metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.77D14, C4⋊Q8.8D7, (C2×C28).294D4, (C2×Q8).43D14, C28.79(C4○D4), C28.6Q8.9C2, C4.25(D4⋊2D7), (C4×C28).129C22, (C2×C28).400C23, Q8⋊Dic7.12C2, (Q8×C14).61C22, C42.D7.8C2, C14.46(C4.4D4), C14.94(C8.C22), C4⋊Dic7.160C22, C2.13(C28.17D4), C2.15(C28.C23), C7⋊4(C42.30C22), (C7×C4⋊Q8).8C2, (C2×C14).531(C2×D4), (C2×C4).72(C7⋊D4), (C2×C7⋊C8).134C22, (C2×C4).497(C22×D7), C22.203(C2×C7⋊D4), SmallGroup(448,616)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.77D14
G = < a,b,c,d | a4=b4=1, c14=b2, d2=a2b2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc13 >
Subgroups: 332 in 90 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×C8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×Q8, C42.30C22, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, Q8×C14, C42.D7, Q8⋊Dic7, C28.6Q8, C7×C4⋊Q8, C42.77D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C8.C22, C7⋊D4, C22×D7, C42.30C22, D4⋊2D7, C2×C7⋊D4, C28.17D4, C28.C23, C42.77D14
►Regular action on 448 points
Generators in S448
(1 346 155 115)(2 116 156 347)(3 348 157 117)(4 118 158 349)(5 350 159 119)(6 120 160 351)(7 352 161 121)(8 122 162 353)(9 354 163 123)(10 124 164 355)(11 356 165 125)(12 126 166 357)(13 358 167 127)(14 128 168 359)(15 360 141 129)(16 130 142 361)(17 362 143 131)(18 132 144 363)(19 364 145 133)(20 134 146 337)(21 338 147 135)(22 136 148 339)(23 340 149 137)(24 138 150 341)(25 342 151 139)(26 140 152 343)(27 344 153 113)(28 114 154 345)(29 287 433 238)(30 239 434 288)(31 289 435 240)(32 241 436 290)(33 291 437 242)(34 243 438 292)(35 293 439 244)(36 245 440 294)(37 295 441 246)(38 247 442 296)(39 297 443 248)(40 249 444 298)(41 299 445 250)(42 251 446 300)(43 301 447 252)(44 225 448 302)(45 303 421 226)(46 227 422 304)(47 305 423 228)(48 229 424 306)(49 307 425 230)(50 231 426 308)(51 281 427 232)(52 233 428 282)(53 283 429 234)(54 235 430 284)(55 285 431 236)(56 237 432 286)(57 393 274 193)(58 194 275 394)(59 395 276 195)(60 196 277 396)(61 397 278 169)(62 170 279 398)(63 399 280 171)(64 172 253 400)(65 401 254 173)(66 174 255 402)(67 403 256 175)(68 176 257 404)(69 405 258 177)(70 178 259 406)(71 407 260 179)(72 180 261 408)(73 409 262 181)(74 182 263 410)(75 411 264 183)(76 184 265 412)(77 413 266 185)(78 186 267 414)(79 415 268 187)(80 188 269 416)(81 417 270 189)(82 190 271 418)(83 419 272 191)(84 192 273 420)(85 224 314 379)(86 380 315 197)(87 198 316 381)(88 382 317 199)(89 200 318 383)(90 384 319 201)(91 202 320 385)(92 386 321 203)(93 204 322 387)(94 388 323 205)(95 206 324 389)(96 390 325 207)(97 208 326 391)(98 392 327 209)(99 210 328 365)(100 366 329 211)(101 212 330 367)(102 368 331 213)(103 214 332 369)(104 370 333 215)(105 216 334 371)(106 372 335 217)(107 218 336 373)(108 374 309 219)(109 220 310 375)(110 376 311 221)(111 222 312 377)(112 378 313 223)
