393,307,407)(338,388,352,374)(340,386,354,372)(342,384,356,370)(344,382,358,368)(346,380,360,366)(348,378,362,392)(350,376,364,390) );
G=PermutationGroup([[(1,376,15,390),(2,391,16,377),(3,378,17,392),(4,365,18,379),(5,380,19,366),(6,367,20,381),(7,382,21,368),(8,369,22,383),(9,384,23,370),(10,371,24,385),(11,386,25,372),(12,373,26,387),(13,388,27,374),(14,375,28,389),(29,394,43,408),(30,409,44,395),(31,396,45,410),(32,411,46,397),(33,398,47,412),(34,413,48,399),(35,400,49,414),(36,415,50,401),(37,402,51,416),(38,417,52,403),(39,404,53,418),(40,419,54,405),(41,406,55,420),(42,393,56,407),(57,224,71,210),(58,211,72,197),(59,198,73,212),(60,213,74,199),(61,200,75,214),(62,215,76,201),(63,202,77,216),(64,217,78,203),(65,204,79,218),(66,219,80,205),(67,206,81,220),(68,221,82,207),(69,208,83,222),(70,223,84,209),(85,183,99,169),(86,170,100,184),(87,185,101,171),(88,172,102,186),(89,187,103,173),(90,174,104,188),(91,189,105,175),(92,176,106,190),(93,191,107,177),(94,178,108,192),(95,193,109,179),(96,180,110,194),(97,195,111,181),(98,182,112,196),(113,154,127,168),(114,141,128,155),(115,156,129,142),(116,143,130,157),(117,158,131,144),(118,145,132,159),(119,160,133,146),(120,147,134,161),(121,162,135,148),(122,149,136,163),(123,164,137,150),(124,151,138,165),(125,166,139,152),(126,153,140,167),(225,260,239,274),(226,275,240,261),(227,262,241,276),(228,277,242,263),(229,264,243,278),(230,279,244,265),(231,266,245,280),(232,253,246,267),(233,268,247,254),(234,255,248,269),(235,270,249,256),(236,257,250,271),(237,272,251,258),(238,259,252,273),(281,435,295,421),(282,422,296,436),(283,437,297,423),(284,424,298,438),(285,439,299,425),(286,426,300,440),(287,441,301,427),(288,428,302,442),(289,443,303,429),(290,430,304,444),(291,445,305,431),(292,432,306,446),(293,447,307,433),(294,434,308,448),(309,354,323,340),(310,341,324,355),(311,356,325,342),(312,343,326,357),(313,358,327,344),(314,345,328,359),(315,360,329,346),(316,347,330,361),(317,362,331,348),(318,349,332,363),(319,364,333,350),(320,351,334,337),(321,338,335,352),(322,353,336,339)], [(1,60,35,131),(2,132,36,61),(3,62,37,133),(4,134,38,63),(5,64,39,135),(6,136,40,65),(7,66,41,137),(8,138,42,67),(9,68,43,139),(10,140,44,69),(11,70,45,113),(12,114,46,71),(13,72,47,115),(14,116,48,73),(15,74,49,117),(16,118,50,75),(17,76,51,119),(18,120,52,77),(19,78,53,121),(20,122,54,79),(21,80,55,123),(22,124,56,81),(23,82,29,125),(24,126,30,83),(25,84,31,127),(26,128,32,57),(27,58,33,129),(28,130,34,59),(85,245,290,340),(86,341,291,246),(87,247,292,342),(88,343,293,248),(89,249,294,344),(90,345,295,250),(91,251,296,346),(92,347,297,252),(93,225,298,348),(94,349,299,226),(95,227,300,350),(96,351,301,228),(97,229,302,352),(98,353,303,230),(99,231,304,354),(100,355,305,232),(101,233,306,356),(102,357,307,234),(103,235,308,358),(104,359,281,236),(105,237,282,360),(106,361,283,238),(107,239,284,362),(108,363,285,240),(109,241,286,364),(110,337,287,242),(111,243,288,338),(112,339,289,244),(141,397,210,373),(142,374,211,398),(143,399,212,375),(144,376,213,400),(145,401,214,377),(146,378,215,402),(147,403,216,379),(148,380,217,404),(149,405,218,381),(150,382,219,406),(151,407,220,383),(152,384,221,408),(153,409,222,385),(154,386,223,410),(155,411,224,387),(156,388,197,412),(157,413,198,389),(158,390,199,414),(159,415,200,391),(160,392,201,416),(161,417,202,365),(162,366,203,418),(163,419,204,367),(164,368,205,420),(165,393,206,369),(166,370,207,394),(167,395,208,371),(168,372,209,396),(169,266,444,323),(170,324,445,267),(171,268,446,325),(172,326,447,269),(173,270,448,327),(174,328,421,271),(175,272,422,329),(176,330,423,273),(177,274,424,331),(178,332,425,275),(179,276,426,333),(180,334,427,277),(181,278,428,335),(182,336,429,279),(183,280,430,309),(184,310,431,253),(185,254,432,311),(186,312,433,255),(187,256,434,313),(188,314,435,257),(189,258,436,315),(190,316,437,259),(191,260,438,317),(192,318,439,261),(193,262,440,319),(194,320,441,263),(195,264,442,321),(196,322,443,265)], [(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)], [(1,276,15,262),(2,439,16,425),(3,274,17,260),(4,437,18,423),(5,272,19,258),(6,435,20,421),(7,270,21,256),(8,433,22,447),(9,268,23,254),(10,431,24,445),(11,266,25,280),(12,429,26,443),(13,264,27,278),(14,427,28,441),