metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7.5C42, C2.3(D7×C42), C7⋊1(C42⋊4C4), (C4×Dic7)⋊10C4, C14.1(C2×C42), (C22×C4).293D14, C2.C42.17D7, C23.245(C22×D7), C14.16(C42⋊C2), C22.27(D4⋊2D7), C14.C42.32C2, (C22×C14).273C23, (C22×C28).328C22, C22.12(Q8⋊2D7), C2.1(C23.11D14), (C22×Dic7).170C22, C22.29(C2×C4×D7), (C2×C4).120(C4×D7), (C2×C4×Dic7).21C2, (C2×C28).137(C2×C4), C2.1(C4⋊C4⋊7D7), (C2×C14).39(C22×C4), (C2×Dic7).74(C2×C4), (C2×C14).122(C4○D4), (C7×C2.C42).19C2, SmallGroup(448,182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7.5C42
G = < a,b,c,d | a14=c4=d4=1, b2=a7, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a7c >
Subgroups: 604 in 178 conjugacy classes, 91 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C22×C4, C22×C4, Dic7, Dic7, C28, C2×C14, C2×C14, C2.C42, C2.C42, C2×C42, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C42⋊4C4, C4×Dic7, C22×Dic7, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, Dic7.5C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, C4○D4, D14, C2×C42, C42⋊C2, C4×D7, C22×D7, C42⋊4C4, C2×C4×D7, D4⋊2D7, Q8⋊2D7, D7×C42, C23.11D14, C4⋊C4⋊7D7, Dic7.5C42
►Regular action on 448 points
Generators in S448
(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)
(1 232 8 225)(2 231 9 238)(3 230 10 237)(4 229 11 236)(5 228 12 235)(6 227 13 234)(7 226 14 233)(15 250 22 243)(16 249 23 242)(17 248 24 241)(18 247 25 240)(19 246 26 239)(20 245 27 252)(21 244 28 251)(29 263 36 256)(30 262 37 255)(31 261 38 254)(32 260 39 253)(33 259 40 266)(34 258 41 265)(35 257 42 264)(43 268 50 275)(44 267 51 274)(45 280 52 273)(46 279 53 272)(47 278 54 271)(48 277 55 270)(49 276 56 269)(57 281 64 288)(58 294 65 287)(59 293 66 286)(60 292 67 285)(61 291 68 284)(62 290 69 283)(63 289 70 282)(71 300 78 307)(72 299 79 306)(73 298 80 305)(74 297 81 304)(75 296 82 303)(76 295 83 302)(77 308 84 301)(85 313 92 320)(86 312 93 319)(87 311 94 318)(88 310 95 317)(89 309 96 316)(90 322 97 315)(91 321 98 314)(99 334 106 327)(100 333 107 326)(101 332 108 325)(102 331 109 324)(103 330 110 323)(104 329 111 336)(105 328 112 335)(113 338 120 345)(114 337 121 344)(115 350 122 343)(116 349 123 342)(117 348 124 341)(118 347 125 340)(119 346 126 339)(127 354 134 361)(128 353 135 360)(129 352 136 359)(130 351 137 358)(131 364 138 357)(132 363 139 356)(133 362 140 355)(141 365 148 372)(142 378 149 371)(143 377 150 370)(144 376 151 369)(145 375 152 368)(146 374 153 367)(147 373 154 366)(155 388 162 381)(156 387 163 380)(157 386 164 379)(158 385 165 392)(159 384 166 391)(160 383 167 390)(161 382 168 389)(169 396 176 403)(170 395 177 402)(171 394 178 401)(172 393 179 400)(173 406 180 399)(174 405 181 398)(175 404 182 397)(183 412 190 419)(184 411 191 418)(185 410 192 417)(186 409 193 416)(187 408 194 415)(188 407 195 414)(189 420 196 413)(197 425 204 432)(198 424 205 431)(199 423 206 430)(200 422 207 429)(201 421 208 428)(202 434 209 427)(203 433 210 426)(211 439 218 446)(212 438 219 445)(213 437 220 444)(214 436 221 443)(215 435 222 442)(216 448 223 441)(217 447 224 440)
