metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊5Dic7, C14.88(C4×D4), C14.24(C4×Q8), C2.6(D4×Dic7), C2.3(Q8×Dic7), C22.24(Q8×D7), (C2×Dic7).18Q8, C22.108(D4×D7), C2.6(D14⋊Q8), (C2×Dic7).176D4, (C22×C4).308D14, C2.7(Dic7.Q8), C14.46(C22⋊Q8), C2.5(D14.5D4), C22.57(C4○D28), C14.20(C42.C2), C23.293(C22×D7), C14.25(C42⋊2C2), C14.45(C42⋊C2), C22.57(D4⋊2D7), C14.C42.27C2, (C22×C14).345C23, (C22×C28).391C22, C7⋊7(C23.63C23), C22.25(Q8⋊2D7), C22.41(C22×Dic7), C14.49(C22.D4), C2.9(C23.21D14), (C22×Dic7).190C22, (C7×C4⋊C4)⋊8C4, (C2×C4⋊C4).18D7, (C14×C4⋊C4).14C2, (C2×C14).80(C2×Q8), (C2×C4×Dic7).36C2, (C2×C28).185(C2×C4), C2.6(C4⋊C4⋊D7), (C2×C14).331(C2×D4), (C2×C4⋊Dic7).17C2, (C2×C4).17(C2×Dic7), (C2×C14).83(C4○D4), (C2×C14).180(C22×C4), SmallGroup(448,515)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊5Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, bab-1=a-1, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 580 in 154 conjugacy classes, 75 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.63C23, C4×Dic7, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×C28, C14.C42, C2×C4×Dic7, C2×C4⋊Dic7, C14×C4⋊C4, C4⋊C4⋊5Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, D14, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×Dic7, C22×D7, C23.63C23, C4○D28, D4×D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C22×Dic7, Dic7.Q8, D14.5D4, D14⋊Q8, C4⋊C4⋊D7, C23.21D14, D4×Dic7, Q8×Dic7, C4⋊C4⋊5Dic7
►Regular action on 448 points
Generators in S448
(1 351 72 216)(2 352 73 217)(3 353 74 218)(4 354 75 219)(5 355 76 220)(6 356 77 221)(7 357 78 222)(8 358 79 223)(9 359 80 224)(10 360 81 211)(11 361 82 212)(12 362 83 213)(13 363 84 214)(14 364 71 215)(15 49 226 337)(16 50 227 338)(17 51 228 339)(18 52 229 340)(19 53 230 341)(20 54 231 342)(21 55 232 343)(22 56 233 344)(23 43 234 345)(24 44 235 346)(25 45 236 347)(26 46 237 348)(27 47 238 349)(28 48 225 350)(29 394 62 332)(30 395 63 333)(31 396 64 334)(32 397 65 335)(33 398 66 336)(34 399 67 323)(35 400 68 324)(36 401 69 325)(37 402 70 326)(38 403 57 327)(39 404 58 328)(40 405 59 329)(41 406 60 330)(42 393 61 331)(85 437 414 380)(86 438 415 381)(87 439 416 382)(88 440 417 383)(89 441 418 384)(90 442 419 385)(91 443 420 386)(92 444 407 387)(93 445 408 388)(94 446 409 389)(95 447 410 390)(96 448 411 391)(97 435 412 392)(98 436 413 379)(99 281 196 153)(100 282 183 154)(101 283 184 141)(102 284 185 142)(103 285 186 143)(104 286 187 144)(105 287 188 145)(106 288 189 146)(107 289 190 147)(108 290 191 148)(109 291 192 149)(110 292 193 150)(111 293 194 151)(112 294 195 152)(113 305 169 128)(114 306 170 129)(115 307 171 130)(116 308 172 131)(117 295 173 132)(118 296 174 133)(119 297 175 134)(120 298 176 135)(121 299 177 136)(122 300 178 137)(123 301 179 138)(124 302 180 139)(125 303 181 140)(126 304 182 127)(155 204 246 265)(156 205 247 266)(157 206 248 253)(158 207 249 254)(159 208 250 255)(160 209 251 256)(161 210 252 257)(162 197 239 258)(163 198 240 259)(164 199 241 260)(165 200 242 261)(166 201 243 262)(167 202 244 263)(168 203 245 264)(267 424 372 316)(268 425 373 317)(269 426 374 318)(270 427 375 319)(271 428 376 320)(272 429 377 321)(273 430 378 322)(274 431 365 309)(275 432 366 310)(276 433 367 311)(277 434 368 312)(278 421 369 313)(279 422 370 314)(280 423 371 315)