(29,311,43,325),(30,170,44,184),(31,309,45,323),(32,196,46,182),(33,335,47,321),(34,194,48,180),(35,333,49,319),(36,192,50,178),(37,331,51,317),(38,190,52,176),(39,329,53,315),(40,188,54,174),(41,327,55,313),(42,186,56,172),(57,265,71,279),(58,181,72,195),(59,263,73,277),(60,179,74,193),(61,261,75,275),(62,177,76,191),(63,259,77,273),(64,175,78,189),(65,257,79,271),(66,173,80,187),(67,255,81,269),(68,171,82,185),(69,253,83,267),(70,169,84,183),(85,154,99,168),(86,371,100,385),(87,152,101,166),(88,369,102,383),(89,150,103,164),(90,367,104,381),(91,148,105,162),(92,365,106,379),(93,146,107,160),(94,391,108,377),(95,144,109,158),(96,389,110,375),(97,142,111,156),(98,387,112,373),(113,444,127,430),(114,336,128,322),(115,442,129,428),(116,334,130,320),(117,440,131,426),(118,332,132,318),(119,438,133,424),(120,330,134,316),(121,436,135,422),(122,328,136,314),(123,434,137,448),(124,326,138,312),(125,432,139,446),(126,324,140,310),(141,230,155,244),(143,228,157,242),(145,226,159,240),(147,252,161,238),(149,250,163,236),(151,248,165,234),(153,246,167,232),(197,302,211,288),(198,337,212,351),(199,300,213,286),(200,363,214,349),(201,298,215,284),(202,361,216,347),(203,296,217,282),(204,359,218,345),(205,294,219,308),(206,357,220,343),(207,292,221,306),(208,355,222,341),(209,290,223,304),(210,353,224,339),(225,402,239,416),(227,400,241,414),(229,398,243,412),(231,396,245,410),(233,394,247,408),(235,420,249,406),(237,418,251,404),(281,405,295,419),(283,403,297,417),(285,401,299,415),(287,399,301,413),(289,397,303,411),(291,395,305,409),(293,393,307,407),(338,388,352,374),(340,386,354,372),(342,384,356,370),(344,382,358,368),(346,380,360,366),(348,378,362,392),(350,376,364,390)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 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 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | D14 | D14 | C7⋊D4 | C8⋊C22 | C8.C22 | Q8×D7 | D4.D14 | C28.C23 |
| kernel | C42.76D14 | C42.D7 | C28.Q8 | C4.Dic14 | C28.6Q8 | C7×C4⋊Q8 | C7⋊C8 | C2×C28 | C4⋊Q8 | C42 | C4⋊C4 | C2×C4 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 12 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C42.76D14 ►in GL8(𝔽113)
| 30 | 105 | 23 | 39 | 0 | 0 | 0 | 0 |
| 9 | 31 | 55 | 4 | 0 | 0 | 0 | 0 |
| 23 | 39 | 83 | 8 | 0 | 0 | 0 | 0 |
| 55 | 4 | 104 | 82 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 91 | 12 | 44 | 89 |
| 0 | 0 | 0 | 0 | 101 | 22 | 24 | 69 |
| 0 | 0 | 0 | 0 | 91 | 12 | 22 | 101 |
| 0 | 0 | 0 | 0 | 101 | 22 | 12 | 91 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 111 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 111 |
| 0 | 0 | 0 | 0 | 1 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 112 |
| 0 | 0 | 10 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 112 | 0 | 0 | 0 | 0 |
| 103 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
| 99 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 74 | 80 | 107 |
| 0 | 0 | 0 | 0 | 39 | 18 | 6 | 58 |
| 0 | 0 | 0 | 0 | 88 | 71 | 65 | 39 |
| 0 | 0 | 0 | 0 | 42 | 47 | 74 | 95 |
| 108 | 76 | 11 | 81 | 0 | 0 | 0 | 0 |
| 72 | 5 | 5 | 102 | 0 | 0 | 0 | 0 |
| 102 | 32 | 108 | 76 | 0 | 0 | 0 | 0 |
| 108 | 11 | 72 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 48 | 86 | 25 |
| 0 | 0 | 0 | 0 | 102 | 88 | 39 | 27 |
| 0 | 0 | 0 | 0 | 68 | 4 | 88 | 65 |
| 0 | 0 | 0 | 0 | 65 | 45 | 11 | 25 |
G:=sub<GL(8,GF(113))| [30,9,23,55,0,0,0,0,105,31,39,4,0,0,0,0,23,55,83,104,0,0,0,0,39,4,8,82,0,0,0,0,0,0,0,0,91,101,91,101,0,0,0,0,12,22,12,22,0,0,0,0,44,24,22,12,0,0,0,0,89,69,101,91],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,111,0,112,0,0,0,0,0,0,111,0,112],[0,0,103,99,0,0,0,0,0,0,25,1,0,0,0,0,10,14,0,0,0,0,0,0,88,112,0,0,0,0,0,0,0,0,0,0,48,39,88,42,0,0,0,0,74,18,71,47,0,0,0,0,80,6,65,74,0,0,0,0,107,58,39,95],[108,72,102,108,0,0,0,0,76,5,32,11,0,0,0,0,11,5,108,72,0,0,0,0,81,102,76,5,0,0,0,0,0,0,0,0,25,102,68,65,0,0,0,0,48,88,4,45,0,0,0,0,86,39,88,11,0,0,0,0,25,27,65,25] >;
C42.76D14 in GAP, Magma, Sage, TeX
C_4^2._{76}D_{14} % in TeX
G:=Group("C4^2.76D14"); // GroupNames label
G:=SmallGroup(448,614);
// by ID
G=gap.SmallGroup(448,614);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,477,64,422,135,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^13>;
// generators/relations