metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.68D14, C7⋊C8⋊3Q8, C7⋊4(C8⋊Q8), C4.33(Q8×D7), C4⋊C4.74D14, (C2×C28).84D4, C28.33(C2×Q8), C14.29(C4⋊Q8), C42.C2.4D7, C28⋊2Q8.16C2, (C4×C28).113C22, (C2×C28).383C23, C28.Q8.14C2, C4.Dic14.15C2, C2.20(D4⋊D14), C42.D7.5C2, C14.121(C8⋊C22), C2.9(Dic7⋊Q8), C4⋊Dic7.153C22, C2.21(D4.9D14), C14.122(C8.C22), (C2×C14).514(C2×D4), (C2×C4).65(C7⋊D4), (C2×C7⋊C8).125C22, (C7×C42.C2).3C2, (C7×C4⋊C4).121C22, (C2×C4).481(C22×D7), C22.187(C2×C7⋊D4), SmallGroup(448,599)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.68D14
G = < a,b,c,d | a4=b4=1, c14=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc13 >
Subgroups: 380 in 90 conjugacy classes, 43 normal (27 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, Dic14, C2×Dic7, C2×C28, C2×C28, C8⋊Q8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C42.D7, C28.Q8, C4.Dic14, C28⋊2Q8, C7×C42.C2, C42.68D14
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, Dic7⋊Q8, D4⋊D14, D4.9D14, C42.68D14
►Regular action on 448 points
Generators in S448
(1 126 15 140)(2 206 16 220)(3 128 17 114)(4 208 18 222)(5 130 19 116)(6 210 20 224)(7 132 21 118)(8 212 22 198)(9 134 23 120)(10 214 24 200)(11 136 25 122)(12 216 26 202)(13 138 27 124)(14 218 28 204)(29 96 43 110)(30 274 44 260)(31 98 45 112)(32 276 46 262)(33 100 47 86)(34 278 48 264)(35 102 49 88)(36 280 50 266)(37 104 51 90)(38 254 52 268)(39 106 53 92)(40 256 54 270)(41 108 55 94)(42 258 56 272)(57 187 71 173)(58 430 72 444)(59 189 73 175)(60 432 74 446)(61 191 75 177)(62 434 76 448)(63 193 77 179)(64 436 78 422)(65 195 79 181)(66 438 80 424)(67 169 81 183)(68 440 82 426)(69 171 83 185)(70 442 84 428)(85 336 99 322)(87 310 101 324)(89 312 103 326)(91 314 105 328)(93 316 107 330)(95 318 109 332)(97 320 111 334)(113 230 127 244)(115 232 129 246)(117 234 131 248)(119 236 133 250)(121 238 135 252)(123 240 137 226)(125 242 139 228)(141 299 155 285)(142 415 156 401)(143 301 157 287)(144 417 158 403)(145 303 159 289)(146 419 160 405)(147 305 161 291)(148 393 162 407)(149 307 163 293)(150 395 164 409)(151 281 165 295)(152 397 166 411)(153 283 167 297)(154 399 168 413)(170 340 184 354)(172 342 186 356)(174 344 188 358)(176 346 190 360)(178 348 192 362)(180 350 194 364)(182 352 196 338)(197 235 211 249)(199 237 213 251)(201 239 215 225)(203 241 217 227)(205 243 219 229)(207 245 221 231)(209 247 223 233)(253 327 267 313)(255 329 269 315)(257 331 271 317)(259 333 273 319)(261 335 275 321)(263 309 277 323)(265 311 279 325)(282 366 296 380)(284 368 298 382)(286 370 300 384)(288 372 302 386)(290 374 304 388)(292 376 306 390)(294 378 308 392)(337 423 351 437)(339 425 353 439)(341 427 355 441)(343 429 357 443)(345 431 359 445)(347 433 361 447)(349 435 363 421)(365 410 379 396)(367 412 381 398)(369 414 383 400)(371 416 385 402)(373 418 387 404)(375 420 389 406)(377 394 391 408)