(1 413 15 399)(2 400 16 414)(3 415 17 401)(4 402 18 416)(5 417 19 403)(6 404 20 418)(7 419 21 405)(8 406 22 420)(9 393 23 407)(10 408 24 394)(11 395 25 409)(12 410 26 396)(13 397 27 411)(14 412 28 398)(29 326 43 312)(30 313 44 327)(31 328 45 314)(32 315 46 329)(33 330 47 316)(34 317 48 331)(35 332 49 318)(36 319 50 333)(37 334 51 320)(38 321 52 335)(39 336 53 322)(40 323 54 309)(41 310 55 324)(42 325 56 311)(57 137 71 123)(58 124 72 138)(59 139 73 125)(60 126 74 140)(61 113 75 127)(62 128 76 114)(63 115 77 129)(64 130 78 116)(65 117 79 131)(66 132 80 118)(67 119 81 133)(68 134 82 120)(69 121 83 135)(70 136 84 122)(85 435 99 421)(86 422 100 436)(87 437 101 423)(88 424 102 438)(89 439 103 425)(90 426 104 440)(91 441 105 427)(92 428 106 442)(93 443 107 429)(94 430 108 444)(95 445 109 431)(96 432 110 446)(97 447 111 433)(98 434 112 448)(141 171 155 185)(142 186 156 172)(143 173 157 187)(144 188 158 174)(145 175 159 189)(146 190 160 176)(147 177 161 191)(148 192 162 178)(149 179 163 193)(150 194 164 180)(151 181 165 195)(152 196 166 182)(153 183 167 169)(154 170 168 184)(197 227 211 241)(198 242 212 228)(199 229 213 243)(200 244 214 230)(201 231 215 245)(202 246 216 232)(203 233 217 247)(204 248 218 234)(205 235 219 249)(206 250 220 236)(207 237 221 251)(208 252 222 238)(209 239 223 225)(210 226 224 240)(253 361 267 347)(254 348 268 362)(255 363 269 349)(256 350 270 364)(257 337 271 351)(258 352 272 338)(259 339 273 353)(260 354 274 340)(261 341 275 355)(262 356 276 342)(263 343 277 357)(264 358 278 344)(265 345 279 359)(266 360 280 346)(281 385 295 371)(282 372 296 386)(283 387 297 373)(284 374 298 388)(285 389 299 375)(286 376 300 390)(287 391 301 377)(288 378 302 392)(289 365 303 379)(290 380 304 366)(291 367 305 381)(292 382 306 368)(293 369 307 383)(294 384 308 370)
(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)
(1 377 141 208)(2 251 142 286)(3 375 143 206)(4 249 144 284)(5 373 145 204)(6 247 146 282)(7 371 147 202)(8 245 148 308)(9 369 149 200)(10 243 150 306)(11 367 151 198)(12 241 152 304)(13 365 153 224)(14 239 154 302)(15 391 155 222)(16 237 156 300)(17 389 157 220)(18 235 158 298)(19 387 159 218)(20 233 160 296)(21 385 161 216)(22 231 162 294)(23 383 163 214)(24 229 164 292)(25 381 165 212)(26 227 166 290)(27 379 167 210)(28 225 168 288)(29 63 447 266)(30 359 448 114)(31 61 421 264)(32 357 422 140)(33 59 423 262)(34 355 424 138)(35 57 425 260)(36 353 426 136)(37 83 427 258)(38 351 428 134)(39 81 429 256)(40 349 430 132)(41 79 431 254)(42 347 432 130)(43 77 433 280)(44 345 434 128)(45 75 435 278)(46 343 436 126)(47 73 437 276)(48 341 438 124)(49 71 439 274)(50 339 440 122)(51 69 441 272)(52 337 442 120)(53 67 443 270)(54 363 444 118)(55 65 445 268)(56 361 446 116)(58 331 261 88)(60 329 263 86)(62 327 265 112)(64 325 267 110)(66 323 269 108)(68 321 271 106)(70 319 273 104)(72 317 275 102)(74 315 277 100)(76 313 279 98)(78 311 253 96)(80 309 255 94)(82 335 257 92)(84 333 259 90)(85 344 328 127)(87 342 330 125)(89 340 332 123)(91 338 334 121)(93 364 336 119)(95 362 310 117)(97 360 312 115)(99 358 314 113)(101 356 316 139)(103 354 318 137)(105 352 320 135)(107 350 322 133)(109 348 324 131)(111 346 326 129)(169 240 411 303)(170 378 412 209)(171 238 413 301)(172 376 414 207)(173 236 415 299)(174 374 416 205)(175 234 417 297)(176 372 418 203)(177 232 419 295)(178 370 420 201)(179 230 393 293)(180 368 394 199)(181 228 395 291)(182 366 396 197)(183 226 397 289)(184 392 398 223)(185 252 399 287)(186 390 400 221)(187 250 401 285)(188 388 402 219)(189 248 403 283)(190 386 404 217)(191 246 405 281)(192 384 406 215)(193 244 407 307)(194 382 408 213)(195 242 409 305)(196 380 410 211)