(1 263 55 389)(2 264 56 390)(3 265 43 391)(4 266 44 392)(5 253 45 379)(6 254 46 380)(7 255 47 381)(8 256 48 382)(9 257 49 383)(10 258 50 384)(11 259 51 385)(12 260 52 386)(13 261 53 387)(14 262 54 388)(15 88 224 161)(16 89 211 162)(17 90 212 163)(18 91 213 164)(19 92 214 165)(20 93 215 166)(21 94 216 167)(22 95 217 168)(23 96 218 155)(24 97 219 156)(25 98 220 157)(26 85 221 158)(27 86 222 159)(28 87 223 160)(29 270 174 289)(30 271 175 290)(31 272 176 291)(32 273 177 292)(33 274 178 293)(34 275 179 294)(35 276 180 281)(36 277 181 282)(37 278 182 283)(38 279 169 284)(39 280 170 285)(40 267 171 286)(41 268 172 287)(42 269 173 288)(57 370 113 142)(58 371 114 143)(59 372 115 144)(60 373 116 145)(61 374 117 146)(62 375 118 147)(63 376 119 148)(64 377 120 149)(65 378 121 150)(66 365 122 151)(67 366 123 152)(68 367 124 153)(69 368 125 154)(70 369 126 141)(71 201 342 445)(72 202 343 446)(73 203 344 447)(74 204 345 448)(75 205 346 435)(76 206 347 436)(77 207 348 437)(78 208 349 438)(79 209 350 439)(80 210 337 440)(81 197 338 441)(82 198 339 442)(83 199 340 443)(84 200 341 444)(99 400 311 139)(100 401 312 140)(101 402 313 127)(102 403 314 128)(103 404 315 129)(104 405 316 130)(105 406 317 131)(106 393 318 132)(107 394 319 133)(108 395 320 134)(109 396 321 135)(110 397 322 136)(111 398 309 137)(112 399 310 138)(183 325 434 303)(184 326 421 304)(185 327 422 305)(186 328 423 306)(187 329 424 307)(188 330 425 308)(189 331 426 295)(190 332 427 296)(191 333 428 297)(192 334 429 298)(193 335 430 299)(194 336 431 300)(195 323 432 301)(196 324 433 302)(225 416 358 251)(226 417 359 252)(227 418 360 239)(228 419 361 240)(229 420 362 241)(230 407 363 242)(231 408 364 243)(232 409 351 244)(233 410 352 245)(234 411 353 246)(235 412 354 247)(236 413 355 248)(237 414 356 249)(238 415 357 250)
(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)
(1 288 8 281)(2 287 9 294)(3 286 10 293)(4 285 11 292)(5 284 12 291)(6 283 13 290)(7 282 14 289)(15 112 22 105)(16 111 23 104)(17 110 24 103)(18 109 25 102)(19 108 26 101)(20 107 27 100)(21 106 28 99)(29 255 36 262)(30 254 37 261)(31 253 38 260)(32 266 39 259)(33 265 40 258)(34 264 41 257)(35 263 42 256)(43 267 50 274)(44 280 51 273)(45 279 52 272)(46 278 53 271)(47 277 54 270)(48 276 55 269)(49 275 56 268)(57 199 64 206)(58 198 65 205)(59 197 66 204)(60 210 67 203)(61 209 68 202)(62 208 69 201)(63 207 70 200)(71 147 78 154)(72 146 79 153)(73 145 80 152)(74 144 81 151)(75 143 82 150)(76 142 83 149)(77 141 84 148)(85 402 92 395)(86 401 93 394)(87 400 94 393)(88 399 95 406)(89 398 96 405)(90 397 97 404)(91 396 98 403)(113 443 120 436)(114 442 121 435)(115 441 122 448)(116 440 123 447)(117 439 124 446)(118 438 125 445)(119 437 126 444)(127 165 134 158)(128 164 135 157)(129 163 136 156)(130 162 137 155)(131 161 138 168)(132 160 139 167)(133 159 140 166)(169 386 176 379)(170 385 177 392)(171 384 178 391)(172 383 179 390)(173 382 180 389)(174 381 181 388)(175 380 182 387)(183 231 190 238)(184 230 191 237)(185 229 192 