metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.147D14, C14.272- 1+4, C4⋊C4.110D14, C42.C2.7D7, Dic7.Q8.3C2, (C4×C28).223C22, (C2×C28).187C23, (C2×C14).230C24, C7⋊(C22.58C24), C28.3Q8.13C2, C28.6Q8.12C2, Dic7⋊C4.85C22, C4⋊Dic7.237C22, C22.251(C23×D7), (C2×Dic7).120C23, (C4×Dic7).138C22, C2.56(D4.10D14), C2.28(Q8.10D14), (C7×C42.C2).6C2, (C7×C4⋊C4).185C22, (C2×C4).202(C22×D7), SmallGroup(448,1139)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.147D14
G = < a,b,c,d | a4=b4=1, c14=a2, d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c13 >
Subgroups: 604 in 172 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2, C4, C22, C7, C2×C4, C2×C4, C2×C4, C14, C14, C42, C42, C4⋊C4, C4⋊C4, Dic7, C28, C2×C14, C42.C2, C42.C2, C2×Dic7, C2×C28, C2×C28, C22.58C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C28.6Q8, Dic7.Q8, C28.3Q8, C7×C42.C2, C42.147D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2- 1+4, C22×D7, C22.58C24, C23×D7, Q8.10D14, D4.10D14, C42.147D14
►Regular action on 448 points
Generators in S448
(1 57 15 71)(2 416 16 402)(3 59 17 73)(4 418 18 404)(5 61 19 75)(6 420 20 406)(7 63 21 77)(8 394 22 408)(9 65 23 79)(10 396 24 410)(11 67 25 81)(12 398 26 412)(13 69 27 83)(14 400 28 414)(29 155 43 141)(30 290 44 304)(31 157 45 143)(32 292 46 306)(33 159 47 145)(34 294 48 308)(35 161 49 147)(36 296 50 282)(37 163 51 149)(38 298 52 284)(39 165 53 151)(40 300 54 286)(41 167 55 153)(42 302 56 288)(58 392 72 378)(60 366 74 380)(62 368 76 382)(64 370 78 384)(66 372 80 386)(68 374 82 388)(70 376 84 390)(85 311 99 325)(86 278 100 264)(87 313 101 327)(88 280 102 266)(89 315 103 329)(90 254 104 268)(91 317 105 331)(92 256 106 270)(93 319 107 333)(94 258 108 272)(95 321 109 335)(96 260 110 274)(97 323 111 309)(98 262 112 276)(113 349 127 363)(114 245 128 231)(115 351 129 337)(116 247 130 233)(117 353 131 339)(118 249 132 235)(119 355 133 341)(120 251 134 237)(121 357 135 343)(122 225 136 239)(123 359 137 345)(124 227 138 241)(125 361 139 347)(126 229 140 243)(142 224 156 210)(144 198 158 212)(146 200 160 214)(148 202 162 216)(150 204 164 218)(152 206 166 220)(154 208 168 222)(169 318 183 332)(170 257 184 271)(171 320 185 334)(172 259 186 273)(173 322 187 336)(174 261 188 275)(175 324 189 310)(176 263 190 277)(177 326 191 312)(178 265 192 279)(179 328 193 314)(180 267 194 253)(181 330 195 316)(182 269 196 255)(197 291 211 305)(199 293 213 307)(201 295 215 281)(203 297 217 283)(205 299 219 285)(207 301 221 287)(209 303 223 289)(226 444 240 430)(228 446 242 432)(230 448 244 434)(232 422 246 436)(234 424 248 438)(236 426 250 440)(238 428 252 442)(338 423 352 437)(340 425 354 439)(342 427 356 441)(344 429 358 443)(346 431 360 445)(348 433 362 447)(350 435 364 421)(365 403 379 417)(367 405 381 419)(369 407 383 393)(371 409 385 395)(373 411 387 397)(375 413 389 399)(377 415 391 401)