(1 130 19 121)(2 131 20 122)(3 132 21 123)(4 133 22 124)(5 134 23 125)(6 135 24 126)(7 136 25 113)(8 137 26 114)(9 138 27 115)(10 139 28 116)(11 140 15 117)(12 127 16 118)(13 128 17 119)(14 129 18 120)(29 161 48 152)(30 162 49 153)(31 163 50 154)(32 164 51 141)(33 165 52 142)(34 166 53 143)(35 167 54 144)(36 168 55 145)(37 155 56 146)(38 156 43 147)(39 157 44 148)(40 158 45 149)(41 159 46 150)(42 160 47 151)(57 195 83 172)(58 196 84 173)(59 183 71 174)(60 184 72 175)(61 185 73 176)(62 186 74 177)(63 187 75 178)(64 188 76 179)(65 189 77 180)(66 190 78 181)(67 191 79 182)(68 192 80 169)(69 193 81 170)(70 194 82 171)(85 211 99 197)(86 212 100 198)(87 213 101 199)(88 214 102 200)(89 215 103 201)(90 216 104 202)(91 217 105 203)(92 218 106 204)(93 219 107 205)(94 220 108 206)(95 221 109 207)(96 222 110 208)(97 223 111 209)(98 224 112 210)(225 358 239 337)(226 359 240 338)(227 360 241 339)(228 361 242 340)(229 362 243 341)(230 363 244 342)(231 364 245 343)(232 351 246 344)(233 352 247 345)(234 353 248 346)(235 354 249 347)(236 355 250 348)(237 356 251 349)(238 357 252 350)(253 386 267 372)(254 387 268 373)(255 388 269 374)(256 389 270 375)(257 390 271 376)(258 391 272 377)(259 392 273 378)(260 379 274 365)(261 380 275 366)(262 381 276 367)(263 382 277 368)(264 383 278 369)(265 384 279 370)(266 385 280 371)(281 414 302 393)(282 415 303 394)(283 416 304 395)(284 417 305 396)(285 418 306 397)(286 419 307 398)(287 420 308 399)(288 407 295 400)(289 408 296 401)(290 409 297 402)(291 410 298 403)(292 411 299 404)(293 412 300 405)(294 413 301 406)(309 435 330 421)(310 436 331 422)(311 437 332 423)(312 438 333 424)(313 439 334 425)(314 440 335 426)(315 441 336 427)(316 442 323 428)(317 443 324 429)(318 444 325 430)(319 445 326 431)(320 446 327 432)(321 447 328 433)(322 448 329 434)
(1 89 32 57)(2 90 33 58)(3 91 34 59)(4 92 35 60)(5 93 36 61)(6 94 37 62)(7 95 38 63)(8 96 39 64)(9 97 40 65)(10 98 41 66)(11 85 42 67)(12 86 29 68)(13 87 30 69)(14 88 31 70)(15 99 47 79)(16 100 48 80)(17 101 49 81)(18 102 50 82)(19 103 51 83)(20 104 52 84)(21 105 53 71)(22 106 54 72)(23 107 55 73)(24 108 56 74)(25 109 43 75)(26 110 44 76)(27 111 45 77)(28 112 46 78)(113 200 147 171)(114 201 148 172)(115 202 149 173)(116 203 150 174)(117 204 151 175)(118 205 152 176)(119 206 153 177)(120 207 154 178)(121 208 141 179)(122 209 142 180)(123 210 143 181)(124 197 144 182)(125 198 145 169)(126 199 146 170)(127 219 161 185)(128 220 162 186)(129 221 163 187)(130 222 164 188)(131 223 165 189)(132 224 166 190)(133 211 167 191)(134 212 168 192)(135 213 155 193)(136 214 156 194)(137 215 157 195)(138 216 158 196)(139 217 159 183)(140 218 160 184)(225 316 253 288)(226 317 254 289)(227 318 255 290)(228 319 256 291)(229 320 257 292)(230 321 258 293)(231 322 259 294)(232 309 260 281)(233 310 261 282)(234 311 262 283)(235 312 263 284)(236 313 264 285)(237 314 265 286)(238 315 266 287)(239 323 267 295)(240 324 268 296)(241 325 269 297)(242 326 270 298)(243 327 271 299)(244 328 272 300)(245 329 273 301)(246 330 274 302)(247 331 275 303)(248 332 276 304)(249 333 277 305)(250 334 278 