(1 393 243 292)(2 293 244 394)(3 395 245 294)(4 295 246 396)(5 397 247 296)(6 297 248 398)(7 399 249 298)(8 299 250 400)(9 401 251 300)(10 301 252 402)(11 403 225 302)(12 303 226 404)(13 405 227 304)(14 305 228 406)(15 407 229 306)(16 307 230 408)(17 409 231 308)(18 281 232 410)(19 411 233 282)(20 283 234 412)(21 413 235 284)(22 285 236 414)(23 415 237 286)(24 287 238 416)(25 417 239 288)(26 289 240 418)(27 419 241 290)(28 291 242 420)(29 58 319 358)(30 359 320 59)(31 60 321 360)(32 361 322 61)(33 62 323 362)(34 363 324 63)(35 64 325 364)(36 337 326 65)(37 66 327 338)(38 339 328 67)(39 68 329 340)(40 341 330 69)(41 70 331 342)(42 343 332 71)(43 72 333 344)(44 345 334 73)(45 74 335 346)(46 347 336 75)(47 76 309 348)(48 349 310 77)(49 78 311 350)(50 351 312 79)(51 80 313 352)(52 353 314 81)(53 82 315 354)(54 355 316 83)(55 84 317 356)(56 357 318 57)(85 191 276 447)(86 448 277 192)(87 193 278 421)(88 422 279 194)(89 195 280 423)(90 424 253 196)(91 169 254 425)(92 426 255 170)(93 171 256 427)(94 428 257 172)(95 173 258 429)(96 430 259 174)(97 175 260 431)(98 432 261 176)(99 177 262 433)(100 434 263 178)(101 179 264 435)(102 436 265 180)(103 181 266 437)(104 438 267 182)(105 183 268 439)(106 440 269 184)(107 185 270 441)(108 442 271 186)(109 187 272 443)(110 444 273 188)(111 189 274 445)(112 446 275 190)(113 391 206 149)(114 150 207 392)(115 365 208 151)(116 152 209 366)(117 367 210 153)(118 154 211 368)(119 369 212 155)(120 156 213 370)(121 371 214 157)(122 158 215 372)(123 373 216 159)(124 160 217 374)(125 375 218 161)(126 162 219 376)(127 377 220 163)(128 164 221 378)(129 379 222 165)(130 166 223 380)(131 381 224 167)(132 168 197 382)(133 383 198 141)(134 142 199 384)(135 385 200 143)(136 144 201 386)(137 387 202 145)(138 146 203 388)(139 389 204 147)(140 148 205 390)
(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)
(1 330 15 316)(2 82 16 68)(3 328 17 314)(4 80 18 66)(5 326 19 312)(6 78 20 64)(7 324 21 310)(8 76 22 62)(9 322 23 336)(10 74 24 60)(11 320 25 334)(12 72 26 58)(13 318 27 332)(14 70 28 84)(29 303 43 289)(30 239 44 225)(31 301 45 287)(32 237 46 251)(33 299 47 285)(34 235 48 249)(35 297 49 283)(36 233 50 247)(37 295 51 281)(38 231 52 245)(39 293 53 307)(40 229 54 243)(41 291 55 305)(42 227 56 241)(57 290 71 304)(59 288 73 302)(61 286 75 300)(63 284 77 298)(65 282 79 296)(67 308 81 294)(69 306 83 292)(85 134 99 120)(86 155 100 141)(87 132 101 118)(88 153 102 167)(89 130 103 116)(90 151 104 165)(91 128 105 114)(92 149 106 163)(93 126 107 140)(94 147 108 161)(95 124 109 138)(96 145 110 159)(97 122 111 136)(98 143 112 157)(113 184 127 170)(115 182 129 196)(117 180 131 194)(119 178 133 192)(121 176 135 190)(123 174 137 188)(125 172 139 186)(142 433 156 447)(144 431 158 445)(146 429 160 443)(148 427 162 441)(150 425 164 439)(152 423 166 437)(154 421 168 435)(169 378 183 392)(171 376 185 390)(173 374 187 388)(175 372 189 386)(177 370 191 384)(179 368 193 382)(181 366 195 380)(197 264 211 278)(198 448 212 434)(199 262 213 276)(200 446 214 432)(201 260 215 274)(202 444 216 430)(203 258 217 272)(204 442 218 428)(205 256 219 270)(206 440 220 426)(207 254 221 268)(208 438 222 424)(209 280 223 266)(210 436 224 422)(226 344 240 358)(228 342 242 356)(230 340 244 354)(232 338 246 352)(234 364 248 350)(236 362 250 348)(238 360 252 346)(253 365 267 379)(255 391 269 377)(257 389 271 375)(259 387 273 373)(261 385 275 371)(263 383 277 369)(265 381 279 367)(309 414 323 400)(311 412 325 398)(313 410 327 396)(315 408 329 394)(317 406 331 420)(319 404 333 418)(321 402 335 416)(337 411 351 397)(339 409 353 395)(341 407 355 393)(343 405 357 419)(345 403 359 417)(347 401 361 415)(349 399 363 413)