236)(186 228 193 235)(187 227 194 234)(188 226 195 233)(189 225 196 232)(211 309 218 316)(212 322 219 315)(213 321 220 314)(214 320 221 313)(215 319 222 312)(216 318 223 311)(217 317 224 310)(239 300 246 307)(240 299 247 306)(241 298 248 305)(242 297 249 304)(243 296 250 303)(244 295 251 302)(245 308 252 301)(323 410 330 417)(324 409 331 416)(325 408 332 415)(326 407 333 414)(327 420 334 413)(328 419 335 412)(329 418 336 411)(337 366 344 373)(338 365 345 372)(339 378 346 371)(340 377 347 370)(341 376 348 369)(342 375 349 368)(343 374 350 367)(351 426 358 433)(352 425 359 432)(353 424 360 431)(354 423 361 430)(355 422 362 429)(356 421 363 428)(357 434 364 427)
G:=sub<Sym(448)| (1,351,72,216)(2,352,73,217)(3,353,74,218)(4,354,75,219)(5,355,76,220)(6,356,77,221)(7,357,78,222)(8,358,79,223)(9,359,80,224)(10,360,81,211)(11,361,82,212)(12,362,83,213)(13,363,84,214)(14,364,71,215)(15,49,226,337)(16,50,227,338)(17,51,228,339)(18,52,229,340)(19,53,230,341)(20,54,231,342)(21,55,232,343)(22,56,233,344)(23,43,234,345)(24,44,235,346)(25,45,236,347)(26,46,237,348)(27,47,238,349)(28,48,225,350)(29,394,62,332)(30,395,63,333)(31,396,64,334)(32,397,65,335)(33,398,66,336)(34,399,67,323)(35,400,68,324)(36,401,69,325)(37,402,70,326)(38,403,57,327)(39,404,58,328)(40,405,59,329)(41,406,60,330)(42,393,61,331)(85,437,414,380)(86,438,415,381)(87,439,416,382)(88,440,417,383)(89,441,418,384)(90,442,419,385)(91,443,420,386)(92,444,407,387)(93,445,408,388)(94,446,409,389)(95,447,410,390)(96,448,411,391)(97,435,412,392)(98,436,413,379)(99,281,196,153)(100,282,183,154)(101,283,184,141)(102,284,185,142)(103,285,186,143)(104,286,187,144)(105,287,188,145)(106,288,189,146)(107,289,190,147)(108,290,191,148)(109,291,192,149)(110,292,193,150)(111,293,194,151)(112,294,195,152)(113,305,169,128)(114,306,170,129)(115,307,171,130)(116,308,172,131)(117,295,173,132)(118,296,174,133)(119,297,175,134)(120,298,176,135)(121,299,177,136)(122,300,178,137)(123,301,179,138)(124,302,180,139)(125,303,181,140)(126,304,182,127)(155,204,246,265)(156,205,247,266)(157,206,248,253)(158,207,249,254)(159,208,250,255)(160,209,251,256)(161,210,252,257)(162,197,239,258)(163,198,240,259)(164,199,241,260)(165,200,242,261)(166,201,243,262)(167,202,244,263)(168,203,245,264)(267,424,372,316)(268,425,373,317)(269,426,374,318)(270,427,375,319)(271,428,376,320)(272,429,377,321)(273,430,378,322)(274,431,365,309)(275,432,366,310)(276,433,367,311)(277,434,368,312)(278,421,369,313)(279,422,370,314)(280,423,371,315), (1,263,55,389)(2,264,56,390)(3,265,43,391)(4,266,44,392)(5,253,45,379)(6,254,46,380)(7,255,47,381)(8,256,48,382)(9,257,49,383)(10,258,50,384)(11,259,51,385)(12,260,52,386)(13,261,53,387)(14,262,54,388)(15,88,224,161)(16,89,211,162)(17,90,212,163)(18,91,213,164)(19,92,214,165)(20,93,215,166)(21,94,216,167)(22,95,217,168)(23,96,218,155)(24,97,219,156)(25,98,220,157)(26,85,221,158)(27,86,222,159)(28,87,223,160)(29,270,174,289)(30,271,175,290)(31,272,176,291)(32,273,177,292)(33,274,178,293)(34,275,179,294)(35,276,180,281)(36,277,181,282)(37,278,182,283)(38,279,169,284)(39,280,170,285)(40,267,171,286)(41,268,172,287)(42,269,173,288)(57,370,113,142)(58