306)(251 335 279 307)(252 336 280 308)(337 421 372 393)(338 422 373 394)(339 423 374 395)(340 424 375 396)(341 425 376 397)(342 426 377 398)(343 427 378 399)(344 428 365 400)(345 429 366 401)(346 430 367 402)(347 431 368 403)(348 432 369 404)(349 433 370 405)(350 434 371 406)(351 442 379 407)(352 443 380 408)(353 444 381 409)(354 445 382 410)(355 446 383 411)(356 447 384 412)(357 448 385 413)(358 435 386 414)(359 436 387 415)(360 437 388 416)(361 438 389 417)(362 439 390 418)(363 440 391 419)(364 441 392 420)
G:=sub<Sym(448)| (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), (1,232,8,225)(2,231,9,238)(3,230,10,237)(4,229,11,236)(5,228,12,235)(6,227,13,234)(7,226,14,233)(15,250,22,243)(16,249,23,242)(17,248,24,241)(18,247,25,240)(19,246,26,239)(20,245,27,252)(21,244,28,251)(29,263,36,256)(30,262,37,255)(31,261,38,254)(32,260,39,253)(33,259,40,266)(34,258,41,265)(35,257,42,264)(43,268,50,275)(44,267,51,274)(45,280,52,273)(46,279,53,272)(47,278,54,271)(48,277,55,270)(49,276,56,269)(57,281,64,288)(58,294,65,287)(59,293,66,286)(60,292,67,285)(61,291,68,284)(62,290,69,283)(63,289,70,282)(71,300,78,307)(72,299,79,306)(73,298,80,305)(74,297,81,304)(75,296,82,303)(76,295,83,302)(77,308,84,301)(85,313,92,320)(86,312,93,319)(87,311,94,318)(88,310,95,317)(89,309,96,316)(90,322,97,315)(91,321,98,314)(99,334,106,327)(100,333,107,326)(101,332,108,325)(102,331,109,324)(103,330,110,323)(104,329,111,336)(105,328,112,335)(113,338,120,345)(114,337,121,344)(115,350,122,343)(116,349,123,342)(117,348,124,341)(118,347,125,340)(119,346,126,339)(127,354,134,361)(128,353,135,360)(129,352,136,359)(130,351,137,358)(131,364,138,357)(132,363,139,356)(133,362,140,355)(141,365,148,372)(142,378,149,371)(143,377,150,370)(144,376,151,369)(145,375,152,368)(146,374,153,367)(147,373,154,366)(155,388,162,381)(156,387,163,380)(157,386,164,379)(158,385,165,392)(159,384,166,391)(160,383,167,390)(161,382,168,389)(169,396,176,403)(170,395,177,402)(171,394,178,401)(172,393,179,400)(173,406,180,399)(174,405,181,398)(175,404,182,397)(183,412,190,419)(184,411,191,418)(185,410,192,417)(186,409,193,416)(187,408,194,415)(188,407,195,414)(189,420,196,413)(197,425,204,432)(198,424,205,431)(199,423,206,430)(200,422,207,429)(201,421,208,428)(202,434,209,427)(203,433,210,426)(211,439,218,446)(212,438,219,445)(213,437,220,444)(214,436,221,443)(215,435,222,442)(216,448,223,441)(217,447,224,440), (1,130,19,121)(2,131,20,122)(3,132,21,123)(4,133,22,124)(5,134,23,125)(6,135,24,126)(7,136,25,113)(8,137,26,114)(9,138,27,115)(10,139,28,116)(11,140,15,117)(12,127,16,118)(13,128,17,119)(14,129,18,120)(29,161,48,152)(30,162,49,153)(31,163,50,154)(32,164,51,141)(33,165,52,142)(34,166,53,143)(35,167,54,144)(36,168,55,145)(37,155,56,146)(38,156,43,147)(39,157,44,148)(40,158,45,149)(41,159,46,150)(42,160,47,151)(57,195,83,172)(58,196,84,173)(59,183,71,174)(60,184,72,175)(61,185,73,176)(62,186,74,177)(63,187,75,178)(64,188,76,179)(65,189,77,180)(66,190,78,181)(67,191,79,182)(68,192,80,169)(69,193,81,170)(70,194,82,171)(85,211,99,197)(86,212,100,198)(87,213,101,199)(88,214