G:=sub<Sym(448)| (1,126,15,140)(2,206,16,220)(3,128,17,114)(4,208,18,222)(5,130,19,116)(6,210,20,224)(7,132,21,118)(8,212,22,198)(9,134,23,120)(10,214,24,200)(11,136,25,122)(12,216,26,202)(13,138,27,124)(14,218,28,204)(29,96,43,110)(30,274,44,260)(31,98,45,112)(32,276,46,262)(33,100,47,86)(34,278,48,264)(35,102,49,88)(36,280,50,266)(37,104,51,90)(38,254,52,268)(39,106,53,92)(40,256,54,270)(41,108,55,94)(42,258,56,272)(57,187,71,173)(58,430,72,444)(59,189,73,175)(60,432,74,446)(61,191,75,177)(62,434,76,448)(63,193,77,179)(64,436,78,422)(65,195,79,181)(66,438,80,424)(67,169,81,183)(68,440,82,426)(69,171,83,185)(70,442,84,428)(85,336,99,322)(87,310,101,324)(89,312,103,326)(91,314,105,328)(93,316,107,330)(95,318,109,332)(97,320,111,334)(113,230,127,244)(115,232,129,246)(117,234,131,248)(119,236,133,250)(121,238,135,252)(123,240,137,226)(125,242,139,228)(141,299,155,285)(142,415,156,401)(143,301,157,287)(144,417,158,403)(145,303,159,289)(146,419,160,405)(147,305,161,291)(148,393,162,407)(149,307,163,293)(150,395,164,409)(151,281,165,295)(152,397,166,411)(153,283,167,297)(154,399,168,413)(170,340,184,354)(172,342,186,356)(174,344,188,358)(176,346,190,360)(178,348,192,362)(180,350,194,364)(182,352,196,338)(197,235,211,249)(199,237,213,251)(201,239,215,225)(203,241,217,227)(205,243,219,229)(207,245,221,231)(209,247,223,233)(253,327,267,313)(255,329,269,315)(257,331,271,317)(259,333,273,319)(261,335,275,321)(263,309,277,323)(265,311,279,325)(282,366,296,380)(284,368,298,382)(286,370,300,384)(288,372,302,386)(290,374,304,388)(292,376,306,390)(294,378,308,392)(337,423,351,437)(339,425,353,439)(341,427,355,441)(343,429,357,443)(345,431,359,445)(347,433,361,447)(349,435,363,421)(365,410,379,396)(367,412,381,398)(369,414,383,400)(371,416,385,402)(373,418,387,404)(375,420,389,406)(377,394,391,408), (1,393,243,292)(2,293,244,394)(3,395,245,294)(4,295,246,396)(5,397,247,296)(6,297,248,398)(7,399,249,298)(8,299,250,400)(9,401,251,300)(10,301,252,402)(11,403,225,302)(12,303,226,404)(13,405,227,304)(14,305,228,406)(15,407,229,306)(16,307,230,408)(17,409,231,308)(18,281,232,410)(19,411,233,282)(20,283,234,412)(21,413,235,284)(22,285,236,414)(23,415,237,286)(24,287,238,416)(25,417,239,288)(26,289,240,418)(27,419,241,290)(28,291,242,420)(29,58,319,358)(30,359,320,59)(31,60,321,360)(32,361,322,61)(33,62,323,362)(34,363,324,63)(35,64,325,364)(36,337,326,65)(37,66,327,338)(38,339,328,67)(39,68,329,340)(40,341,330,69)(41,70,331,342)(42,343,332,71)(43,72,333,344)(44,345,334,73)(45,74,335,346)(46,347,336,75)(47,76,309,348)(48,349,310,77)(49,78,311,350)(50,351,312,79)(51,80,313,352)(52,353,314,81)(53,82,315,354)(54,355,316,83)(55,84,317,356)(56,357,318,57)(85,191,276,447)(86,448,277,192)(87,193,278,421)(88,422,279,194)(89,195,280,423)(90,424,253,196)(91,169,254,425)(92,426,255,170)(93,171,256,427)(94,428,257,172)(95,173,258