G:=sub<Sym(448)| (1,346,155,115)(2,116,156,347)(3,348,157,117)(4,118,158,349)(5,350,159,119)(6,120,160,351)(7,352,161,121)(8,122,162,353)(9,354,163,123)(10,124,164,355)(11,356,165,125)(12,126,166,357)(13,358,167,127)(14,128,168,359)(15,360,141,129)(16,130,142,361)(17,362,143,131)(18,132,144,363)(19,364,145,133)(20,134,146,337)(21,338,147,135)(22,136,148,339)(23,340,149,137)(24,138,150,341)(25,342,151,139)(26,140,152,343)(27,344,153,113)(28,114,154,345)(29,287,433,238)(30,239,434,288)(31,289,435,240)(32,241,436,290)(33,291,437,242)(34,243,438,292)(35,293,439,244)(36,245,440,294)(37,295,441,246)(38,247,442,296)(39,297,443,248)(40,249,444,298)(41,299,445,250)(42,251,446,300)(43,301,447,252)(44,225,448,302)(45,303,421,226)(46,227,422,304)(47,305,423,228)(48,229,424,306)(49,307,425,230)(50,231,426,308)(51,281,427,232)(52,233,428,282)(53,283,429,234)(54,235,430,284)(55,285,431,236)(56,237,432,286)(57,393,274,193)(58,194,275,394)(59,395,276,195)(60,196,277,396)(61,397,278,169)(62,170,279,398)(63,399,280,171)(64,172,253,400)(65,401,254,173)(66,174,255,402)(67,403,256,175)(68,176,257,404)(69,405,258,177)(70,178,259,406)(71,407,260,179)(72,180,261,408)(73,409,262,181)(74,182,263,410)(75,411,264,183)(76,184,265,412)(77,413,266,185)(78,186,267,414)(79,415,268,187)(80,188,269,416)(81,417,270,189)(82,190,271,418)(83,419,272,191)(84,192,273,420)(85,224,314,379)(86,380,315,197)(87,198,316,381)(88,382,317,199)(89,200,318,383)(90,384,319,201)(91,202,320,385)(92,386,321,203)(93,204,322,387)(94,388,323,205)(95,206,324,389)(96,390,325,207)(97,208,326,391)(98,392,327,209)(99,210,328,365)(100,366,329,211)(101,212,330,367)(102,368,331,213)(103,214,332,369)(104,370,333,215)(105,216,334,371)(106,372,335,217)(107,218,336,373)(108,374,309,219)(109,220,310,375)(110,376,311,221)(111,222,312,377)(112,378,313,223), (1,413,15,399)(2,400,16,414)(3,415,17,401)(4,402,18,416)(5,417,19,403)(6,404,20,418)(7,419,21,405)(8,406,22,420)(9,393,23,407)(10,408,24,394)(11,395,25,409)(12,410,26,396)(13,397,27,411)(14,412,28,398)(29,326,43,312)(30,313,44,327)(31,328,45,314)(32,315,46,329)(33,330,47,316)(34,317,48,331)(35,332,49,318)(36,319,50,333)(37,334,51,320)(38,321,52,335)(39,336,53,322)(40,323,54,309)(41,310,55,324)(42,325,56,311)(57,137,71,123)(58,124,72,138)(59,139,73,125)(60,126,74,140)(61,113,75,127)(62,128,76,114)(63,115,77,129)(64,130,78,116)(65,117,79,131)(66,132,80,118)(67,119,81,133)(68,134,82,120)(69,121,83,135)(70,136,84,122)(85,435,99,421)(86,422,100,436)(87,437,101,423)(88,424,102,438)(89,439,103,425)(90,426,104,440)(91,441,105,427)(92,428,106,442)(93,443,107,429)(94,430,108,444)(95,445,109,431)(96,432,110,446)(97,447,111,433)(98,434,112,448)(141,171,155,185)(142,186,156,172)(143,173,157,187)(144,188,158,174)(145,175,159,189)(146,190,160,176)(147,177,161,191)(148,192,162,178)(149,179,163,193)(150,194,164,180)(151,181,165,195)(152,196,166,182)(153,183,167,169)(154,170,168,184)(197,227,211,241)(198,242,212,228)(199,229,213,243)(200,244,214,230)(201,231,215,245)(202,246,216,232)(203,233,217,247)(204,248,218,234)(205,235,219,249)(206,250,220,236)(207,237,221,251)(208,252,222,238)(209,239,223,225)(210,226,224,240)(