,371,114,143)(59,372,115,144)(60,373,116,145)(61,374,117,146)(62,375,118,147)(63,376,119,148)(64,377,120,149)(65,378,121,150)(66,365,122,151)(67,366,123,152)(68,367,124,153)(69,368,125,154)(70,369,126,141)(71,201,342,445)(72,202,343,446)(73,203,344,447)(74,204,345,448)(75,205,346,435)(76,206,347,436)(77,207,348,437)(78,208,349,438)(79,209,350,439)(80,210,337,440)(81,197,338,441)(82,198,339,442)(83,199,340,443)(84,200,341,444)(99,400,311,139)(100,401,312,140)(101,402,313,127)(102,403,314,128)(103,404,315,129)(104,405,316,130)(105,406,317,131)(106,393,318,132)(107,394,319,133)(108,395,320,134)(109,396,321,135)(110,397,322,136)(111,398,309,137)(112,399,310,138)(183,325,434,303)(184,326,421,304)(185,327,422,305)(186,328,423,306)(187,329,424,307)(188,330,425,308)(189,331,426,295)(190,332,427,296)(191,333,428,297)(192,334,429,298)(193,335,430,299)(194,336,431,300)(195,323,432,301)(196,324,433,302)(225,416,358,251)(226,417,359,252)(227,418,360,239)(228,419,361,240)(229,420,362,241)(230,407,363,242)(231,408,364,243)(232,409,351,244)(233,410,352,245)(234,411,353,246)(235,412,354,247)(236,413,355,248)(237,414,356,249)(238,415,357,250), (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), (1,288,8,281)(2,287,9,294)(3,286,10,293)(4,285,11,292)(5,284,12,291)(6,283,13,290)(7,282,14,289)(15,112,22,105)(16,111,23,104)(17,110,24,103)(18,109,25,102)(19,108,26,101)(20,107,27,100)(21,106,28,99)(29,255,36,262)(30,254,37,261)(31,253,38,260)(32,266,39,259)(33,265,40,258)(34,264,41,257)(35,263,42,256)(43,267,50,274)(44,280,51,273)(45,279,52,272)(46,278,53,271)(47,277,54,270)(48,276,55,269)(49,275,56,268)(57,199,64,206)(58,198,65,205)(59,197,66,204)(60,210,67,203)(61,209,68,202)(62,208,69,201)(63,207,70,200)(71,147,78,154)(72,146,79,153)(73,145,80,152)(74,144,81,151)(75,143,82,150)(76,142,83,149)(77,141,84,148)(85,402,92,395)(86,401,93,394)(87,400,94,393)(88,399,95,406)(89,398,96,405)(90,397,97,404)(91,396,98,403)(113,443,120,436)(114,442,121,435)(115,441,122,448)(116,440,123,447)(117,439,124,446)(118,438,125,445)(119,437,126,444)(127,165,134,158)(128,164,135,157)(129,163,136,156)(130,162,137,155)(131,161,138,168)(132,160,139,167)(133,159,140,166)(169,386,176,379)(170,385,177,392)(171,384,178,391)(172,383,179,390)(173,382,180,389)(174,381,181,388)(175,380,182,387)(183,231,190,238)(184,230,191,237)(185,229,192,236)(186,228,193,235)(187,227,194,234)(188,226,195,233)(189,225,196,232)(211,309,218,316)(212,322,219,315)(213,321,220,314)(214,320,221,313)(215,319,222,312)(216,318,223,311)(217,317,224,310)(239,300,246,307)(240,299,247,306)(241,298,248,305)(242,297,249,304)(243,296,250,303)(244,295,251,302)(245,308,252,301)(323,410,330,417)(324,409,331,416)(325,408,332,415)(326,407,333,414)(327,420,334,413)(328,419,335,412)(329,418,336,411)(337,366,344,373)(338,365,345,372)(339,378,346,371)(340,377,347,370)(341,376,348,369)(342,375,349,368)(343,374,350,367)(351,426,358,433)(352,425,359,432)(353,424,360,431)(354,423,361,430)(355,422,362,429)(356,421,363,428)(357,434,364,427)>;