,429)(96,430,259,174)(97,175,260,431)(98,432,261,176)(99,177,262,433)(100,434,263,178)(101,179,264,435)(102,436,265,180)(103,181,266,437)(104,438,267,182)(105,183,268,439)(106,440,269,184)(107,185,270,441)(108,442,271,186)(109,187,272,443)(110,444,273,188)(111,189,274,445)(112,446,275,190)(113,391,206,149)(114,150,207,392)(115,365,208,151)(116,152,209,366)(117,367,210,153)(118,154,211,368)(119,369,212,155)(120,156,213,370)(121,371,214,157)(122,158,215,372)(123,373,216,159)(124,160,217,374)(125,375,218,161)(126,162,219,376)(127,377,220,163)(128,164,221,378)(129,379,222,165)(130,166,223,380)(131,381,224,167)(132,168,197,382)(133,383,198,141)(134,142,199,384)(135,385,200,143)(136,144,201,386)(137,387,202,145)(138,146,203,388)(139,389,204,147)(140,148,205,390), (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), (1,330,15,316)(2,82,16,68)(3,328,17,314)(4,80,18,66)(5,326,19,312)(6,78,20,64)(7,324,21,310)(8,76,22,62)(9,322,23,336)(10,74,24,60)(11,320,25,334)(12,72,26,58)(13,318,27,332)(14,70,28,84)(29,303,43,289)(30,239,44,225)(31,301,45,287)(32,237,46,251)(33,299,47,285)(34,235,48,249)(35,297,49,283)(36,233,50,247)(37,295,51,281)(38,231,52,245)(39,293,53,307)(40,229,54,243)(41,291,55,305)(42,227,56,241)(57,290,71,304)(59,288,73,302)(61,286,75,300)(63,284,77,298)(65,282,79,296)(67,308,81,294)(69,306,83,292)(85,134,99,120)(86,155,100,141)(87,132,101,118)(88,153,102,167)(89,130,103,116)(90,151,104,165)(91,128,105,114)(92,149,106,163)(93,126,107,140)(94,147,108,161)(95,124,109,138)(96,145,110,159)(97,122,111,136)(98,143,112,157)(113,184,127,170)(115,182,129,196)(117,180,131,194)(119,178,133,192)(121,176,135,190)(123,174,137,188)(125,172,139,186)(142,433,156,447)(144,431,158,445)(146,429,160,443)(148,427,162,441)(150,425,164,439)(152,423,166,437)(154,421,168,435)(169,378,183,392)(171,376,185,390)(173,374,187,388)(175,372,189,386)(177,370,191,384)(179,368,193,382)(181,366,195,380)(197,264,211,278)(198,448,212,434)(199,262,213,276)(200,446,214,432)(201,260,215,274)(202,444,216,430)(203,258,217,272)(204,442,218,428)(205,256,219,270)(206,440,220,426)(207,254,221,268)(208,438,222,424)(209,280,223,266)(210,436,224,422)(226,344,240,358)(228,342,242,356)(230,340,244,354)(232,338,246,352)(234,364,248,350)(236,362,250,348)(238,360,252,346)(253,365,267,379)(255,391,269,377)(257,389,271,375)(259,387,273,373)(261,385,275,371)(263,383,277,369)(265,381,279,367)(309,414,323,400)(311,412,325,398)(313,410,327,396)(315,408,329,394)(317,406,331,420)(319,404,333,418)(321,402,335,416)(337,411,351,397)(339,409,353,395)(341,407,355,393)(343,405,357,419)(345,403,359,417)(347,401,361,415)(349,399,363,413)>;
G:=Group( (1,126,15,140)(2,206,16,220)(3,128,17,114)(4,208,18,222)(5,130,19,116)(6,210,20,224)(7,132,21,118)(8,212,22,198)(9,134,23,120)(10,214,24,200)(11,136,25,122)(12,216,26,202)(13,138,27,124)(14,218,28,204)(29,96,43,110)(30,274,44,260)(31,98,45,112)(32,276,46,262)(33,100,47,86)(34,278,48,264)(35,102,49,88)(36,280,50,266)(37,104,51,90)(38,254,52,268)(39,106,53,92)(40,256,54,270)(41,108,55,94)(42,258,56,272)(57,187,71,173)(58,430,72,444)(59,189,73,175)(60,432,74,446)(61,191,75,177)