253,361,267,347)(254,348,268,362)(255,363,269,349)(256,350,270,364)(257,337,271,351)(258,352,272,338)(259,339,273,353)(260,354,274,340)(261,341,275,355)(262,356,276,342)(263,343,277,357)(264,358,278,344)(265,345,279,359)(266,360,280,346)(281,385,295,371)(282,372,296,386)(283,387,297,373)(284,374,298,388)(285,389,299,375)(286,376,300,390)(287,391,301,377)(288,378,302,392)(289,365,303,379)(290,380,304,366)(291,367,305,381)(292,382,306,368)(293,369,307,383)(294,384,308,370), (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), (1,377,141,208)(2,251,142,286)(3,375,143,206)(4,249,144,284)(5,373,145,204)(6,247,146,282)(7,371,147,202)(8,245,148,308)(9,369,149,200)(10,243,150,306)(11,367,151,198)(12,241,152,304)(13,365,153,224)(14,239,154,302)(15,391,155,222)(16,237,156,300)(17,389,157,220)(18,235,158,298)(19,387,159,218)(20,233,160,296)(21,385,161,216)(22,231,162,294)(23,383,163,214)(24,229,164,292)(25,381,165,212)(26,227,166,290)(27,379,167,210)(28,225,168,288)(29,63,447,266)(30,359,448,114)(31,61,421,264)(32,357,422,140)(33,59,423,262)(34,355,424,138)(35,57,425,260)(36,353,426,136)(37,83,427,258)(38,351,428,134)(39,81,429,256)(40,349,430,132)(41,79,431,254)(42,347,432,130)(43,77,433,280)(44,345,434,128)(45,75,435,278)(46,343,436,126)(47,73,437,276)(48,341,438,124)(49,71,439,274)(50,339,440,122)(51,69,441,272)(52,337,442,120)(53,67,443,270)(54,363,444,118)(55,65,445,268)(56,361,446,116)(58,331,261,88)(60,329,263,86)(62,327,265,112)(64,325,267,110)(66,323,269,108)(68,321,271,106)(70,319,273,104)(72,317,275,102)(74,315,277,100)(76,313,279,98)(78,311,253,96)(80,309,255,94)(82,335,257,92)(84,333,259,90)(85,344,328,127)(87,342,330,125)(89,340,332,123)(91,338,334,121)(93,364,336,119)(95,362,310,117)(97,360,312,115)(99,358,314,113)(101,356,316,139)(103,354,318,137)(105,352,320,135)(107,350,322,133)(109,348,324,131)(111,346,326,129)(169,240,411,303)(170,378,412,209)(171,238,413,301)(172,376,414,207)(173,236,415,299)(174,374,416,205)(175,234,417,297)(176,372,418,203)(177,232,419,295)(178,370,420,201)(179,230,393,293)(180,368,394,199)(181,228,395,291)(182,366,396,197)(183,226,397,289)(184,392,398,223)(185,252,399,287)(186,390,400,221)(187,250,401,285)(188,388,402,219)(189,248,403,283)(190,386,404,217)(191,246,405,281)(192,384,406,215)(193,244,407,307)(194,382,408,213)(195,242,409,305)(196,380,410,211)>;
G:=Group( (1,346,155,115)(2,116,156,347)(3,348,157,117)(4,118,158,349)(5,350,159,119)(6,120,160,351)(7,352,161,121)(8,122,162,353)(9,354,163,123)(10,124,164,355)(11,356,165,125)(12,126,166,357)(13,358,167,127)(14,128,168,359)(15,360,141,129)(16,130,142,361)(17,362,143,131)(18,132,144,363)(19,364,145,133)(20,134,146,337)(21,338,147,135)(22,136,148,339)(23,340,149,137)(24,138,150,341)(25,342,151,139)(26,140,152,343)(27,344,153,113)(28,114,154,345)(29,287,433,238)(30,239,434,288)(31,289,435,240)(32,241,436,290)(33,291,437,242)(34,243,438,292)(35,293,439,244)(36,245,440,294)(37,295,441,246)(38,247,442,296)(39,297,443,248)(40,249,444,298)(41,299,445,250)(42,251,446,300)(43,301,447,252)(44,225,448,302)(45,303,421,226)(46,227,422,304)(47,305,423,228)(48,229,424,306)(49,307,425,230)(50,231,426,308)(51,281,427,232)(52,233,428,282)(53,283,429,234)(54,235,430,284)(55,285,431,236