(1 155 391 303)(2 304 392 156)(3 157 365 305)(4 306 366 158)(5 159 367 307)(6 308 368 160)(7 161 369 281)(8 282 370 162)(9 163 371 283)(10 284 372 164)(11 165 373 285)(12 286 374 166)(13 167 375 287)(14 288 376 168)(15 141 377 289)(16 290 378 142)(17 143 379 291)(18 292 380 144)(19 145 381 293)(20 294 382 146)(21 147 383 295)(22 296 384 148)(23 149 385 297)(24 298 386 150)(25 151 387 299)(26 300 388 152)(27 153 389 301)(28 302 390 154)(29 415 209 71)(30 72 210 416)(31 417 211 73)(32 74 212 418)(33 419 213 75)(34 76 214 420)(35 393 215 77)(36 78 216 394)(37 395 217 79)(38 80 218 396)(39 397 219 81)(40 82 220 398)(41 399 221 83)(42 84 222 400)(43 401 223 57)(44 58 224 402)(45 403 197 59)(46 60 198 404)(47 405 199 61)(48 62 200 406)(49 407 201 63)(50 64 202 408)(51 409 203 65)(52 66 204 410)(53 411 205 67)(54 68 206 412)(55 413 207 69)(56 70 208 414)(85 433 176 140)(86 113 177 434)(87 435 178 114)(88 115 179 436)(89 437 180 116)(90 117 181 438)(91 439 182 118)(92 119 183 440)(93 441 184 120)(94 121 185 442)(95 443 186 122)(96 123 187 444)(97 445 188 124)(98 125 189 446)(99 447 190 126)(100 127 191 448)(101 421 192 128)(102 129 193 422)(103 423 194 130)(104 131 195 424)(105 425 196 132)(106 133 169 426)(107 427 170 134)(108 135 171 428)(109 429 172 136)(110 137 173 430)(111 431 174 138)(112 139 175 432)(225 321 344 273)(226 274 345 322)(227 323 346 275)(228 276 347 324)(229 325 348 277)(230 278 349 326)(231 327 350 279)(232 280 351 328)(233 329 352 253)(234 254 353 330)(235 331 354 255)(236 256 355 332)(237 333 356 257)(238 258 357 334)(239 335 358 259)(240 260 359 336)(241 309 360 261)(242 262 361 310)(243 311 362 263)(244 264 363 312)(245 313 364 265)(246 266 337 314)(247 315 338 267)(248 268 339 316)(249 317 340 269)(250 270 341 318)(251 319 342 271)(252 272 343 320)
(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)
(1 114 377 421)(2 127 378 434)(3 140 379 447)(4 125 380 432)(5 138 381 445)(6 123 382 430)(7 136 383 443)(8 121 384 428)(9 134 385 441)(10 119 386 426)(11 132 387 439)(12 117 388 424)(13 130 389 437)(14 115 390 422)(15 128 391 435)(16 113 392 448)(17 126 365 433)(18 139 366 446)(19 124 367 431)(20 137 368 444)(21 122 369 429)(22 135 370 442)(23 120 371 427)(24 133 372 440)(25 118 373 425)(26 131 374 438)(27 116 375 423)(28 129 376 436)(29 279 223 313)(30 264 224 326)(31 277 197 311)(32 262 198 324)(33 275 199 309)(34 260 200 322)(35 273 201 335)(36 258 202 320)(37 271 203 333)(38 256 204 318)(39 269 205 331)(40 254 206 316)(41 267 207 329)(42 280 208 314)(43 265 209 327)(44 278 210 312)(45 263 211 325)(46 276 212 310)(47 261 213 323)(48 274 214 336)(49 259 215 321)(50 272 216 334)(51 257 217 319)(52 270 218 332)(53 255 219 317)(54 268 220 330)(55 253 221 315)(56 266 222 328)(57 231 415 364)(58 244 416 349)(59 229 417 362)(60 242 418 347)(61 227 419 360)(62 240 420 345)(63 225 393 358)(64 238 394 343)(65 251 395 356)(66 236 396 341)(67 249 397 354)(68 234 398 339)(69 247 399 352)(70 232 400 337)(71 245 401 350)(72 230 402 363)(73 243 403 348)(74 228 404 361)(75 241 405 346)(76 226 406 359)(77 239 407 344)(78 252 408 357)(79 237 409 342)(80 250 410 355)(81 235 411 340)(82 248 412 353)(83 233 413 338)(84 246 414 351)(85 157 190 291)(86 142 191 304)(87 155 192 289)(88 168 193 302)(89 153 194 287)(90 166 195 300)(91 151 196 285)(92 164 169 298)(93 149 170 283)(94 162 171 296)(95 147 172 281)(96 160 173 294)(97 145 174 307)(98 158 175 292)(99 143 176 305)(100 156 177 290)(101 141 178 303)(102 154 179 288)(103 167 180 301)(104 152 181 286)(105 165 182 299)(106 150 183 284)(107 163 184 297)(108 148 185 282)(109 161 186 295)(110 146 187 308)(111 159 188 293)(112 144 189 306)