,102,200)(89,215,103,201)(90,216,104,202)(91,217,105,203)(92,218,106,204)(93,219,107,205)(94,220,108,206)(95,221,109,207)(96,222,110,208)(97,223,111,209)(98,224,112,210)(225,358,239,337)(226,359,240,338)(227,360,241,339)(228,361,242,340)(229,362,243,341)(230,363,244,342)(231,364,245,343)(232,351,246,344)(233,352,247,345)(234,353,248,346)(235,354,249,347)(236,355,250,348)(237,356,251,349)(238,357,252,350)(253,386,267,372)(254,387,268,373)(255,388,269,374)(256,389,270,375)(257,390,271,376)(258,391,272,377)(259,392,273,378)(260,379,274,365)(261,380,275,366)(262,381,276,367)(263,382,277,368)(264,383,278,369)(265,384,279,370)(266,385,280,371)(281,414,302,393)(282,415,303,394)(283,416,304,395)(284,417,305,396)(285,418,306,397)(286,419,307,398)(287,420,308,399)(288,407,295,400)(289,408,296,401)(290,409,297,402)(291,410,298,403)(292,411,299,404)(293,412,300,405)(294,413,301,406)(309,435,330,421)(310,436,331,422)(311,437,332,423)(312,438,333,424)(313,439,334,425)(314,440,335,426)(315,441,336,427)(316,442,323,428)(317,443,324,429)(318,444,325,430)(319,445,326,431)(320,446,327,432)(321,447,328,433)(322,448,329,434), (1,89,32,57)(2,90,33,58)(3,91,34,59)(4,92,35,60)(5,93,36,61)(6,94,37,62)(7,95,38,63)(8,96,39,64)(9,97,40,65)(10,98,41,66)(11,85,42,67)(12,86,29,68)(13,87,30,69)(14,88,31,70)(15,99,47,79)(16,100,48,80)(17,101,49,81)(18,102,50,82)(19,103,51,83)(20,104,52,84)(21,105,53,71)(22,106,54,72)(23,107,55,73)(24,108,56,74)(25,109,43,75)(26,110,44,76)(27,111,45,77)(28,112,46,78)(113,200,147,171)(114,201,148,172)(115,202,149,173)(116,203,150,174)(117,204,151,175)(118,205,152,176)(119,206,153,177)(120,207,154,178)(121,208,141,179)(122,209,142,180)(123,210,143,181)(124,197,144,182)(125,198,145,169)(126,199,146,170)(127,219,161,185)(128,220,162,186)(129,221,163,187)(130,222,164,188)(131,223,165,189)(132,224,166,190)(133,211,167,191)(134,212,168,192)(135,213,155,193)(136,214,156,194)(137,215,157,195)(138,216,158,196)(139,217,159,183)(140,218,160,184)(225,316,253,288)(226,317,254,289)(227,318,255,290)(228,319,256,291)(229,320,257,292)(230,321,258,293)(231,322,259,294)(232,309,260,281)(233,310,261,282)(234,311,262,283)(235,312,263,284)(236,313,264,285)(237,314,265,286)(238,315,266,287)(239,323,267,295)(240,324,268,296)(241,325,269,297)(242,326,270,298)(243,327,271,299)(244,328,272,300)(245,329,273,301)(246,330,274,302)(247,331,275,303)(248,332,276,304)(249,333,277,305)(250,334,278,306)(251,335,279,307)(252,336,280,308)(337,421,372,393)(338,422,373,394)(339,423,374,395)(340,424,375,396)(341,425,376,397)(342,426,377,398)(343,427,378,399)(344,428,365,400)(345,429,366,401)(346,430,367,402)(347,431,368,403)(348,432,369,404)(349,433,370,405)(350,434,371,406)(351,442,379,407)(352,443,380,408)(353,444,381,409)(354,445,382,410)(355,446,383,411)(356,447,384,412)(357,448,385,413)(358,435,386,414)(359,436,387,415)(360,437,388,416)(361,438,389,417)(362,439,390,418)(363,440,391,419)(364,441,392,420)>;