)(56,237,432,286)(57,393,274,193)(58,194,275,394)(59,395,276,195)(60,196,277,396)(61,397,278,169)(62,170,279,398)(63,399,280,171)(64,172,253,400)(65,401,254,173)(66,174,255,402)(67,403,256,175)(68,176,257,404)(69,405,258,177)(70,178,259,406)(71,407,260,179)(72,180,261,408)(73,409,262,181)(74,182,263,410)(75,411,264,183)(76,184,265,412)(77,413,266,185)(78,186,267,414)(79,415,268,187)(80,188,269,416)(81,417,270,189)(82,190,271,418)(83,419,272,191)(84,192,273,420)(85,224,314,379)(86,380,315,197)(87,198,316,381)(88,382,317,199)(89,200,318,383)(90,384,319,201)(91,202,320,385)(92,386,321,203)(93,204,322,387)(94,388,323,205)(95,206,324,389)(96,390,325,207)(97,208,326,391)(98,392,327,209)(99,210,328,365)(100,366,329,211)(101,212,330,367)(102,368,331,213)(103,214,332,369)(104,370,333,215)(105,216,334,371)(106,372,335,217)(107,218,336,373)(108,374,309,219)(109,220,310,375)(110,376,311,221)(111,222,312,377)(112,378,313,223), (1,413,15,399)(2,400,16,414)(3,415,17,401)(4,402,18,416)(5,417,19,403)(6,404,20,418)(7,419,21,405)(8,406,22,420)(9,393,23,407)(10,408,24,394)(11,395,25,409)(12,410,26,396)(13,397,27,411)(14,412,28,398)(29,326,43,312)(30,313,44,327)(31,328,45,314)(32,315,46,329)(33,330,47,316)(34,317,48,331)(35,332,49,318)(36,319,50,333)(37,334,51,320)(38,321,52,335)(39,336,53,322)(40,323,54,309)(41,310,55,324)(42,325,56,311)(57,137,71,123)(58,124,72,138)(59,139,73,125)(60,126,74,140)(61,113,75,127)(62,128,76,114)(63,115,77,129)(64,130,78,116)(65,117,79,131)(66,132,80,118)(67,119,81,133)(68,134,82,120)(69,121,83,135)(70,136,84,122)(85,435,99,421)(86,422,100,436)(87,437,101,423)(88,424,102,438)(89,439,103,425)(90,426,104,440)(91,441,105,427)(92,428,106,442)(93,443,107,429)(94,430,108,444)(95,445,109,431)(96,432,110,446)(97,447,111,433)(98,434,112,448)(141,171,155,185)(142,186,156,172)(143,173,157,187)(144,188,158,174)(145,175,159,189)(146,190,160,176)(147,177,161,191)(148,192,162,178)(149,179,163,193)(150,194,164,180)(151,181,165,195)(152,196,166,182)(153,183,167,169)(154,170,168,184)(197,227,211,241)(198,242,212,228)(199,229,213,243)(200,244,214,230)(201,231,215,245)(202,246,216,232)(203,233,217,247)(204,248,218,234)(205,235,219,249)(206,250,220,236)(207,237,221,251)(208,252,222,238)(209,239,223,225)(210,226,224,240)(253,361,267,347)(254,348,268,362)(255,363,269,349)(256,350,270,364)(257,337,271,351)(258,352,272,338)(259,339,273,353)(260,354,274,340)(261,341,275,355)(262,356,276,342)(263,343,277,357)(264,358,278,344)(265,345,279,359)(266,360,280,346)(281,385,295,371)(282,372,296,386)(283,387,297,373)(284,374,298,388)(285,389,299,375)(286,376,300,390)(287,391,301,377)(288,378,302,392)(289,365,303,379)(290,380,304,366)(291,367,305,381)(292,382,306,368)(293,369,307,383)(294,384,308,370), (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), (1,377,141,208)(2,251,142,286)(3,375,143,206)(4,249,144,284)(5,373,145,204)(6,247,146,282)(7,371,147,202)(8,245,148,308)(9,369,149,200)(10,243,150,306)(11,367,151,198)(12,241,152,304)(13,365,153,224)(14,239,154,302)(15,391,155,222)(16,237,156,300)(17,389,157,220)(18,235,158,298)(19,387,159,218)(20,233,160,296)(21,385,161,216)(22,231,162,294)(23,383,163,214)(24,229,164,292)(25,381,165,212)(26,227,166,290)(27,379,167,210)(28,