G:=sub<Sym(448)| (1,57,15,71)(2,416,16,402)(3,59,17,73)(4,418,18,404)(5,61,19,75)(6,420,20,406)(7,63,21,77)(8,394,22,408)(9,65,23,79)(10,396,24,410)(11,67,25,81)(12,398,26,412)(13,69,27,83)(14,400,28,414)(29,155,43,141)(30,290,44,304)(31,157,45,143)(32,292,46,306)(33,159,47,145)(34,294,48,308)(35,161,49,147)(36,296,50,282)(37,163,51,149)(38,298,52,284)(39,165,53,151)(40,300,54,286)(41,167,55,153)(42,302,56,288)(58,392,72,378)(60,366,74,380)(62,368,76,382)(64,370,78,384)(66,372,80,386)(68,374,82,388)(70,376,84,390)(85,311,99,325)(86,278,100,264)(87,313,101,327)(88,280,102,266)(89,315,103,329)(90,254,104,268)(91,317,105,331)(92,256,106,270)(93,319,107,333)(94,258,108,272)(95,321,109,335)(96,260,110,274)(97,323,111,309)(98,262,112,276)(113,349,127,363)(114,245,128,231)(115,351,129,337)(116,247,130,233)(117,353,131,339)(118,249,132,235)(119,355,133,341)(120,251,134,237)(121,357,135,343)(122,225,136,239)(123,359,137,345)(124,227,138,241)(125,361,139,347)(126,229,140,243)(142,224,156,210)(144,198,158,212)(146,200,160,214)(148,202,162,216)(150,204,164,218)(152,206,166,220)(154,208,168,222)(169,318,183,332)(170,257,184,271)(171,320,185,334)(172,259,186,273)(173,322,187,336)(174,261,188,275)(175,324,189,310)(176,263,190,277)(177,326,191,312)(178,265,192,279)(179,328,193,314)(180,267,194,253)(181,330,195,316)(182,269,196,255)(197,291,211,305)(199,293,213,307)(201,295,215,281)(203,297,217,283)(205,299,219,285)(207,301,221,287)(209,303,223,289)(226,444,240,430)(228,446,242,432)(230,448,244,434)(232,422,246,436)(234,424,248,438)(236,426,250,440)(238,428,252,442)(338,423,352,437)(340,425,354,439)(342,427,356,441)(344,429,358,443)(346,431,360,445)(348,433,362,447)(350,435,364,421)(365,403,379,417)(367,405,381,419)(369,407,383,393)(371,409,385,395)(373,411,387,397)(375,413,389,399)(377,415,391,401), (1,155,391,303)(2,304,392,156)(3,157,365,305)(4,306,366,158)(5,159,367,307)(6,308,368,160)(7,161,369,281)(8,282,370,162)(9,163,371,283)(10,284,372,164)(11,165,373,285)(12,286,374,166)(13,167,375,287)(14,288,376,168)(15,141,377,289)(16,290,378,142)(17,143,379,291)(18,292,380,144)(19,145,381,293)(20,294,382,146)(21,147,383,295)(22,296,384,148)(23,149,385,297)(24,298,386,150)(25,151,387,299)(26,300,388,152)(27,153,389,301)(28,302,390,154)(29,415,209,71)(30,72,210,416)(31,417,211,73)(32,74,212,418)(33,419,213,75)(34,76,214,420)(35,393,215,77)(36,78,216,394)(37,395,217,79)(38,80,218,396)(39,397,219,81)(40,82,220,398)(41,399,221,83)(42,84,222,400)(43,401,223,57)(44,58,224,402)(45,403,197,59)(46,60,198,404)(47,405,199,61)(48,62,200,406)(49,407,201,63)(50,64,202,408)(51,409,203,65)(52,66,204,410)(53,411,205,67)(54,68,206,412)(55,413,207,69)(56,70,208,414)(85,433,176,140)(86,113,177,434)(87,435,178,114)(88,115,179,436)(89,437,180,116)(90,117,181,438)(91,439,182,118)(92,119,183,440)(93,441,184,120)(94,121,185,442)(95,443,186,122)(96,123,187,444)(97,445,188,124)(98,125,189,446)(99,447,190,126)(100,127,191,448)(101,421,192,128)(102,129,193,422)(103,423,194,130)(104,131,195,424)(105,425,196,132)(106,133,169,426)(107,427,170,134)(108,135,171,428)(109,429,172,136)(110,137,173