G:=Group( (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), (1,232,8,225)(2,231,9,238)(3,230,10,237)(4,229,11,236)(5,228,12,235)(6,227,13,234)(7,226,14,233)(15,250,22,243)(16,249,23,242)(17,248,24,241)(18,247,25,240)(19,246,26,239)(20,245,27,252)(21,244,28,251)(29,263,36,256)(30,262,37,255)(31,261,38,254)(32,260,39,253)(33,259,40,266)(34,258,41,265)(35,257,42,264)(43,268,50,275)(44,267,51,274)(45,280,52,273)(46,279,53,272)(47,278,54,271)(48,277,55,270)(49,276,56,269)(57,281,64,288)(58,294,65,287)(59,293,66,286)(60,292,67,285)(61,291,68,284)(62,290,69,283)(63,289,70,282)(71,300,78,307)(72,299,79,306)(73,298,80,305)(74,297,81,304)(75,296,82,303)(76,295,83,302)(77,308,84,301)(85,313,92,320)(86,312,93,319)(87,311,94,318)(88,310,95,317)(89,309,96,316)(90,322,97,315)(91,321,98,314)(99,334,106,327)(100,333,107,326)(101,332,108,325)(102,331,109,324)(103,330,110,323)(104,329,111,336)(105,328,112,335)(113,338,120,345)(114,337,121,344)(115,350,122,343)(116,349,123,342)(117,348,124,341)(118,347,125,340)(119,346,126,339)(127,354,134,361)(128,353,135,360)(129,352,136,359)(130,351,137,358)(131,364,138,357)(132,363,139,356)(133,362,140,355)(141,365,148,372)(142,378,149,371)(143,377,150,370)(144,376,151,369)(145,375,152,368)(146,374,153,367)(147,373,154,366)(155,388,162,381)(156,387,163,380)(157,386,164,379)(158,385,165,392)(159,384,166,391)(160,383,167,390)(161,382,168,389)(169,396,176,403)(170,395,177,402)(171,394,178,401)(172,393,179,400)(173,406,180,399)(174,405,181,398)(175,404,182,397)(183,412,190,419)(184,411,191,418)(185,410,192,417)(186,409,193,416)(187,408,194,415)(188,407,195,414)(189,420,196,413)(197,425,204,432)(198,424,205,431)(199,423,206,430)(200,422,207,429)(201,421,208,428)(202,434,209,427)(203,433,210,426)(211,439,218,446)(212,438,219,445)(213,437,220,444)(214,436,221,443)(215,435,222,442)(216,448,223,441)(217,447,224,440), (1,130,19,121)(2,131,20,122)(3,132,21,123)(4,133,22,124)(5,134,23,125)(6,135,24,126)(7,136,25,113)(8,137,26,114)(9,138,27,115)(10,139,28,116)(11,140,15,117)(12,127,16,118)(13,128,17,119)(14,129,18,120)(29,161,48,152)(30,162,49,153)(31,163,50,154)(32,164,51,141)(33,165,52,142)(34,166,53,143)(35,167,54,144)(36,168,55,145)(37,155,56,146)(38,156,43,147)(39,157,44,148)(40,158,45,149)(41,159,46,150)(42,160,47,151)(57,195,83,172)(58,196,84,173)(59,183,71,174)(60,184,72,175)(61,185,73,176)(62,186,74,177)(63,187,75,178)(64,188,76,179)(65,189,77,180)(66,190,78,181)(67,191,79,182)(68,192,80,169)(69,193,81,170)(70,194,82,171)(85,211,99,197)(86,212,100,198)(87,213,101,199)(88,214,102,200)(89,215,103,201)(90,216,104,202)(91,217,105,203)(92,218,106,204)(93,219,107,205)(94,220,108,206)(95,221,109,207)(96,222,110,208)(97,223,111,209)(98,224,112,210)(225,358,239,337)(226,359,240,338)(227,360,241,339)(228,361,242,340)(229,362,243,341)(230,363,244,342)(231,364,245,343)(232,351,246,344)(233,352,247,345)(234,353,248,346)(235,354,249,347)(236,355,250,348)(237,356,251,349)(238,357,252,350)(253,386,267,372)(254,387,268,373)(255,388,269,374)(256,389,270,375)(257,390,271,376)(258,391,272,377)(259,392,273,378)(260,379,274,365)(261,380,275,366)(262,381,276,367)(263,382,277,368)(264,383,278,369)(265,384,279,370)(266,385,280,371)(281,414,