225,168,288)(29,63,447,266)(30,359,448,114)(31,61,421,264)(32,357,422,140)(33,59,423,262)(34,355,424,138)(35,57,425,260)(36,353,426,136)(37,83,427,258)(38,351,428,134)(39,81,429,256)(40,349,430,132)(41,79,431,254)(42,347,432,130)(43,77,433,280)(44,345,434,128)(45,75,435,278)(46,343,436,126)(47,73,437,276)(48,341,438,124)(49,71,439,274)(50,339,440,122)(51,69,441,272)(52,337,442,120)(53,67,443,270)(54,363,444,118)(55,65,445,268)(56,361,446,116)(58,331,261,88)(60,329,263,86)(62,327,265,112)(64,325,267,110)(66,323,269,108)(68,321,271,106)(70,319,273,104)(72,317,275,102)(74,315,277,100)(76,313,279,98)(78,311,253,96)(80,309,255,94)(82,335,257,92)(84,333,259,90)(85,344,328,127)(87,342,330,125)(89,340,332,123)(91,338,334,121)(93,364,336,119)(95,362,310,117)(97,360,312,115)(99,358,314,113)(101,356,316,139)(103,354,318,137)(105,352,320,135)(107,350,322,133)(109,348,324,131)(111,346,326,129)(169,240,411,303)(170,378,412,209)(171,238,413,301)(172,376,414,207)(173,236,415,299)(174,374,416,205)(175,234,417,297)(176,372,418,203)(177,232,419,295)(178,370,420,201)(179,230,393,293)(180,368,394,199)(181,228,395,291)(182,366,396,197)(183,226,397,289)(184,392,398,223)(185,252,399,287)(186,390,400,221)(187,250,401,285)(188,388,402,219)(189,248,403,283)(190,386,404,217)(191,246,405,281)(192,384,406,215)(193,244,407,307)(194,382,408,213)(195,242,409,305)(196,380,410,211) );
G=PermutationGroup([[(1,346,155,115),(2,116,156,347),(3,348,157,117),(4,118,158,349),(5,350,159,119),(6,120,160,351),(7,352,161,121),(8,122,162,353),(9,354,163,123),(10,124,164,355),(11,356,165,125),(12,126,166,357),(13,358,167,127),(14,128,168,359),(15,360,141,129),(16,130,142,361),(17,362,143,131),(18,132,144,363),(19,364,145,133),(20,134,146,337),(21,338,147,135),(22,136,148,339),(23,340,149,137),(24,138,150,341),(25,342,151,139),(26,140,152,343),(27,344,153,113),(28,114,154,345),(29,287,433,238),(30,239,434,288),(31,289,435,240),(32,241,436,290),(33,291,437,242),(34,243,438,292),(35,293,439,244),(36,245,440,294),(37,295,441,246),(38,247,442,296),(39,297,443,248),(40,249,444,298),(41,299,445,250),(42,251,446,300),(43,301,447,252),(44,225,448,302),(45,303,421,226),(46,227,422,304),(47,305,423,228),(48,229,424,306),(49,307,425,230),(50,231,426,308),(51,281,427,232),(52,233,428,282),(53,283,429,234),(54,235,430,284),(55,285,431,236),(56,237,432,286),(57,393,274,193),(58,194,275,394),(59,395,276,195),(60,196,277,396),(61,397,278,169),(62,170,279,398),(63,399,280,171),(64,172,253,400),(65,401,254,173),(66,174,255,402),(67,403,256,175),(68,176,257,404),(69,405,258,177),(70,178,259,406),(71,407,260,179),(72,180,261,408),(73,409,262,181),(74,182,263,410),(75,411,264,183),(76,184,265,412),(77,413,266,185),(78,186,267,414),(79,415,268,187),(80,188,269,416),(81,417,270,189),(82,190,271,418),(83,419,272,191),(84,192,273,420),(85,224,314,379),(86,380,315,197),(87,198,316,381),(88,382,317,199),(89,200,318,383),(90,384,319,201),(91,202,320,385),(92,386,321,203),(93,204,322,387),(94,388,323,205),(95,206,324,389),(96,390,325,207),(97,208,326,391),(98,392,327,209),(99,210,328,365),(100,366,329,211),(101,212,330,367),(102,368,331,213),(103,214,332,369),(104,370,333,215),(105,216,334,371),(106,372,335,217),(107,218,336,373),