,430)(111,431,174,138)(112,139,175,432)(225,321,344,273)(226,274,345,322)(227,323,346,275)(228,276,347,324)(229,325,348,277)(230,278,349,326)(231,327,350,279)(232,280,351,328)(233,329,352,253)(234,254,353,330)(235,331,354,255)(236,256,355,332)(237,333,356,257)(238,258,357,334)(239,335,358,259)(240,260,359,336)(241,309,360,261)(242,262,361,310)(243,311,362,263)(244,264,363,312)(245,313,364,265)(246,266,337,314)(247,315,338,267)(248,268,339,316)(249,317,340,269)(250,270,341,318)(251,319,342,271)(252,272,343,320), (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), (1,114,377,421)(2,127,378,434)(3,140,379,447)(4,125,380,432)(5,138,381,445)(6,123,382,430)(7,136,383,443)(8,121,384,428)(9,134,385,441)(10,119,386,426)(11,132,387,439)(12,117,388,424)(13,130,389,437)(14,115,390,422)(15,128,391,435)(16,113,392,448)(17,126,365,433)(18,139,366,446)(19,124,367,431)(20,137,368,444)(21,122,369,429)(22,135,370,442)(23,120,371,427)(24,133,372,440)(25,118,373,425)(26,131,374,438)(27,116,375,423)(28,129,376,436)(29,279,223,313)(30,264,224,326)(31,277,197,311)(32,262,198,324)(33,275,199,309)(34,260,200,322)(35,273,201,335)(36,258,202,320)(37,271,203,333)(38,256,204,318)(39,269,205,331)(40,254,206,316)(41,267,207,329)(42,280,208,314)(43,265,209,327)(44,278,210,312)(45,263,211,325)(46,276,212,310)(47,261,213,323)(48,274,214,336)(49,259,215,321)(50,272,216,334)(51,257,217,319)(52,270,218,332)(53,255,219,317)(54,268,220,330)(55,253,221,315)(56,266,222,328)(57,231,415,364)(58,244,416,349)(59,229,417,362)(60,242,418,347)(61,227,419,360)(62,240,420,345)(63,225,393,358)(64,238,394,343)(65,251,395,356)(66,236,396,341)(67,249,397,354)(68,234,398,339)(69,247,399,352)(70,232,400,337)(71,245,401,350)(72,230,402,363)(73,243,403,348)(74,228,404,361)(75,241,405,346)(76,226,406,359)(77,239,407,344)(78,252,408,357)(79,237,409,342)(80,250,410,355)(81,235,411,340)(82,248,412,353)(83,233,413,338)(84,246,414,351)(85,157,190,291)(86,142,191,304)(87,155,192,289)(88,168,193,302)(89,153,194,287)(90,166,195,300)(91,151,196,285)(92,164,169,298)(93,149,170,283)(94,162,171,296)(95,147,172,281)(96,160,173,294)(97,145,174,307)(98,158,175,292)(99,143,176,305)(100,156,177,290)(101,141,178,303)(102,154,179,288)(103,167,180,301)(104,152,181,286)(105,165,182,299)(106,150,183,284)(107,163,184,297)(108,148,185,282)(109,161,186,295)(110,146,187,308)(111,159,188,293)(112,144,189,306)>;
G:=Group( (1,57,15,71)(2,416,16,402)(3,59,17,73)(4,418,18,404)(5,61,19,75)(6,420,20,406)(7,63,21,77)(8,394,22,408)(9,65,23,79)(10,396,24,410)(11,67,25,81)(12,398,26,412)(13,69,27,83)(14,400,28,414)(29,155,43,141)(30,290,44,304)(31,157,45,143)(32,292,46,306)(33,159,47,145)(34,294,48,308)(35,161,49,147)(36,296,50,282)(37,163,51,149)(38,298,52,284)(39,165,53,151)(40,300,54,286)(41,167,55,153)(42,302,56,288)(58,392,72,378)(60,366,74,380)(62,368,76,382)(64,370,78,384)(66,372,80,386)(68,374,82,388)(70,376,84,390)(85,311,99,325)(86,278,100,264)(87,313,101,327)(88,280,102,266)(89,315,103,329)(90,254,104,268)(91,317,105,331)(92,256,106,270)(93,319,107,333)(94,258,108,272)(95,321,109,335)(96,260,110,274)(97,323,111,309)(98,262,112,276)(113,349,127,363)(114,245,128,231)(115,351,129,337)(116,247,130,233)(117,353,131,339)