(62,434,76,448)(63,193,77,179)(64,436,78,422)(65,195,79,181)(66,438,80,424)(67,169,81,183)(68,440,82,426)(69,171,83,185)(70,442,84,428)(85,336,99,322)(87,310,101,324)(89,312,103,326)(91,314,105,328)(93,316,107,330)(95,318,109,332)(97,320,111,334)(113,230,127,244)(115,232,129,246)(117,234,131,248)(119,236,133,250)(121,238,135,252)(123,240,137,226)(125,242,139,228)(141,299,155,285)(142,415,156,401)(143,301,157,287)(144,417,158,403)(145,303,159,289)(146,419,160,405)(147,305,161,291)(148,393,162,407)(149,307,163,293)(150,395,164,409)(151,281,165,295)(152,397,166,411)(153,283,167,297)(154,399,168,413)(170,340,184,354)(172,342,186,356)(174,344,188,358)(176,346,190,360)(178,348,192,362)(180,350,194,364)(182,352,196,338)(197,235,211,249)(199,237,213,251)(201,239,215,225)(203,241,217,227)(205,243,219,229)(207,245,221,231)(209,247,223,233)(253,327,267,313)(255,329,269,315)(257,331,271,317)(259,333,273,319)(261,335,275,321)(263,309,277,323)(265,311,279,325)(282,366,296,380)(284,368,298,382)(286,370,300,384)(288,372,302,386)(290,374,304,388)(292,376,306,390)(294,378,308,392)(337,423,351,437)(339,425,353,439)(341,427,355,441)(343,429,357,443)(345,431,359,445)(347,433,361,447)(349,435,363,421)(365,410,379,396)(367,412,381,398)(369,414,383,400)(371,416,385,402)(373,418,387,404)(375,420,389,406)(377,394,391,408), (1,393,243,292)(2,293,244,394)(3,395,245,294)(4,295,246,396)(5,397,247,296)(6,297,248,398)(7,399,249,298)(8,299,250,400)(9,401,251,300)(10,301,252,402)(11,403,225,302)(12,303,226,404)(13,405,227,304)(14,305,228,406)(15,407,229,306)(16,307,230,408)(17,409,231,308)(18,281,232,410)(19,411,233,282)(20,283,234,412)(21,413,235,284)(22,285,236,414)(23,415,237,286)(24,287,238,416)(25,417,239,288)(26,289,240,418)(27,419,241,290)(28,291,242,420)(29,58,319,358)(30,359,320,59)(31,60,321,360)(32,361,322,61)(33,62,323,362)(34,363,324,63)(35,64,325,364)(36,337,326,65)(37,66,327,338)(38,339,328,67)(39,68,329,340)(40,341,330,69)(41,70,331,342)(42,343,332,71)(43,72,333,344)(44,345,334,73)(45,74,335,346)(46,347,336,75)(47,76,309,348)(48,349,310,77)(49,78,311,350)(50,351,312,79)(51,80,313,352)(52,353,314,81)(53,82,315,354)(54,355,316,83)(55,84,317,356)(56,357,318,57)(85,191,276,447)(86,448,277,192)(87,193,278,421)(88,422,279,194)(89,195,280,423)(90,424,253,196)(91,169,254,425)(92,426,255,170)(93,171,256,427)(94,428,257,172)(95,173,258,429)(96,430,259,174)(97,175,260,431)(98,432,261,176)(99,177,262,433)(100,434,263,178)(101,179,264,435)(102,436,265,180)(103,181,266,437)(104,438,267,182)(105,183,268,439)(106,440,269,184)(107,185,270,441)(108,442,271,186)(109,187,272,443)(110,444,273,188)(111,189,274,445)(112,446,275,190)(113,391,206,149)(114,150,207,392)(115,365,208,151)(116,152,209,366)(117,367,210,153)(118,154,211,368)(119,369,212,155)(120,156,213,370)(121,371,214,157)(122,158,215,372)(123,373,216,159)(124,160,217,374)(125,375,218,161)(126,162,219,376)(127,377,220,163)(128,164,221,378)(129,379,222,165)(130,166,223,380)(131,381,224,167)(132,168,197,382)(133,383,198,141)(134,142,199,384)(135,385,200,143)(136,144,201,386)(137,387,202,145)(138,146,203,388)(139,389,204,147)(140,148,205,390