302,393)(282,415,303,394)(283,416,304,395)(284,417,305,396)(285,418,306,397)(286,419,307,398)(287,420,308,399)(288,407,295,400)(289,408,296,401)(290,409,297,402)(291,410,298,403)(292,411,299,404)(293,412,300,405)(294,413,301,406)(309,435,330,421)(310,436,331,422)(311,437,332,423)(312,438,333,424)(313,439,334,425)(314,440,335,426)(315,441,336,427)(316,442,323,428)(317,443,324,429)(318,444,325,430)(319,445,326,431)(320,446,327,432)(321,447,328,433)(322,448,329,434), (1,89,32,57)(2,90,33,58)(3,91,34,59)(4,92,35,60)(5,93,36,61)(6,94,37,62)(7,95,38,63)(8,96,39,64)(9,97,40,65)(10,98,41,66)(11,85,42,67)(12,86,29,68)(13,87,30,69)(14,88,31,70)(15,99,47,79)(16,100,48,80)(17,101,49,81)(18,102,50,82)(19,103,51,83)(20,104,52,84)(21,105,53,71)(22,106,54,72)(23,107,55,73)(24,108,56,74)(25,109,43,75)(26,110,44,76)(27,111,45,77)(28,112,46,78)(113,200,147,171)(114,201,148,172)(115,202,149,173)(116,203,150,174)(117,204,151,175)(118,205,152,176)(119,206,153,177)(120,207,154,178)(121,208,141,179)(122,209,142,180)(123,210,143,181)(124,197,144,182)(125,198,145,169)(126,199,146,170)(127,219,161,185)(128,220,162,186)(129,221,163,187)(130,222,164,188)(131,223,165,189)(132,224,166,190)(133,211,167,191)(134,212,168,192)(135,213,155,193)(136,214,156,194)(137,215,157,195)(138,216,158,196)(139,217,159,183)(140,218,160,184)(225,316,253,288)(226,317,254,289)(227,318,255,290)(228,319,256,291)(229,320,257,292)(230,321,258,293)(231,322,259,294)(232,309,260,281)(233,310,261,282)(234,311,262,283)(235,312,263,284)(236,313,264,285)(237,314,265,286)(238,315,266,287)(239,323,267,295)(240,324,268,296)(241,325,269,297)(242,326,270,298)(243,327,271,299)(244,328,272,300)(245,329,273,301)(246,330,274,302)(247,331,275,303)(248,332,276,304)(249,333,277,305)(250,334,278,306)(251,335,279,307)(252,336,280,308)(337,421,372,393)(338,422,373,394)(339,423,374,395)(340,424,375,396)(341,425,376,397)(342,426,377,398)(343,427,378,399)(344,428,365,400)(345,429,366,401)(346,430,367,402)(347,431,368,403)(348,432,369,404)(349,433,370,405)(350,434,371,406)(351,442,379,407)(352,443,380,408)(353,444,381,409)(354,445,382,410)(355,446,383,411)(356,447,384,412)(357,448,385,413)(358,435,386,414)(359,436,387,415)(360,437,388,416)(361,438,389,417)(362,439,390,418)(363,440,391,419)(364,441,392,420) );
G=PermutationGroup([[(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)], [(1,232,8,225),(2,231,9,238),(3,230,10,237),(4,229,11,236),(5,228,12,235),(6,227,13,234),(7,226,14,233),(15,250,22,243),(16,249,23,242),(17,248,24,241),(18,247,25,240),(19,246,26,239),(20,245,27,252),(21,244,28,251),(29,263,36,256),(30,262,37,255),(31,261,38,254),(32,260,39,253),(33,259,40,266),(34,258,41,265),(35,257,42,264),(43,268,50,275),(44,267,51,274),(45,280,52,273),(46,279,53,272),(47,278,54,271),(48,277,55,270),(49,276,56,269),(57,281,64,288),(58,294,65,287),(59,293,66,286),(60,292,67,285),(61,291,68,284),(62,290,69,283),(63,289,70,282),(71,300,78,307),(72,299,79,306),(73,298,80,305),(74,297,81,304),(75,296,82,303),(76,295,83,302),(77,308,84,301),(85,313,92,320),(86,312,93,319),(87,311,94,318),