(118,249,132,235)(119,355,133,341)(120,251,134,237)(121,357,135,343)(122,225,136,239)(123,359,137,345)(124,227,138,241)(125,361,139,347)(126,229,140,243)(142,224,156,210)(144,198,158,212)(146,200,160,214)(148,202,162,216)(150,204,164,218)(152,206,166,220)(154,208,168,222)(169,318,183,332)(170,257,184,271)(171,320,185,334)(172,259,186,273)(173,322,187,336)(174,261,188,275)(175,324,189,310)(176,263,190,277)(177,326,191,312)(178,265,192,279)(179,328,193,314)(180,267,194,253)(181,330,195,316)(182,269,196,255)(197,291,211,305)(199,293,213,307)(201,295,215,281)(203,297,217,283)(205,299,219,285)(207,301,221,287)(209,303,223,289)(226,444,240,430)(228,446,242,432)(230,448,244,434)(232,422,246,436)(234,424,248,438)(236,426,250,440)(238,428,252,442)(338,423,352,437)(340,425,354,439)(342,427,356,441)(344,429,358,443)(346,431,360,445)(348,433,362,447)(350,435,364,421)(365,403,379,417)(367,405,381,419)(369,407,383,393)(371,409,385,395)(373,411,387,397)(375,413,389,399)(377,415,391,401), (1,155,391,303)(2,304,392,156)(3,157,365,305)(4,306,366,158)(5,159,367,307)(6,308,368,160)(7,161,369,281)(8,282,370,162)(9,163,371,283)(10,284,372,164)(11,165,373,285)(12,286,374,166)(13,167,375,287)(14,288,376,168)(15,141,377,289)(16,290,378,142)(17,143,379,291)(18,292,380,144)(19,145,381,293)(20,294,382,146)(21,147,383,295)(22,296,384,148)(23,149,385,297)(24,298,386,150)(25,151,387,299)(26,300,388,152)(27,153,389,301)(28,302,390,154)(29,415,209,71)(30,72,210,416)(31,417,211,73)(32,74,212,418)(33,419,213,75)(34,76,214,420)(35,393,215,77)(36,78,216,394)(37,395,217,79)(38,80,218,396)(39,397,219,81)(40,82,220,398)(41,399,221,83)(42,84,222,400)(43,401,223,57)(44,58,224,402)(45,403,197,59)(46,60,198,404)(47,405,199,61)(48,62,200,406)(49,407,201,63)(50,64,202,408)(51,409,203,65)(52,66,204,410)(53,411,205,67)(54,68,206,412)(55,413,207,69)(56,70,208,414)(85,433,176,140)(86,113,177,434)(87,435,178,114)(88,115,179,436)(89,437,180,116)(90,117,181,438)(91,439,182,118)(92,119,183,440)(93,441,184,120)(94,121,185,442)(95,443,186,122)(96,123,187,444)(97,445,188,124)(98,125,189,446)(99,447,190,126)(100,127,191,448)(101,421,192,128)(102,129,193,422)(103,423,194,130)(104,131,195,424)(105,425,196,132)(106,133,169,426)(107,427,170,134)(108,135,171,428)(109,429,172,136)(110,137,173,430)(111,431,174,138)(112,139,175,432)(225,321,344,273)(226,274,345,322)(227,323,346,275)(228,276,347,324)(229,325,348,277)(230,278,349,326)(231,327,350,279)(232,280,351,328)(233,329,352,253)(234,254,353,330)(235,331,354,255)(236,256,355,332)(237,333,356,257)(238,258,357,334)(239,335,358,259)(240,260,359,336)(241,309,360,261)(242,262,361,310)(243,311,362,263)(244,264,363,312)(245,313,364,265)(246,266,337,314)(247,315,338,267)(248,268,339,316)(249,317,340,269)(250,270,341,318)(251,319,342,271)(252,272,343,320), (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), (1,114,377,421)(2,127,378,434)(3,140,379,447)(4,125,380,432)(5,138,381,445)(6,123,382,430)(7,136,383,443)(8,121,384,428)(9,134,385,441)(10,119,386,426)(11,132,387,439)(12,117,388,424)(13,130,389,437)(14,115,390,422)(15,128,391,435)(16,113,392,448)(17,126,365,433)(18,139,366,446)(19,124,367,431)(20,137,368,444)(21,122,369,429)(22,135,370,442)(23,120,