), (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), (1,330,15,316)(2,82,16,68)(3,328,17,314)(4,80,18,66)(5,326,19,312)(6,78,20,64)(7,324,21,310)(8,76,22,62)(9,322,23,336)(10,74,24,60)(11,320,25,334)(12,72,26,58)(13,318,27,332)(14,70,28,84)(29,303,43,289)(30,239,44,225)(31,301,45,287)(32,237,46,251)(33,299,47,285)(34,235,48,249)(35,297,49,283)(36,233,50,247)(37,295,51,281)(38,231,52,245)(39,293,53,307)(40,229,54,243)(41,291,55,305)(42,227,56,241)(57,290,71,304)(59,288,73,302)(61,286,75,300)(63,284,77,298)(65,282,79,296)(67,308,81,294)(69,306,83,292)(85,134,99,120)(86,155,100,141)(87,132,101,118)(88,153,102,167)(89,130,103,116)(90,151,104,165)(91,128,105,114)(92,149,106,163)(93,126,107,140)(94,147,108,161)(95,124,109,138)(96,145,110,159)(97,122,111,136)(98,143,112,157)(113,184,127,170)(115,182,129,196)(117,180,131,194)(119,178,133,192)(121,176,135,190)(123,174,137,188)(125,172,139,186)(142,433,156,447)(144,431,158,445)(146,429,160,443)(148,427,162,441)(150,425,164,439)(152,423,166,437)(154,421,168,435)(169,378,183,392)(171,376,185,390)(173,374,187,388)(175,372,189,386)(177,370,191,384)(179,368,193,382)(181,366,195,380)(197,264,211,278)(198,448,212,434)(199,262,213,276)(200,446,214,432)(201,260,215,274)(202,444,216,430)(203,258,217,272)(204,442,218,428)(205,256,219,270)(206,440,220,426)(207,254,221,268)(208,438,222,424)(209,280,223,266)(210,436,224,422)(226,344,240,358)(228,342,242,356)(230,340,244,354)(232,338,246,352)(234,364,248,350)(236,362,250,348)(238,360,252,346)(253,365,267,379)(255,391,269,377)(257,389,271,375)(259,387,273,373)(261,385,275,371)(263,383,277,369)(265,381,279,367)(309,414,323,400)(311,412,325,398)(313,410,327,396)(315,408,329,394)(317,406,331,420)(319,404,333,418)(321,402,335,416)(337,411,351,397)(339,409,353,395)(341,407,355,393)(343,405,357,419)(345,403,359,417)(347,401,361,415)(349,399,363,413) );
G=PermutationGroup([[(1,126,15,140),(2,206,16,220),(3,128,17,114),(4,208,18,222),(5,130,19,116),(6,210,20,224),(7,132,21,118),(8,212,22,198),(9,134,23,120),(10,214,24,200),(11,136,25,122),(12,216,26,202),(13,138,27,124),(14,218,28,204),(29,96,43,110),(30,274,44,260),(31,98,45,112),(32,276,46,262),(33,100,47,86),(34,278,48,264),(35,102,49,88),(36,280,50,266),(37,104,51,90),(38,254,52,268),(39,106,53,92),(40,256,54,270),(41,108,55,94),(42,258,56,272),(57,187,71,173),(58,430,72,444),(59,189,73,175),(60,432,74,446),(61,191,75,177),(62,434,76,448),(63,193,77,179),(64,436,78,422),(65,195,79,181),(66,438,80,424),(67,169,81,183),(68,440,82,426),(69,171,83,185),(70,442,84,428),(85,336,99,322),(87,310,101,324),(89,312,103,326),(91,314,105,328),(93,316,107,330),(95,318,109,332),(97,320,111,334),(113,230,127,244),(115,232,129,246),(117,234,131,248),(119,236,133,250),(121,238,135,252),(123,240,137,226),(125,242,139,228),(141,299,155,285),(142,415,156,401),(143,301,157,287),(144,417,158,403),(145,303,159,289),(146,419,160,405),(147,305,161,291),(148,393,162,407),(149,307,163,293),(150,395,164,409),(151,281,165,295),(152,397,166,411),(153,283,167,297),(154,399,168,413),(170,340,184,354),(172,342,186,356),(174,344,188,358),(176,346,190,360),(178,348,192