G:=Group( (1,351,72,216)(2,352,73,217)(3,353,74,218)(4,354,75,219)(5,355,76,220)(6,356,77,221)(7,357,78,222)(8,358,79,223)(9,359,80,224)(10,360,81,211)(11,361,82,212)(12,362,83,213)(13,363,84,214)(14,364,71,215)(15,49,226,337)(16,50,227,338)(17,51,228,339)(18,52,229,340)(19,53,230,341)(20,54,231,342)(21,55,232,343)(22,56,233,344)(23,43,234,345)(24,44,235,346)(25,45,236,347)(26,46,237,348)(27,47,238,349)(28,48,225,350)(29,394,62,332)(30,395,63,333)(31,396,64,334)(32,397,65,335)(33,398,66,336)(34,399,67,323)(35,400,68,324)(36,401,69,325)(37,402,70,326)(38,403,57,327)(39,404,58,328)(40,405,59,329)(41,406,60,330)(42,393,61,331)(85,437,414,380)(86,438,415,381)(87,439,416,382)(88,440,417,383)(89,441,418,384)(90,442,419,385)(91,443,420,386)(92,444,407,387)(93,445,408,388)(94,446,409,389)(95,447,410,390)(96,448,411,391)(97,435,412,392)(98,436,413,379)(99,281,196,153)(100,282,183,154)(101,283,184,141)(102,284,185,142)(103,285,186,143)(104,286,187,144)(105,287,188,145)(106,288,189,146)(107,289,190,147)(108,290,191,148)(109,291,192,149)(110,292,193,150)(111,293,194,151)(112,294,195,152)(113,305,169,128)(114,306,170,129)(115,307,171,130)(116,308,172,131)(117,295,173,132)(118,296,174,133)(119,297,175,134)(120,298,176,135)(121,299,177,136)(122,300,178,137)(123,301,179,138)(124,302,180,139)(125,303,181,140)(126,304,182,127)(155,204,246,265)(156,205,247,266)(157,206,248,253)(158,207,249,254)(159,208,250,255)(160,209,251,256)(161,210,252,257)(162,197,239,258)(163,198,240,259)(164,199,241,260)(165,200,242,261)(166,201,243,262)(167,202,244,263)(168,203,245,264)(267,424,372,316)(268,425,373,317)(269,426,374,318)(270,427,375,319)(271,428,376,320)(272,429,377,321)(273,430,378,322)(274,431,365,309)(275,432,366,310)(276,433,367,311)(277,434,368,312)(278,421,369,313)(279,422,370,314)(280,423,371,315), (1,263,55,389)(2,264,56,390)(3,265,43,391)(4,266,44,392)(5,253,45,379)(6,254,46,380)(7,255,47,381)(8,256,48,382)(9,257,49,383)(10,258,50,384)(11,259,51,385)(12,260,52,386)(13,261,53,387)(14,262,54,388)(15,88,224,161)(16,89,211,162)(17,90,212,163)(18,91,213,164)(19,92,214,165)(20,93,215,166)(21,94,216,167)(22,95,217,168)(23,96,218,155)(24,97,219,156)(25,98,220,157)(26,85,221,158)(27,86,222,159)(28,87,223,160)(29,270,174,289)(30,271,175,290)(31,272,176,291)(32,273,177,292)(33,274,178,293)(34,275,179,294)(35,276,180,281)(36,277,181,282)(37,278,182,283)(38,279,169,284)(39,280,170,285)(40,267,171,286)(41,268,172,287)(42,269,173,288)(57,370,113,142)(58,371,114,143)(59,372,115,144)(60,373,116,145)(61,374,117,146)(62,375,118,147)(63,376,119,148)(64,377,120,149)(65,378,121,150)(66,365,122,151)(67,366,123,152)(68,367,124,153)(69,368,125,154)(70,369,126,141)(71,201,342,445)(72,202,343,446)(73,203,344,447)(74,204,345,448)(75,205,346,435)(76,206,347,436)(77,207,348,437)(78,208,349,438)(79,209,350,439)(80,210,337,440)(81,197,338,441)(82,198,339,442)(83,199,340,443)(84,200,341,444)(99,400,311,139)(100,401,312,140)(101,402,313,127)(102,403,314,128)(103,404,315,129)(104,405,316,130)(105,406,317,131)(106,393,318,132)(107,394,319,133)(108,395,320,134)(109,396,321,135)(110,397,322,136)(111,398,309,137)(112,399,310,138)(183,325,434,303)(184,326,421,304)(185,327,422,305)(186,328,423,306)(187,329,424,307)(188,330,425,308)(189,331,426,295)(190,332,427,296)(191,333