371,427)(24,133,372,440)(25,118,373,425)(26,131,374,438)(27,116,375,423)(28,129,376,436)(29,279,223,313)(30,264,224,326)(31,277,197,311)(32,262,198,324)(33,275,199,309)(34,260,200,322)(35,273,201,335)(36,258,202,320)(37,271,203,333)(38,256,204,318)(39,269,205,331)(40,254,206,316)(41,267,207,329)(42,280,208,314)(43,265,209,327)(44,278,210,312)(45,263,211,325)(46,276,212,310)(47,261,213,323)(48,274,214,336)(49,259,215,321)(50,272,216,334)(51,257,217,319)(52,270,218,332)(53,255,219,317)(54,268,220,330)(55,253,221,315)(56,266,222,328)(57,231,415,364)(58,244,416,349)(59,229,417,362)(60,242,418,347)(61,227,419,360)(62,240,420,345)(63,225,393,358)(64,238,394,343)(65,251,395,356)(66,236,396,341)(67,249,397,354)(68,234,398,339)(69,247,399,352)(70,232,400,337)(71,245,401,350)(72,230,402,363)(73,243,403,348)(74,228,404,361)(75,241,405,346)(76,226,406,359)(77,239,407,344)(78,252,408,357)(79,237,409,342)(80,250,410,355)(81,235,411,340)(82,248,412,353)(83,233,413,338)(84,246,414,351)(85,157,190,291)(86,142,191,304)(87,155,192,289)(88,168,193,302)(89,153,194,287)(90,166,195,300)(91,151,196,285)(92,164,169,298)(93,149,170,283)(94,162,171,296)(95,147,172,281)(96,160,173,294)(97,145,174,307)(98,158,175,292)(99,143,176,305)(100,156,177,290)(101,141,178,303)(102,154,179,288)(103,167,180,301)(104,152,181,286)(105,165,182,299)(106,150,183,284)(107,163,184,297)(108,148,185,282)(109,161,186,295)(110,146,187,308)(111,159,188,293)(112,144,189,306) );
G=PermutationGroup([[(1,57,15,71),(2,416,16,402),(3,59,17,73),(4,418,18,404),(5,61,19,75),(6,420,20,406),(7,63,21,77),(8,394,22,408),(9,65,23,79),(10,396,24,410),(11,67,25,81),(12,398,26,412),(13,69,27,83),(14,400,28,414),(29,155,43,141),(30,290,44,304),(31,157,45,143),(32,292,46,306),(33,159,47,145),(34,294,48,308),(35,161,49,147),(36,296,50,282),(37,163,51,149),(38,298,52,284),(39,165,53,151),(40,300,54,286),(41,167,55,153),(42,302,56,288),(58,392,72,378),(60,366,74,380),(62,368,76,382),(64,370,78,384),(66,372,80,386),(68,374,82,388),(70,376,84,390),(85,311,99,325),(86,278,100,264),(87,313,101,327),(88,280,102,266),(89,315,103,329),(90,254,104,268),(91,317,105,331),(92,256,106,270),(93,319,107,333),(94,258,108,272),(95,321,109,335),(96,260,110,274),(97,323,111,309),(98,262,112,276),(113,349,127,363),(114,245,128,231),(115,351,129,337),(116,247,130,233),(117,353,131,339),(118,249,132,235),(119,355,133,341),(120,251,134,237),(121,357,135,343),(122,225,136,239),(123,359,137,345),(124,227,138,241),(125,361,139,347),(126,229,140,243),(142,224,156,210),(144,198,158,212),(146,200,160,214),(148,202,162,216),(150,204,164,218),(152,206,166,220),(154,208,168,222),(169,318,183,332),(170,257,184,271),(171,320,185,334),(172,259,186,273),(173,322,187,336),(174,261,188,275),(175,324,189,310),(176,263,190,277),(177,326,191,312),(178,265,192,279),(179,328,193,314),(180,267,194,253),(181,330,195,316),(182,269,196,255),(197,291,211,305),(199,293,213,307),(201,295,215,281),(203,297,217,283),(205,299,219,285),(207,301,221,287),(209,303,223,289),(226,444,240,430),(228,446,242,432),(230,448,244,434),(232,422,246,436),(234,424,248,438),(236,426,250,440),(238,428,252,442),(338,423,352,437),(340,425,354,439),(342,427,356,441),(344,429,358,443