,362),(180,350,194,364),(182,352,196,338),(197,235,211,249),(199,237,213,251),(201,239,215,225),(203,241,217,227),(205,243,219,229),(207,245,221,231),(209,247,223,233),(253,327,267,313),(255,329,269,315),(257,331,271,317),(259,333,273,319),(261,335,275,321),(263,309,277,323),(265,311,279,325),(282,366,296,380),(284,368,298,382),(286,370,300,384),(288,372,302,386),(290,374,304,388),(292,376,306,390),(294,378,308,392),(337,423,351,437),(339,425,353,439),(341,427,355,441),(343,429,357,443),(345,431,359,445),(347,433,361,447),(349,435,363,421),(365,410,379,396),(367,412,381,398),(369,414,383,400),(371,416,385,402),(373,418,387,404),(375,420,389,406),(377,394,391,408)], [(1,393,243,292),(2,293,244,394),(3,395,245,294),(4,295,246,396),(5,397,247,296),(6,297,248,398),(7,399,249,298),(8,299,250,400),(9,401,251,300),(10,301,252,402),(11,403,225,302),(12,303,226,404),(13,405,227,304),(14,305,228,406),(15,407,229,306),(16,307,230,408),(17,409,231,308),(18,281,232,410),(19,411,233,282),(20,283,234,412),(21,413,235,284),(22,285,236,414),(23,415,237,286),(24,287,238,416),(25,417,239,288),(26,289,240,418),(27,419,241,290),(28,291,242,420),(29,58,319,358),(30,359,320,59),(31,60,321,360),(32,361,322,61),(33,62,323,362),(34,363,324,63),(35,64,325,364),(36,337,326,65),(37,66,327,338),(38,339,328,67),(39,68,329,340),(40,341,330,69),(41,70,331,342),(42,343,332,71),(43,72,333,344),(44,345,334,73),(45,74,335,346),(46,347,336,75),(47,76,309,348),(48,349,310,77),(49,78,311,350),(50,351,312,79),(51,80,313,352),(52,353,314,81),(53,82,315,354),(54,355,316,83),(55,84,317,356),(56,357,318,57),(85,191,276,447),(86,448,277,192),(87,193,278,421),(88,422,279,194),(89,195,280,423),(90,424,253,196),(91,169,254,425),(92,426,255,170),(93,171,256,427),(94,428,257,172),(95,173,258,429),(96,430,259,174),(97,175,260,431),(98,432,261,176),(99,177,262,433),(100,434,263,178),(101,179,264,435),(102,436,265,180),(103,181,266,437),(104,438,267,182),(105,183,268,439),(106,440,269,184),(107,185,270,441),(108,442,271,186),(109,187,272,443),(110,444,273,188),(111,189,274,445),(112,446,275,190),(113,391,206,149),(114,150,207,392),(115,365,208,151),(116,152,209,366),(117,367,210,153),(118,154,211,368),(119,369,212,155),(120,156,213,370),(121,371,214,157),(122,158,215,372),(123,373,216,159),(124,160,217,374),(125,375,218,161),(126,162,219,376),(127,377,220,163),(128,164,221,378),(129,379,222,165),(130,166,223,380),(131,381,224,167),(132,168,197,382),(133,383,198,141),(134,142,199,384),(135,385,200,143),(136,144,201,386),(137,387,202,145),(138,146,203,388),(139,389,204,147),(140,148,205,390)], [(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)], [(1,330,15,316),(2,82,16,68),(3,328,17,314),(4,80,18,66),(5,326,19,312),(6,78,20,64),(7,324,21,310),(8,76,22,62),(9,322,23,336),(10,74,24,60),(11,320,25,334),(12,72,26,58),(13,318,27,332),(14,70,28,84),(29,303,43,289),(30,239,44,225),(31,301,45,287),(32,237,46,251),(33,299,47,285),(34,235,48,249),(35,297,49,283),(36,233,50,247),(37,295,51,281),(38,231,52,245),(39,293,53,307),(40,229,54,243),(41,291,55,305),(42,227,56,241),(57,290,71,304),(59,288,73,302),(61,286,75,