,428,297)(192,334,429,298)(193,335,430,299)(194,336,431,300)(195,323,432,301)(196,324,433,302)(225,416,358,251)(226,417,359,252)(227,418,360,239)(228,419,361,240)(229,420,362,241)(230,407,363,242)(231,408,364,243)(232,409,351,244)(233,410,352,245)(234,411,353,246)(235,412,354,247)(236,413,355,248)(237,414,356,249)(238,415,357,250), (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), (1,288,8,281)(2,287,9,294)(3,286,10,293)(4,285,11,292)(5,284,12,291)(6,283,13,290)(7,282,14,289)(15,112,22,105)(16,111,23,104)(17,110,24,103)(18,109,25,102)(19,108,26,101)(20,107,27,100)(21,106,28,99)(29,255,36,262)(30,254,37,261)(31,253,38,260)(32,266,39,259)(33,265,40,258)(34,264,41,257)(35,263,42,256)(43,267,50,274)(44,280,51,273)(45,279,52,272)(46,278,53,271)(47,277,54,270)(48,276,55,269)(49,275,56,268)(57,199,64,206)(58,198,65,205)(59,197,66,204)(60,210,67,203)(61,209,68,202)(62,208,69,201)(63,207,70,200)(71,147,78,154)(72,146,79,153)(73,145,80,152)(74,144,81,151)(75,143,82,150)(76,142,83,149)(77,141,84,148)(85,402,92,395)(86,401,93,394)(87,400,94,393)(88,399,95,406)(89,398,96,405)(90,397,97,404)(91,396,98,403)(113,443,120,436)(114,442,121,435)(115,441,122,448)(116,440,123,447)(117,439,124,446)(118,438,125,445)(119,437,126,444)(127,165,134,158)(128,164,135,157)(129,163,136,156)(130,162,137,155)(131,161,138,168)(132,160,139,167)(133,159,140,166)(169,386,176,379)(170,385,177,392)(171,384,178,391)(172,383,179,390)(173,382,180,389)(174,381,181,388)(175,380,182,387)(183,231,190,238)(184,230,191,237)(185,229,192,236)(186,228,193,235)(187,227,194,234)(188,226,195,233)(189,225,196,232)(211,309,218,316)(212,322,219,315)(213,321,220,314)(214,320,221,313)(215,319,222,312)(216,318,223,311)(217,317,224,310)(239,300,246,307)(240,299,247,306)(241,298,248,305)(242,297,249,304)(243,296,250,303)(244,295,251,302)(245,308,252,301)(323,410,330,417)(324,409,331,416)(325,408,332,415)(326,407,333,414)(327,420,334,413)(328,419,335,412)(329,418,336,411)(337,366,344,373)(338,365,345,372)(339,378,346,371)(340,377,347,370)(341,376,348,369)(342,375,349,368)(343,374,350,367)(351,426,358,433)(352,425,359,432)(353,424,360,431)(354,423,361,430)(355,422,362,429)(356,421,363,428)(357,434,364,427) );
G=PermutationGroup([[(1,351,72,216),(2,352,73,217),(3,353,74,218),(4,354,75,219),(5,355,76,220),(6,356,77,221),(7,357,78,222),(8,358,79,223),(9,359,80,224),(10,360,81,211),(11,361,82,212),(12,362,83,213),(13,363,84,214),(14,364,71,215),(15,49,226,337),(16,50,227,338),(17,51,228,339),(18,52,229,340),(19,53,230,341),(20,54,231,342),(21,55,232,343),(22,56,233,344),(23,43,234,345),(24,44,235,346),(25,45,236,347),(26,46,237,348),(27,47,238,349),(28,48,225,350),(29,394,62,332),(30,395,63,333),(31,396,64,334),(32,397,65,335),(33,398,66,336),(34,399,67,323),(35,400,68,324),(36,401,69,325),(37,402,70,326),(38,403,57,327),(39,404,58,328),(40,405,59,329),(41,406,60,330),(42,393,61,331),(85,437,414,380),(86,438,415,381),(87,439,416,382),(88,440,417,383),(89,441,418,384),(90,442,419,385),(91,443,420,386),(92,444,407,387),(93,445,408,388),(94,446,409,389),(95,447,410,390),(96,448,411,391),(97,435,412,392),(98,436,413,379),(99,281,196,153),(100,