(108,374,309,219),(109,220,310,375),(110,376,311,221),(111,222,312,377),(112,378,313,223)], [(1,413,15,399),(2,400,16,414),(3,415,17,401),(4,402,18,416),(5,417,19,403),(6,404,20,418),(7,419,21,405),(8,406,22,420),(9,393,23,407),(10,408,24,394),(11,395,25,409),(12,410,26,396),(13,397,27,411),(14,412,28,398),(29,326,43,312),(30,313,44,327),(31,328,45,314),(32,315,46,329),(33,330,47,316),(34,317,48,331),(35,332,49,318),(36,319,50,333),(37,334,51,320),(38,321,52,335),(39,336,53,322),(40,323,54,309),(41,310,55,324),(42,325,56,311),(57,137,71,123),(58,124,72,138),(59,139,73,125),(60,126,74,140),(61,113,75,127),(62,128,76,114),(63,115,77,129),(64,130,78,116),(65,117,79,131),(66,132,80,118),(67,119,81,133),(68,134,82,120),(69,121,83,135),(70,136,84,122),(85,435,99,421),(86,422,100,436),(87,437,101,423),(88,424,102,438),(89,439,103,425),(90,426,104,440),(91,441,105,427),(92,428,106,442),(93,443,107,429),(94,430,108,444),(95,445,109,431),(96,432,110,446),(97,447,111,433),(98,434,112,448),(141,171,155,185),(142,186,156,172),(143,173,157,187),(144,188,158,174),(145,175,159,189),(146,190,160,176),(147,177,161,191),(148,192,162,178),(149,179,163,193),(150,194,164,180),(151,181,165,195),(152,196,166,182),(153,183,167,169),(154,170,168,184),(197,227,211,241),(198,242,212,228),(199,229,213,243),(200,244,214,230),(201,231,215,245),(202,246,216,232),(203,233,217,247),(204,248,218,234),(205,235,219,249),(206,250,220,236),(207,237,221,251),(208,252,222,238),(209,239,223,225),(210,226,224,240),(253,361,267,347),(254,348,268,362),(255,363,269,349),(256,350,270,364),(257,337,271,351),(258,352,272,338),(259,339,273,353),(260,354,274,340),(261,341,275,355),(262,356,276,342),(263,343,277,357),(264,358,278,344),(265,345,279,359),(266,360,280,346),(281,385,295,371),(282,372,296,386),(283,387,297,373),(284,374,298,388),(285,389,299,375),(286,376,300,390),(287,391,301,377),(288,378,302,392),(289,365,303,379),(290,380,304,366),(291,367,305,381),(292,382,306,368),(293,369,307,383),(294,384,308,370)], [(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)], [(1,377,141,208),(2,251,142,286),(3,375,143,206),(4,249,144,284),(5,373,145,204),(6,247,146,282),(7,371,147,202),(8,245,148,308),(9,369,149,200),(10,243,150,306),(11,367,151,198),(12,241,152,304),(13,365,153,224),(14,239,154,302),(15,391,155,222),(16,237,156,300),(17,389,157,220),(18,235,158,298),(19,387,159,218),(20,233,160,296),(21,385,161,216),(22,231,162,294),(23,383,163,214),(24,229,164,292),(25,381,165,212),(26,227,166,290),(27,379,167,210),(28,225,168,288),(29,63,447,266),(30,359,448,114),(31,61,421,264),(32,357,422,140),(33,59,423,262),(34,355,424,138),(35,57,425,260),(36,353,426,136),(37,83,427,258),(38,351,428,134),(39,81,429,256),(40,349,430,132),(41,79,431,254),(42,347,432,130),(43,77,433,280),(44,345,434,128),(45,75,435,278),(46,343,436,126),(47,73,437,276),(48,341,438,124),(49,71,439,274),(50,339,440,122),(51,69,441,272),(52,337,442,120),(53,67,443,270),(54,363,444,118),(55,65,445,268),(56,361,446,116),(58,331,261,88),(60,329,263,86),(62,327,265,112),(64,325,267,110),(66,323,269,108),(68,321,271,106),(70,319,273,104),(72,317,275,102),(74,315,277,100),(76,313,279,98),(78,311,253,96),(80,309,255,94),(82,335,257,92),(84