300),(63,284,77,298),(65,282,79,296),(67,308,81,294),(69,306,83,292),(85,134,99,120),(86,155,100,141),(87,132,101,118),(88,153,102,167),(89,130,103,116),(90,151,104,165),(91,128,105,114),(92,149,106,163),(93,126,107,140),(94,147,108,161),(95,124,109,138),(96,145,110,159),(97,122,111,136),(98,143,112,157),(113,184,127,170),(115,182,129,196),(117,180,131,194),(119,178,133,192),(121,176,135,190),(123,174,137,188),(125,172,139,186),(142,433,156,447),(144,431,158,445),(146,429,160,443),(148,427,162,441),(150,425,164,439),(152,423,166,437),(154,421,168,435),(169,378,183,392),(171,376,185,390),(173,374,187,388),(175,372,189,386),(177,370,191,384),(179,368,193,382),(181,366,195,380),(197,264,211,278),(198,448,212,434),(199,262,213,276),(200,446,214,432),(201,260,215,274),(202,444,216,430),(203,258,217,272),(204,442,218,428),(205,256,219,270),(206,440,220,426),(207,254,221,268),(208,438,222,424),(209,280,223,266),(210,436,224,422),(226,344,240,358),(228,342,242,356),(230,340,244,354),(232,338,246,352),(234,364,248,350),(236,362,250,348),(238,360,252,346),(253,365,267,379),(255,391,269,377),(257,389,271,375),(259,387,273,373),(261,385,275,371),(263,383,277,369),(265,381,279,367),(309,414,323,400),(311,412,325,398),(313,410,327,396),(315,408,329,394),(317,406,331,420),(319,404,333,418),(321,402,335,416),(337,411,351,397),(339,409,353,395),(341,407,355,393),(343,405,357,419),(345,403,359,417),(347,401,361,415),(349,399,363,413)]])
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 | D4.9D14 |
| kernel | C42.68D14 | C42.D7 | C28.Q8 | C4.Dic14 | C28⋊2Q8 | C7×C42.C2 | C7⋊C8 | C2×C28 | C42.C2 | 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 C42.68D14 ►in GL6(𝔽113)
| 47 | 29 | 0 | 0 | 0 | 0 |
| 29 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 75 | 0 | 0 |
| 0 | 0 | 38 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 75 |
| 0 | 0 | 0 | 0 | 38 | 55 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 111 | 0 |
| 0 | 0 | 0 | 112 | 0 | 111 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 80 | 104 | 50 | 94 |
| 0 | 0 | 9 | 112 | 19 | 105 |
| 0 | 0 | 58 | 56 | 33 | 9 |
| 0 | 0 | 57 | 110 | 104 | 1 |
| 29 | 66 | 0 | 0 | 0 | 0 |
| 66 | 84 | 0 | 0 | 0 | 0 |
| 0 | 0 | 81 | 21 | 68 | 26 |
| 0 | 0 | 72 | 32 | 73 | 45 |
| 0 | 0 | 66 | 105 | 32 | 92 |
| 0 | 0 | 21 | 47 | 41 | 81 |
G:=sub<GL(6,GF(113))| [47,29,0,0,0,0,29,66,0,0,0,0,0,0,58,38,0,0,0,0,75,55,0,0,0,0,0,0,58,38,0,0,0,0,75,55],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,1,0,0,0,0,112,0,1,0,0,111,0,1,0,0,0,0,111,0,1],[0,1,0,0,0,0,112,0,0,0,0,0,0,0,80,9,58,57,0,0,104,112,56,110,0,0,50,19,33,104,0,0,94,105,9,1],[29,66,0,0,0,0,66,84,0,0,0,0,0,0,81,72,66,21,0,0,21,32,105,47,0,0,68,73,32,41,0,0,26,45,92,81] >;
C42.68D14 in GAP, Magma, Sage, TeX
C_4^2._{68}D_{14} % in TeX
G:=Group("C4^2.68D14"); // GroupNames label
G:=SmallGroup(448,599);
// by ID
G=gap.SmallGroup(448,599);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,477,64,422,471,58,438,102,18822]);
// 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*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^13>;
// generators/relations