(88,310,95,317),(89,309,96,316),(90,322,97,315),(91,321,98,314),(99,334,106,327),(100,333,107,326),(101,332,108,325),(102,331,109,324),(103,330,110,323),(104,329,111,336),(105,328,112,335),(113,338,120,345),(114,337,121,344),(115,350,122,343),(116,349,123,342),(117,348,124,341),(118,347,125,340),(119,346,126,339),(127,354,134,361),(128,353,135,360),(129,352,136,359),(130,351,137,358),(131,364,138,357),(132,363,139,356),(133,362,140,355),(141,365,148,372),(142,378,149,371),(143,377,150,370),(144,376,151,369),(145,375,152,368),(146,374,153,367),(147,373,154,366),(155,388,162,381),(156,387,163,380),(157,386,164,379),(158,385,165,392),(159,384,166,391),(160,383,167,390),(161,382,168,389),(169,396,176,403),(170,395,177,402),(171,394,178,401),(172,393,179,400),(173,406,180,399),(174,405,181,398),(175,404,182,397),(183,412,190,419),(184,411,191,418),(185,410,192,417),(186,409,193,416),(187,408,194,415),(188,407,195,414),(189,420,196,413),(197,425,204,432),(198,424,205,431),(199,423,206,430),(200,422,207,429),(201,421,208,428),(202,434,209,427),(203,433,210,426),(211,439,218,446),(212,438,219,445),(213,437,220,444),(214,436,221,443),(215,435,222,442),(216,448,223,441),(217,447,224,440)], [(1,130,19,121),(2,131,20,122),(3,132,21,123),(4,133,22,124),(5,134,23,125),(6,135,24,126),(7,136,25,113),(8,137,26,114),(9,138,27,115),(10,139,28,116),(11,140,15,117),(12,127,16,118),(13,128,17,119),(14,129,18,120),(29,161,48,152),(30,162,49,153),(31,163,50,154),(32,164,51,141),(33,165,52,142),(34,166,53,143),(35,167,54,144),(36,168,55,145),(37,155,56,146),(38,156,43,147),(39,157,44,148),(40,158,45,149),(41,159,46,150),(42,160,47,151),(57,195,83,172),(58,196,84,173),(59,183,71,174),(60,184,72,175),(61,185,73,176),(62,186,74,177),(63,187,75,178),(64,188,76,179),(65,189,77,180),(66,190,78,181),(67,191,79,182),(68,192,80,169),(69,193,81,170),(70,194,82,171),(85,211,99,197),(86,212,100,198),(87,213,101,199),(88,214,102,200),(89,215,103,201),(90,216,104,202),(91,217,105,203),(92,218,106,204),(93,219,107,205),(94,220,108,206),(95,221,109,207),(96,222,110,208),(97,223,111,209),(98,224,112,210),(225,358,239,337),(226,359,240,338),(227,360,241,339),(228,361,242,340),(229,362,243,341),(230,363,244,342),(231,364,245,343),(232,351,246,344),(233,352,247,345),(234,353,248,346),(235,354,249,347),(236,355,250,348),(237,356,251,349),(238,357,252,350),(253,386,267,372),(254,387,268,373),(255,388,269,374),(256,389,270,375),(257,390,271,376),(258,391,272,377),(259,392,273,378),(260,379,274,365),(261,380,275,366),(262,381,276,367),(263,382,277,368),(264,383,278,369),(265,384,279,370),(266,385,280,371),(281,414,302,393),(282,415,303,394),(283,416,304,395),(284,417,305,396),(285,418,306,397),(286,419,307,398),(287,420,308,399),(288,407,295,400),(289,408,296,401),(290,409,297,402),(291,410,298,403),(292,411,299,404),(293,412,300,405),(294,413,301,406),(309,435,330,421),(310,436,331,422),(311,437,332,423),(312,438,333,424),(313,439,334,425),(314,440,335,426),(315,441,336,427),(316,442,323,428),(317,443,324,429),(318,444,325,430),(319,445,326,431),(320,