,333,259,90),(85,344,328,127),(87,342,330,125),(89,340,332,123),(91,338,334,121),(93,364,336,119),(95,362,310,117),(97,360,312,115),(99,358,314,113),(101,356,316,139),(103,354,318,137),(105,352,320,135),(107,350,322,133),(109,348,324,131),(111,346,326,129),(169,240,411,303),(170,378,412,209),(171,238,413,301),(172,376,414,207),(173,236,415,299),(174,374,416,205),(175,234,417,297),(176,372,418,203),(177,232,419,295),(178,370,420,201),(179,230,393,293),(180,368,394,199),(181,228,395,291),(182,366,396,197),(183,226,397,289),(184,392,398,223),(185,252,399,287),(186,390,400,221),(187,250,401,285),(188,388,402,219),(189,248,403,283),(190,386,404,217),(191,246,405,281),(192,384,406,215),(193,244,407,307),(194,382,408,213),(195,242,409,305),(196,380,410,211)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C8.C22 | D4⋊2D7 | C28.C23 |
| kernel | C42.77D14 | C42.D7 | Q8⋊Dic7 | C28.6Q8 | C7×C4⋊Q8 | C2×C28 | C4⋊Q8 | C28 | C42 | C2×Q8 | C2×C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 12 | 2 | 6 | 12 |
Matrix representation of C42.77D14 ►in GL8(𝔽113)
| 46 | 106 | 73 | 53 | 0 | 0 | 0 | 0 |
| 7 | 96 | 60 | 98 | 0 | 0 | 0 | 0 |
| 40 | 60 | 67 | 7 | 0 | 0 | 0 | 0 |
| 53 | 15 | 106 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 91 | 7 | 21 | 17 |
| 0 | 0 | 0 | 0 | 44 | 22 | 42 | 104 |
| 0 | 0 | 0 | 0 | 18 | 34 | 72 | 106 |
| 0 | 0 | 0 | 0 | 84 | 71 | 14 | 41 |
| 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 | 58 | 15 |
| 0 | 0 | 0 | 0 | 0 | 1 | 40 | 98 |
| 0 | 0 | 0 | 0 | 83 | 83 | 112 | 0 |
| 0 | 0 | 0 | 0 | 33 | 3 | 0 | 112 |
| 0 | 0 | 33 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 80 | 104 | 0 | 0 | 0 | 0 |
| 33 | 33 | 0 | 0 | 0 | 0 | 0 | 0 |
| 80 | 104 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 48 | 96 | 62 |
| 0 | 0 | 0 | 0 | 108 | 29 | 62 | 73 |
| 0 | 0 | 0 | 0 | 15 | 0 | 99 | 65 |
| 0 | 0 | 0 | 0 | 91 | 15 | 96 | 64 |
| 43 | 24 | 78 | 12 | 0 | 0 | 0 | 0 |
| 72 | 70 | 36 | 35 | 0 | 0 | 0 | 0 |
| 35 | 101 | 70 | 89 | 0 | 0 | 0 | 0 |
| 77 | 78 | 41 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 8 | 45 | 2 |
| 0 | 0 | 0 | 0 | 90 | 65 | 11 | 86 |
| 0 | 0 | 0 | 0 | 100 | 66 | 103 | 65 |
| 0 | 0 | 0 | 0 | 24 | 97 | 5 | 10 |
G:=sub<GL(8,GF(113))| [46,7,40,53,0,0,0,0,106,96,60,15,0,0,0,0,73,60,67,106,0,0,0,0,53,98,7,17,0,0,0,0,0,0,0,0,91,44,18,84,0,0,0,0,7,22,34,71,0,0,0,0,21,42,72,14,0,0,0,0,17,104,106,41],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,83,33,0,0,0,0,0,1,83,3,0,0,0,0,58,40,112,0,0,0,0,0,15,98,0,112],[0,0,33,80,0,0,0,0,0,0,33,104,0,0,0,0,33,80,0,0,0,0,0,0,33,104,0,0,0,0,0,0,0,0,0,0,34,108,15,91,0,0,0,0,48,29,0,15,0,0,0,0,96,62,99,96,0,0,0,0,62,73,65,64],[43,72,35,77,0,0,0,0,24,70,101,78,0,0,0,0,78,36,70,41,0,0,0,0,12,35,89,43,0,0,0,0,0,0,0,0,48,90,100,24,0,0,0,0,8,65,66,97,0,0,0,0,45,11,103,5,0,0,0,0,2,86,65,10] >;
C42.77D14 in GAP, Magma, Sage, TeX
C_4^2._{77}D_{14} % in TeX
G:=Group("C4^2.77D14"); // GroupNames label
G:=SmallGroup(448,616);
// by ID
G=gap.SmallGroup(448,616);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,477,64,590,135,184,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=a^2*b^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=a^2*b*c^13>;
// generators/relations