),(346,431,360,445),(348,433,362,447),(350,435,364,421),(365,403,379,417),(367,405,381,419),(369,407,383,393),(371,409,385,395),(373,411,387,397),(375,413,389,399),(377,415,391,401)], [(1,155,391,303),(2,304,392,156),(3,157,365,305),(4,306,366,158),(5,159,367,307),(6,308,368,160),(7,161,369,281),(8,282,370,162),(9,163,371,283),(10,284,372,164),(11,165,373,285),(12,286,374,166),(13,167,375,287),(14,288,376,168),(15,141,377,289),(16,290,378,142),(17,143,379,291),(18,292,380,144),(19,145,381,293),(20,294,382,146),(21,147,383,295),(22,296,384,148),(23,149,385,297),(24,298,386,150),(25,151,387,299),(26,300,388,152),(27,153,389,301),(28,302,390,154),(29,415,209,71),(30,72,210,416),(31,417,211,73),(32,74,212,418),(33,419,213,75),(34,76,214,420),(35,393,215,77),(36,78,216,394),(37,395,217,79),(38,80,218,396),(39,397,219,81),(40,82,220,398),(41,399,221,83),(42,84,222,400),(43,401,223,57),(44,58,224,402),(45,403,197,59),(46,60,198,404),(47,405,199,61),(48,62,200,406),(49,407,201,63),(50,64,202,408),(51,409,203,65),(52,66,204,410),(53,411,205,67),(54,68,206,412),(55,413,207,69),(56,70,208,414),(85,433,176,140),(86,113,177,434),(87,435,178,114),(88,115,179,436),(89,437,180,116),(90,117,181,438),(91,439,182,118),(92,119,183,440),(93,441,184,120),(94,121,185,442),(95,443,186,122),(96,123,187,444),(97,445,188,124),(98,125,189,446),(99,447,190,126),(100,127,191,448),(101,421,192,128),(102,129,193,422),(103,423,194,130),(104,131,195,424),(105,425,196,132),(106,133,169,426),(107,427,170,134),(108,135,171,428),(109,429,172,136),(110,137,173,430),(111,431,174,138),(112,139,175,432),(225,321,344,273),(226,274,345,322),(227,323,346,275),(228,276,347,324),(229,325,348,277),(230,278,349,326),(231,327,350,279),(232,280,351,328),(233,329,352,253),(234,254,353,330),(235,331,354,255),(236,256,355,332),(237,333,356,257),(238,258,357,334),(239,335,358,259),(240,260,359,336),(241,309,360,261),(242,262,361,310),(243,311,362,263),(244,264,363,312),(245,313,364,265),(246,266,337,314),(247,315,338,267),(248,268,339,316),(249,317,340,269),(250,270,341,318),(251,319,342,271),(252,272,343,320)], [(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)], [(1,114,377,421),(2,127,378,434),(3,140,379,447),(4,125,380,432),(5,138,381,445),(6,123,382,430),(7,136,383,443),(8,121,384,428),(9,134,385,441),(10,119,386,426),(11,132,387,439),(12,117,388,424),(13,130,389,437),(14,115,390,422),(15,128,391,435),(16,113,392,448),(17,126,365,433),(18,139,366,446),(19,124,367,431),(20,137,368,444),(21,122,369,429),(22,135,370,442),(23,120,371,427),(24,133,372,440),(25,118,373,425),(26,131,374,438),(27,116,375,423),(28,129,376,436),(29,279,223,313),(30,264,224,326),(31,277,197,311),(32,262,198,324),(33,275,199,309),(34,260,200,322),(35,273,201,335),(36,258,202,320),(37,271,203,333),(38,256,204,318),(39,269,205,331),(40,254,206,316),(41,267,207,329),(42,280,208,314),(43,265,209,327),(44,278,210,312),(45,263,211,325),(46,276,212,310),(47,261,213,323),(48,274,214,336),(49,259,215,321),(50,272,216,334),(51,257,217,319),(52,270,218,332),(53,255,219,317),(54,268,220,330),(55,253,221,315),(56,266,222,328),(57,231,415,364),(58,244,416,349),(59,229,417,362),(60,242,418,347),(61,227,419,360),(62,240,420