446,327,432),(321,447,328,433),(322,448,329,434)], [(1,89,32,57),(2,90,33,58),(3,91,34,59),(4,92,35,60),(5,93,36,61),(6,94,37,62),(7,95,38,63),(8,96,39,64),(9,97,40,65),(10,98,41,66),(11,85,42,67),(12,86,29,68),(13,87,30,69),(14,88,31,70),(15,99,47,79),(16,100,48,80),(17,101,49,81),(18,102,50,82),(19,103,51,83),(20,104,52,84),(21,105,53,71),(22,106,54,72),(23,107,55,73),(24,108,56,74),(25,109,43,75),(26,110,44,76),(27,111,45,77),(28,112,46,78),(113,200,147,171),(114,201,148,172),(115,202,149,173),(116,203,150,174),(117,204,151,175),(118,205,152,176),(119,206,153,177),(120,207,154,178),(121,208,141,179),(122,209,142,180),(123,210,143,181),(124,197,144,182),(125,198,145,169),(126,199,146,170),(127,219,161,185),(128,220,162,186),(129,221,163,187),(130,222,164,188),(131,223,165,189),(132,224,166,190),(133,211,167,191),(134,212,168,192),(135,213,155,193),(136,214,156,194),(137,215,157,195),(138,216,158,196),(139,217,159,183),(140,218,160,184),(225,316,253,288),(226,317,254,289),(227,318,255,290),(228,319,256,291),(229,320,257,292),(230,321,258,293),(231,322,259,294),(232,309,260,281),(233,310,261,282),(234,311,262,283),(235,312,263,284),(236,313,264,285),(237,314,265,286),(238,315,266,287),(239,323,267,295),(240,324,268,296),(241,325,269,297),(242,326,270,298),(243,327,271,299),(244,328,272,300),(245,329,273,301),(246,330,274,302),(247,331,275,303),(248,332,276,304),(249,333,277,305),(250,334,278,306),(251,335,279,307),(252,336,280,308),(337,421,372,393),(338,422,373,394),(339,423,374,395),(340,424,375,396),(341,425,376,397),(342,426,377,398),(343,427,378,399),(344,428,365,400),(345,429,366,401),(346,430,367,402),(347,431,368,403),(348,432,369,404),(349,433,370,405),(350,434,371,406),(351,442,379,407),(352,443,380,408),(353,444,381,409),(354,445,382,410),(355,446,383,411),(356,447,384,412),(357,448,385,413),(358,435,386,414),(359,436,387,415),(360,437,388,416),(361,438,389,417),(362,439,390,418),(363,440,391,419),(364,441,392,420)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AF | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D7 | C4○D4 | D14 | C4×D7 | D4⋊2D7 | Q8⋊2D7 |
| kernel | Dic7.5C42 | C14.C42 | C7×C2.C42 | C2×C4×Dic7 | C4×Dic7 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 3 | 1 | 3 | 24 | 3 | 8 | 9 | 36 | 9 | 3 |
Matrix representation of Dic7.5C42 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 28 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 17 |
| 0 | 0 | 0 | 24 | 17 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 18 | 27 | 0 | 0 |
| 0 | 2 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,1,3],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,24,0,0,0,17,17],[17,0,0,0,0,0,18,2,0,0,0,27,11,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12] >;
Dic7.5C42 in GAP, Magma, Sage, TeX
{\rm Dic}_7._5C_4^2 % in TeX
G:=Group("Dic7.5C4^2"); // GroupNames label
G:=SmallGroup(448,182);
// by ID
G=gap.SmallGroup(448,182);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^4=d^4=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^7*c>;
// generators/relations