282,183,154),(101,283,184,141),(102,284,185,142),(103,285,186,143),(104,286,187,144),(105,287,188,145),(106,288,189,146),(107,289,190,147),(108,290,191,148),(109,291,192,149),(110,292,193,150),(111,293,194,151),(112,294,195,152),(113,305,169,128),(114,306,170,129),(115,307,171,130),(116,308,172,131),(117,295,173,132),(118,296,174,133),(119,297,175,134),(120,298,176,135),(121,299,177,136),(122,300,178,137),(123,301,179,138),(124,302,180,139),(125,303,181,140),(126,304,182,127),(155,204,246,265),(156,205,247,266),(157,206,248,253),(158,207,249,254),(159,208,250,255),(160,209,251,256),(161,210,252,257),(162,197,239,258),(163,198,240,259),(164,199,241,260),(165,200,242,261),(166,201,243,262),(167,202,244,263),(168,203,245,264),(267,424,372,316),(268,425,373,317),(269,426,374,318),(270,427,375,319),(271,428,376,320),(272,429,377,321),(273,430,378,322),(274,431,365,309),(275,432,366,310),(276,433,367,311),(277,434,368,312),(278,421,369,313),(279,422,370,314),(280,423,371,315)], [(1,263,55,389),(2,264,56,390),(3,265,43,391),(4,266,44,392),(5,253,45,379),(6,254,46,380),(7,255,47,381),(8,256,48,382),(9,257,49,383),(10,258,50,384),(11,259,51,385),(12,260,52,386),(13,261,53,387),(14,262,54,388),(15,88,224,161),(16,89,211,162),(17,90,212,163),(18,91,213,164),(19,92,214,165),(20,93,215,166),(21,94,216,167),(22,95,217,168),(23,96,218,155),(24,97,219,156),(25,98,220,157),(26,85,221,158),(27,86,222,159),(28,87,223,160),(29,270,174,289),(30,271,175,290),(31,272,176,291),(32,273,177,292),(33,274,178,293),(34,275,179,294),(35,276,180,281),(36,277,181,282),(37,278,182,283),(38,279,169,284),(39,280,170,285),(40,267,171,286),(41,268,172,287),(42,269,173,288),(57,370,113,142),(58,371,114,143),(59,372,115,144),(60,373,116,145),(61,374,117,146),(62,375,118,147),(63,376,119,148),(64,377,120,149),(65,378,121,150),(66,365,122,151),(67,366,123,152),(68,367,124,153),(69,368,125,154),(70,369,126,141),(71,201,342,445),(72,202,343,446),(73,203,344,447),(74,204,345,448),(75,205,346,435),(76,206,347,436),(77,207,348,437),(78,208,349,438),(79,209,350,439),(80,210,337,440),(81,197,338,441),(82,198,339,442),(83,199,340,443),(84,200,341,444),(99,400,311,139),(100,401,312,140),(101,402,313,127),(102,403,314,128),(103,404,315,129),(104,405,316,130),(105,406,317,131),(106,393,318,132),(107,394,319,133),(108,395,320,134),(109,396,321,135),(110,397,322,136),(111,398,309,137),(112,399,310,138),(183,325,434,303),(184,326,421,304),(185,327,422,305),(186,328,423,306),(187,329,424,307),(188,330,425,308),(189,331,426,295),(190,332,427,296),(191,333,428,297),(192,334,429,298),(193,335,430,299),(194,336,431,300),(195,323,432,301),(196,324,433,302),(225,416,358,251),(226,417,359,252),(227,418,360,239),(228,419,361,240),(229,420,362,241),(230,407,363,242),(231,408,364,243),(232,409,351,244),(233,410,352,245),(234,411,353,246),(235,412,354,247),(236,413,355,248),(237,414,356,249),(238,415,357,250)], [(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)], [(1,288,8,281),(2,287,9,294),(3,286,10,293),(4,285,11,292),(5,284,12,291),(6,283,13,290),(7,282,14,289),(15,112,22,105),(16,111,23,104),(17,110,24,103),(18,109,25,102),(19,108,26