,345),(63,225,393,358),(64,238,394,343),(65,251,395,356),(66,236,396,341),(67,249,397,354),(68,234,398,339),(69,247,399,352),(70,232,400,337),(71,245,401,350),(72,230,402,363),(73,243,403,348),(74,228,404,361),(75,241,405,346),(76,226,406,359),(77,239,407,344),(78,252,408,357),(79,237,409,342),(80,250,410,355),(81,235,411,340),(82,248,412,353),(83,233,413,338),(84,246,414,351),(85,157,190,291),(86,142,191,304),(87,155,192,289),(88,168,193,302),(89,153,194,287),(90,166,195,300),(91,151,196,285),(92,164,169,298),(93,149,170,283),(94,162,171,296),(95,147,172,281),(96,160,173,294),(97,145,174,307),(98,158,175,292),(99,143,176,305),(100,156,177,290),(101,141,178,303),(102,154,179,288),(103,167,180,301),(104,152,181,286),(105,165,182,299),(106,150,183,284),(107,163,184,297),(108,148,185,282),(109,161,186,295),(110,146,187,308),(111,159,188,293),(112,144,189,306)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4G | 4H | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | 2- 1+4 | Q8.10D14 | D4.10D14 |
| kernel | C42.147D14 | C28.6Q8 | Dic7.Q8 | C28.3Q8 | C7×C42.C2 | C42.C2 | C42 | C4⋊C4 | C14 | C2 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 3 | 3 | 18 | 3 | 6 | 12 |
Matrix representation of C42.147D14 ►in GL8(𝔽29)
| 1 | 0 | 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 1 | 11 | 15 | 0 | 0 | 0 | 0 |
| 23 | 13 | 28 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 | 25 | 22 |
| 0 | 0 | 0 | 0 | 18 | 2 | 7 | 4 |
| 0 | 0 | 0 | 0 | 2 | 18 | 2 | 18 |
| 0 | 0 | 0 | 0 | 11 | 27 | 11 | 27 |
| 11 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 27 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 19 | 11 | 2 | 0 | 0 | 0 | 0 |
| 15 | 5 | 22 | 3 | 0 | 0 | 0 | 0 |
| 12 | 18 | 6 | 10 | 0 | 0 | 0 | 0 |
| 18 | 27 | 19 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 15 | 23 | 5 |
| 0 | 0 | 0 | 0 | 14 | 4 | 24 | 9 |
| 0 | 0 | 0 | 0 | 9 | 2 | 12 | 14 |
| 0 | 0 | 0 | 0 | 27 | 15 | 15 | 25 |
| 18 | 1 | 26 | 13 | 0 | 0 | 0 | 0 |
| 7 | 11 | 25 | 5 | 0 | 0 | 0 | 0 |
| 10 | 3 | 14 | 19 | 0 | 0 | 0 | 0 |
| 8 | 23 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 19 | 28 |
| 0 | 0 | 0 | 0 | 6 | 28 | 27 | 10 |
| 0 | 0 | 0 | 0 | 23 | 11 | 28 | 26 |
| 0 | 0 | 0 | 0 | 22 | 6 | 23 | 1 |
G:=sub<GL(8,GF(29))| [1,0,23,16,0,0,0,0,0,1,13,0,0,0,0,0,0,11,28,0,0,0,0,0,18,15,0,28,0,0,0,0,0,0,0,0,27,18,2,11,0,0,0,0,11,2,18,27,0,0,0,0,25,7,2,11,0,0,0,0,22,4,18,27],[11,21,0,0,0,0,0,0,15,18,0,0,0,0,0,0,0,0,20,14,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,28,0,1,0,0,0,0,0,0,28,0,1,0,0,0,0,27,0,1,0,0,0,0,0,0,27,0,1],[0,15,12,18,0,0,0,0,19,5,18,27,0,0,0,0,11,22,6,19,0,0,0,0,2,3,10,18,0,0,0,0,0,0,0,0,17,14,9,27,0,0,0,0,15,4,2,15,0,0,0,0,23,24,12,15,0,0,0,0,5,9,14,25],[18,7,10,8,0,0,0,0,1,11,3,23,0,0,0,0,26,25,14,1,0,0,0,0,13,5,19,15,0,0,0,0,0,0,0,0,1,6,23,22,0,0,0,0,3,28,11,6,0,0,0,0,19,27,28,23,0,0,0,0,28,10,26,1] >;
C42.147D14 in GAP, Magma, Sage, TeX
C_4^2._{147}D_{14} % in TeX
G:=Group("C4^2.147D14"); // GroupNames label
G:=SmallGroup(448,1139);
// by ID
G=gap.SmallGroup(448,1139);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,758,555,100,675,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^13>;
// generators/relations