,101),(20,107,27,100),(21,106,28,99),(29,255,36,262),(30,254,37,261),(31,253,38,260),(32,266,39,259),(33,265,40,258),(34,264,41,257),(35,263,42,256),(43,267,50,274),(44,280,51,273),(45,279,52,272),(46,278,53,271),(47,277,54,270),(48,276,55,269),(49,275,56,268),(57,199,64,206),(58,198,65,205),(59,197,66,204),(60,210,67,203),(61,209,68,202),(62,208,69,201),(63,207,70,200),(71,147,78,154),(72,146,79,153),(73,145,80,152),(74,144,81,151),(75,143,82,150),(76,142,83,149),(77,141,84,148),(85,402,92,395),(86,401,93,394),(87,400,94,393),(88,399,95,406),(89,398,96,405),(90,397,97,404),(91,396,98,403),(113,443,120,436),(114,442,121,435),(115,441,122,448),(116,440,123,447),(117,439,124,446),(118,438,125,445),(119,437,126,444),(127,165,134,158),(128,164,135,157),(129,163,136,156),(130,162,137,155),(131,161,138,168),(132,160,139,167),(133,159,140,166),(169,386,176,379),(170,385,177,392),(171,384,178,391),(172,383,179,390),(173,382,180,389),(174,381,181,388),(175,380,182,387),(183,231,190,238),(184,230,191,237),(185,229,192,236),(186,228,193,235),(187,227,194,234),(188,226,195,233),(189,225,196,232),(211,309,218,316),(212,322,219,315),(213,321,220,314),(214,320,221,313),(215,319,222,312),(216,318,223,311),(217,317,224,310),(239,300,246,307),(240,299,247,306),(241,298,248,305),(242,297,249,304),(243,296,250,303),(244,295,251,302),(245,308,252,301),(323,410,330,417),(324,409,331,416),(325,408,332,415),(326,407,333,414),(327,420,334,413),(328,419,335,412),(329,418,336,411),(337,366,344,373),(338,365,345,372),(339,378,346,371),(340,377,347,370),(341,376,348,369),(342,375,349,368),(343,374,350,367),(351,426,358,433),(352,425,359,432),(353,424,360,431),(354,423,361,430),(355,422,362,429),(356,421,363,428),(357,434,364,427)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | C4○D4 | Dic7 | D14 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 |
| kernel | C4⋊C4⋊5Dic7 | C14.C42 | C2×C4×Dic7 | C2×C4⋊Dic7 | C14×C4⋊C4 | C7×C4⋊C4 | C2×Dic7 | C2×Dic7 | C2×C4⋊C4 | C2×C14 | C4⋊C4 | C22×C4 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 2 | 3 | 8 | 12 | 9 | 24 | 3 | 3 | 3 | 3 |
Matrix representation of C4⋊C4⋊5Dic7 ►in GL8(𝔽29)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 16 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 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 | 28 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 19 |
| 16 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 22 |
G:=sub<GL(8,GF(29))| [0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[16,27,0,0,0,0,0,0,27,13,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,9,0,0,0,0,0,0,1,19],[16,27,0,0,0,0,0,0,27,13,0,0,0,0,0,0,0,0,2,11,0,0,0,0,0,0,18,27,0,0,0,0,0,0,0,0,27,2,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,0,4,22] >;
C4⋊C4⋊5Dic7 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_5{\rm Dic}_7 % in TeX
G:=Group("C4:C4:5Dic7"); // GroupNames label
G:=SmallGroup(448,515);
// by ID
G=gap.SmallGroup(448,515);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,422,387,184,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c^7,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations