G = C4⋊(C4⋊Dic7) order 448 = 26·7
The semidirect product of C4 and C4⋊Dic7 acting via C4⋊Dic7/C2×Dic7=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊3(C4⋊C4), C4⋊C4⋊6Dic7, C4⋊1(C4⋊Dic7), C14.89(C4×D4), C2.5(C28⋊Q8), C14.25(C4×Q8), (C2×C28).20Q8, C2.4(Q8×Dic7), C2.7(D4×Dic7), (C2×C4).142D28, (C2×C28).140D4, C14.23(C4⋊Q8), C22.25(Q8×D7), C2.3(C4⋊D28), (C2×Dic7).19Q8, (C2×C4).31Dic14, C22.109(D4×D7), C22.46(C2×D28), C14.52(C4⋊D4), C2.4(D14⋊2Q8), (C2×Dic7).109D4, (C22×C4).102D14, C14.48(C22⋊Q8), C2.5(C28.3Q8), (C22×C28).66C22, C14.24(C42.C2), C22.29(C2×Dic14), C23.294(C22×D7), C22.58(D4⋊2D7), C14.C42.29C2, (C22×C14).349C23, C7⋊5(C23.65C23), C22.26(Q8⋊2D7), C22.42(C22×Dic7), (C22×Dic7).191C22, (C7×C4⋊C4)⋊9C4, C14.38(C2×C4⋊C4), (C2×C4⋊C4).22D7, (C14×C4⋊C4).15C2, C2.8(C2×C4⋊Dic7), (C2×C4×Dic7).8C2, (C2×C28).45(C2×C4), (C2×C14).38(C2×Q8), (C2×C14).333(C2×D4), (C2×C4⋊Dic7).35C2, (C2×C4).18(C2×Dic7), (C2×C14).187(C4○D4), (C2×C14).181(C22×C4), SmallGroup(448,519)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊(C4⋊Dic7)
G = < a,b,c,d | a4=b4=c14=1, d2=c7, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 644 in 170 conjugacy classes, 91 normal (41 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.65C23, C4×Dic7, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C2×C4×Dic7, C2×C4⋊Dic7, C2×C4⋊Dic7, C14×C4⋊C4, C4⋊(C4⋊Dic7)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, D14, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, Dic14, D28, C2×Dic7, C22×D7, C23.65C23, C4⋊Dic7, C2×Dic14, C2×D28, D4×D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C22×Dic7, C28⋊Q8, C28.3Q8, C4⋊D28, D14⋊2Q8, C2×C4⋊Dic7, D4×Dic7, Q8×Dic7, C4⋊(C4⋊Dic7)
►Regular action on 448 points
Generators in S448
(1 168 391 338)(2 155 392 339)(3 156 379 340)(4 157 380 341)(5 158 381 342)(6 159 382 343)(7 160 383 344)(8 161 384 345)(9 162 385 346)(10 163 386 347)(11 164 387 348)(12 165 388 349)(13 166 389 350)(14 167 390 337)(15 361 317 397)(16 362 318 398)(17 363 319 399)(18 364 320 400)(19 351 321 401)(20 352 322 402)(21 353 309 403)(22 354 310 404)(23 355 311 405)(24 356 312 406)(25 357 313 393)(26 358 314 394)(27 359 315 395)(28 360 316 396)(29 264 145 276)(30 265 146 277)(31 266 147 278)(32 253 148 279)(33 254 149 280)(34 255 150 267)(35 256 151 268)(36 257 152 269)(37 258 153 270)(38 259 154 271)(39 260 141 272)(40 261 142 273)(41 262 143 274)(42 263 144 275)(43 375 243 222)(44 376 244 223)(45 377 245 224)(46 378 246 211)(47 365 247 212)(48 366 248 213)(49 367 249 214)(50 368 250 215)(51 369 251 216)(52 370 252 217)(53 371 239 218)(54 372 240 219)(55 373 241 220)(56 374 242 221)(57 238 325 433)(58 225 326 434)(59 226 327 421)(60 227 328 422)(61 228 329 423)(62 229 330 424)(63 230 331 425)(64 231 332 426)(65 232 333 427)(66 233 334 428)(67 234 335 429)(68 235 336 430)(69 236 323 431)(70 237 324 432)(71 448 136 295)(72 435 137 296)(73 436 138 297)(74 437 139 298)(75 438 140 299)(76 439 127 300)(77 440 128 301)(78 441 129 302)(79 442 130 303)(80 443 131 304)(81 444 132 305)(82 445 133 306)(83 446 134 307)(84 447 135 308)(85 419 174 114)(86 420 175 115)(87 407 176 116)(88 408 177 117)(89 409 178 118)(90 410 179 119)(91 411 180 120)(92 412 181 121)(93 413 182 122)(94 414 169 123)(95 415 170 124)(96 416 171 125)(97 417 172 126)(98 418 173 113)(99 183 292 206)(100 184 293 207)(101 185 294 208)(102 186 281 209)(103 187 282 210)(104 188 283 197)(105 189 284 198)(106 190 285 199)(107 191 286 200)(108 192 287 201)(109 193 288 202)(110 194 289 203)(111 195 290 204)(112 196 291 205)
(1 374 433 205)(2 375 434 206)(3 376 421 207)(4 377 422 208)(5 378 423 209)(6 365 424 210)(7 366 425 197)(8 367 426 198)(9 368 427 199)(10 369 428 200)(11 370 429 201)(12 371 430 202)(13 372 431 203)(14 373 432 204)(15 149 415 437)(16 150 416 438)(17 151 417 439)(18 152 418 440)(19 153 419 441)(20 154 420 442)(21 141 407 443)(22 142 408 444)(23 143 409 445)(24 144 410 446)(25 145 411 447)(26 146 412 448)(27 147 413 435)(28 148 414 436)(29 120 308 313)(30 121 295 314)(31 122 296 315)(32 123 297 316)(33 124 298 317)(34 125 299 318)(35 126 300 319)(36 113 301 320)(37 114 302 321)(38 115 303 322)(39 116 304 309)(40 117 305 310)(41 118 306 311)(42 119 307 312)(43 58 292 155)(44 59 293 156)(45 60 294 157)(46 61 281 158)(47 62 282 159)(48 63 283 160)(49 64 284 161)(50 65 285 162)(51 66 286 163)(52 67 287 164)(53 68 288 165)(54 69 289 166)(55 70 290 167)(56 57 291 168)(71 358 265 181)(72 359 266 182)(73 360 253 169)(74 361 254 170)(75 362 255 171)(76 363 256 172)(77 364 257 173)(78 351 258 174)(79 352 259 175)(80 353 260 176)(81 354 261 177)(82 355 262 178)(83 356 263 179)(84 357 264 180)(85 129 401 270)(86 130 402 271)(87 131 403 272)(88 132 404 273)(89 133 405 274)(90 134 406 275)(91 135 393 276)(92 136 394 277)(93 137 395 278)(94 138 396 279)(95 139 397 280)(96 140 398 267)(97 127 399 268)(98 128 400 269)(99 339 243 326)(100 340 244 327)(101 341 245 328)(102 342 246 329)(103 343 247 330)(104 344 248 331)(105 345 249 332)(106 346 250 333)(107 347 251 334)(108 348 252 335)(109 349 239 336)(110 350 240 323)(111 337 241 324)(112 338 242 325)(183 392 222 225)(184 379 223 226)(185 380 224 227)(186 381 211 228)(187 382 212 229)(188 383 213 230)(189 384 214 231)(190 385 215 232)(191 386 216 233)(192 387 217 234)(193 388 218 235)(194 389 219 236)(195 390 220 237)(196 391 221 238)
(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)
(1 436 8 443)(2 435 9 442)(3 448 10 441)(4 447 11 440)(5 446 12 439)(6 445 13 438)(7 444 14 437)(15 197 22 204)(16 210 23 203)(17 209 24 202)(18 208 25 201)(19 207 26 200)(20 206 27 199)(21 205 28 198)(29 234 36 227)(30 233 37 226)(31 232 38 225)(32 231 39 238)(33 230 40 237)(34 229 41 236)(35 228 42 235)(43 93 50 86)(44 92 51 85)(45 91 52 98)(46 90 53 97)(47 89 54 96)(48 88 55 95)(49 87 56 94)(57 279 64 272)(58 278 65 271)(59 277 66 270)(60 276 67 269)(61 275 68 268)(62 274 69 267)(63 273 70 280)(71 347 78 340)(72 346 79 339)(73 345 80 338)(74 344 81 337)(75 343 82 350)(76 342 83 349)(77 341 84 348)(99 359 106 352)(100 358 107 351)(101 357 108 364)(102 356 109 363)(103 355 110 362)(104 354 111 361)(105 353 112 360)(113 224 120 217)(114 223 121 216)(115 222 122 215)(116 221 123 214)(117 220 124 213)(118 219 125 212)(119 218 126 211)(127 158 134 165)(128 157 135 164)(129 156 136 163)(130 155 137 162)(131 168 138 161)(132 167 139 160)(133 166 140 159)(141 433 148 426)(142 432 149 425)(143 431 150 424)(144 430 151 423)(145 429 152 422)(146 428 153 421)(147 427 154 434)(169 249 176 242)(170 248 177 241)(171 247 178 240)(172 246 179 239)(173 245 180 252)(174 244 181 251)(175 243 182 250)(183 315 190 322)(184 314 191 321)(185 313 192 320)(186 312 193 319)(187 311 194 318)(188 310 195 317)(189 309 196 316)(253 332 260 325)(254 331 261 324)(255 330 262 323)(256 329 263 336)(257 328 264 335)(258 327 265 334)(259 326 266 333)(281 406 288 399)(282 405 289 398)(283 404 290 397)(284 403 291 396)(285 402 292 395)(286 401 293 394)(287 400 294 393)(295 386 302 379)(296 385 303 392)(297 384 304 391)(298 383 305 390)(299 382 306 389)(300 381 307 388)(301 380 308 387)(365 409 372 416)(366 408 373 415)(367 407 374 414)(368 420 375 413)(369 419 376 412)(370 418 377 411)(371 417 378 410)
G:=sub<Sym(448)| (1,168,391,338)(2,155,392,339)(3,156,379,340)(4,157,380,341)(5,158,381,342)(6,159,382,343)(7,160,383,344)(8,161,384,345)(9,162,385,346)(10,163,386,347)(11,164,387,348)(12,165,388,349)(13,166,389,350)(14,167,390,337)(15,361,317,397)(16,362,318,398)(17,363,319,399)(18,364,320,400)(19,351,321,401)(20,352,322,402)(21,353,309,403)(22,354,310,404)(23,355,311,405)(24,356,312,406)(25,357,313,393)(26,358,314,394)(27,359,315,395)(28,360,316,396)(29,264,145,276)(30,265,146,277)(31,266,147,278)(32,253,148,279)(33,254,149,280)(34,255,150,267)(35,256,151,268)(36,257,152,269)(37,258,153,270)(38,259,154,271)(39,260,141,272)(40,261,142,273)(41,262,143,274)(42,263,144,275)(43,375,243,222)(44,376,244,223)(45,377,245,224)(46,378,246,211)(47,365,247,212)(48,366,248,213)(49,367,249,214)(50,368,250,215)(51,369,251,216)(52,370,252,217)(53,371,239,218)(54,372,240,219)(55,373,241,220)(56,374,242,221)(57,238,325,433)(58,225,326,434)(59,226,327,421)(60,227,328,422)(61,228,329,423)(62,229,330,424)(63,230,331,425)(64,231,332,426)(65,232,333,427)(66,233,334,428)(67,234,335,429)(68,235,336,430)(69,236,323,431)(70,237,324,432)(71,448,136,295)(72,435,137,296)(73,436,138,297)(74,437,139,298)(75,438,140,299)(76,439,127,300)(77,440,128,301)(78,441,129,302)(79,442,130,303)(80,443,131,304)(81,444,132,305)(82,445,133,306)(83,446,134,307)(84,447,135,308)(85,419,174,114)(86,420,175,115)(87,407,176,116)(88,408,177,117)(89,409,178,118)(90,410,179,119)(91,411,180,120)(92,412,181,121)(93,413,182,122)(94,414,169,123)(95,415,170,124)(96,416,171,125)(97,417,172,126)(98,418,173,113)(99,183,292,206)(100,184,293,207)(101,185,294,208)(102,186,281,209)(103,187,282,210)(104,188,283,197)(105,189,284,198)(106,190,285,199)(107,191,286,200)(108,192,287,201)(109,193,288,202)(110,194,289,203)(111,195,290,204)(112,196,291,205), (1,374,433,205)(2,375,434,206)(3,376,421,207)(4,377,422,208)(5,378,423,209)(6,365,424,210)(7,366,425,197)(8,367,426,198)(9,368,427,199)(10,369,428,200)(11,370,429,201)(12,371,430,202)(13,372,431,203)(14,373,432,204)(15,149,415,437)(16,150,416,438)(17,151,417,439)(18,152,418,440)(19,153,419,441)(20,154,420,442)(21,141,407,443)(22,142,408,444)(23,143,409,445)(24,144,410,446)(25,145,411,447)(26,146,412,448)(27,147,413,435)(28,148,414,436)(29,120,308,313)(30,121,295,314)(31,122,296,315)(32,123,297,316)(33,124,298,317)(34,125,299,318)(35,126,300,319)(36,113,301,320)(37,114,302,321)(38,115,303,322)(39,116,304,309)(40,117,305,310)(41,118,306,311)(42,119,307,312)(43,58,292,155)(44,59,293,156)(45,60,294,157)(46,61,281,158)(47,62,282,159)(48,63,283,160)(49,64,284,161)(50,65,285,162)(51,66,286,163)(52,67,287,164)(53,68,288,165)(54,69,289,166)(55,70,290,167)(56,57,291,168)(71,358,265,181)(72,359,266,182)(73,360,253,169)(74,361,254,170)(75,362,255,171)(76,363,256,172)(77,364,257,173)(78,351,258,174)(79,352,259,175)(80,353,260,176)(81,354,261,177)(82,355,262,178)(83,356,263,179)(84,357,264,180)(85,129,401,270)(86,130,402,271)(87,131,403,272)(88,132,404,273)(89,133,405,274)(90,134,406,275)(91,135,393,276)(92,136,394,277)(93,137,395,278)(94,138,396,279)(95,139,397,280)(96,140,398,267)(97,127,399,268)(98,128,400,269)(99,339,243,326)(100,340,244,327)(101,341,245,328)(102,342,246,329)(103,343,247,330)(104,344,248,331)(105,345,249,332)(106,346,250,333)(107,347,251,334)(108,348,252,335)(109,349,239,336)(110,350,240,323)(111,337,241,324)(112,338,242,325)(183,392,222,225)(184,379,223,226)(185,380,224,227)(186,381,211,228)(187,382,212,229)(188,383,213,230)(189,384,214,231)(190,385,215,232)(191,386,216,233)(192,387,217,234)(193,388,218,235)(194,389,219,236)(195,390,220,237)(196,391,221,238), (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), (1,436,8,443)(2,435,9,442)(3,448,10,441)(4,447,11,440)(5,446,12,439)(6,445,13,438)(7,444,14,437)(15,197,22,204)(16,210,23,203)(17,209,24,202)(18,208,25,201)(19,207,26,200)(20,206,27,199)(21,205,28,198)(29,234,36,227)(30,233,37,226)(31,232,38,225)(32,231,39,238)(33,230,40,237)(34,229,41,236)(35,228,42,235)(43,93,50,86)(44,92,51,85)(45,91,52,98)(46,90,53,97)(47,89,54,96)(48,88,55,95)(49,87,56,94)(57,279,64,272)(58,278,65,271)(59,277,66,270)(60,276,67,269)(61,275,68,268)(62,274,69,267)(63,273,70,280)(71,347,78,340)(72,346,79,339)(73,345,80,338)(74,344,81,337)(75,343,82,350)(76,342,83,349)(77,341,84,348)(99,359,106,352)(100,358,107,351)(101,357,108,364)(102,356,109,363)(103,355,110,362)(104,354,111,361)(105,353,112,360)(113,224,120,217)(114,223,121,216)(115,222,122,215)(116,221,123,214)(117,220,124,213)(118,219,125,212)(119,218,126,211)(127,158,134,165)(128,157,135,164)(129,156,136,163)(130,155,137,162)(131,168,138,161)(132,167,139,160)(133,166,140,159)(141,433,148,426)(142,432,149,425)(143,431,150,424)(144,430,151,423)(145,429,152,422)(146,428,153,421)(147,427,154,434)(169,249,176,242)(170,248,177,241)(171,247,178,240)(172,246,179,239)(173,245,180,252)(174,244,181,251)(175,243,182,250)(183,315,190,322)(184,314,191,321)(185,313,192,320)(186,312,193,319)(187,311,194,318)(188,310,195,317)(189,309,196,316)(253,332,260,325)(254,331,261,324)(255,330,262,323)(256,329,263,336)(257,328,264,335)(258,327,265,334)(259,326,266,333)(281,406,288,399)(282,405,289,398)(283,404,290,397)(284,403,291,396)(285,402,292,395)(286,401,293,394)(287,400,294,393)(295,386,302,379)(296,385,303,392)(297,384,304,391)(298,383,305,390)(299,382,306,389)(300,381,307,388)(301,380,308,387)(365,409,372,416)(366,408,373,415)(367,407,374,414)(368,420,375,413)(369,419,376,412)(370,418,377,411)(371,417,378,410)>;
G:=Group( (1,168,391,338)(2,155,392,339)(3,156,379,340)(4,157,380,341)(5,158,381,342)(6,159,382,343)(7,160,383,344)(8,161,384,345)(9,162,385,346)(10,163,386,347)(11,164,387,348)(12,165,388,349)(13,166,389,350)(14,167,390,337)(15,361,317,397)(16,362,318,398)(17,363,319,399)(18,364,320,400)(19,351,321,401)(20,352,322,402)(21,353,309,403)(22,354,310,404)(23,355,311,405)(24,356,312,406)(25,357,313,393)(26,358,314,394)(27,359,315,395)(28,360,316,396)(29,264,145,276)(30,265,146,277)(31,266,147,278)(32,253,148,279)(33,254,149,280)(34,255,150,267)(35,256,151,268)(36,257,152,269)(37,258,153,270)(38,259,154,271)(39,260,141,272)(40,261,142,273)(41,262,143,274)(42,263,144,275)(43,375,243,222)(44,376,244,223)(45,377,245,224)(46,378,246,211)(47,365,247,212)(48,366,248,213)(49,367,249,214)(50,368,250,215)(51,369,251,216)(52,370,252,217)(53,371,239,218)(54,372,240,219)(55,373,241,220)(56,374,242,221)(57,238,325,433)(58,225,326,434)(59,226,327,421)(60,227,328,422)(61,228,329,423)(62,229,330,424)(63,230,331,425)(64,231,332,426)(65,232,333,427)(66,233,334,428)(67,234,335,429)(68,235,336,430)(69,236,323,431)(70,237,324,432)(71,448,136,295)(72,435,137,296)(73,436,138,297)(74,437,139,298)(75,438,140,299)(76,439,127,300)(77,440,128,301)(78,441,129,302)(79,442,130,303)(80,443,131,304)(81,444,132,305)(82,445,133,306)(83,446,134,307)(84,447,135,308)(85,419,174,114)(86,420,175,115)(87,407,176,116)(88,408,177,117)(89,409,178,118)(90,410,179,119)(91,411,180,120)(92,412,181,121)(93,413,182,122)(94,414,169,123)(95,415,170,124)(96,416,171,125)(97,417,172,126)(98,418,173,113)(99,183,292,206)(100,184,293,207)(101,185,294,208)(102,186,281,209)(103,187,282,210)(104,188,283,197)(105,189,284,198)(106,190,285,199)(107,191,286,200)(108,192,287,201)(109,193,288,202)(110,194,289,203)(111,195,290,204)(112,196,291,205), (1,374,433,205)(2,375,434,206)(3,376,421,207)(4,377,422,208)(5,378,423,209)(6,365,424,210)(7,366,425,197)(8,367,426,198)(9,368,427,199)(10,369,428,200)(11,370,429,201)(12,371,430,202)(13,372,431,203)(14,373,432,204)(15,149,415,437)(16,150,416,438)(17,151,417,439)(18,152,418,440)(19,153,419,441)(20,154,420,442)(21,141,407,443)(22,142,408,444)(23,143,409,445)(24,144,410,446)(25,145,411,447)(26,146,412,448)(27,147,413,435)(28,148,414,436)(29,120,308,313)(30,121,295,314)(31,122,296,315)(32,123,297,316)(33,124,298,317)(34,125,299,318)(35,126,300,319)(36,113,301,320)(37,114,302,321)(38,115,303,322)(39,116,304,309)(40,117,305,310)(41,118,306,311)(42,119,307,312)(43,58,292,155)(44,59,293,156)(45,60,294,157)(46,61,281,158)(47,62,282,159)(48,63,283,160)(49,64,284,161)(50,65,285,162)(51,66,286,163)(52,67,287,164)(53,68,288,165)(54,69,289,166)(55,70,290,167)(56,57,291,168)(71,358,265,181)(72,359,266,182)(73,360,253,169)(74,361,254,170)(75,362,255,171)(76,363,256,172)(77,364,257,173)(78,351,258,174)(79,352,259,175)(80,353,260,176)(81,354,261,177)(82,355,262,178)(83,356,263,179)(84,357,264,180)(85,129,401,270)(86,130,402,271)(87,131,403,272)(88,132,404,273)(89,133,405,274)(90,134,406,275)(91,135,393,276)(92,136,394,277)(93,137,395,278)(94,138,396,279)(95,139,397,280)(96,140,398,267)(97,127,399,268)(98,128,400,269)(99,339,243,326)(100,340,244,327)(101,341,245,328)(102,342,246,329)(103,343,247,330)(104,344,248,331)(105,345,249,332)(106,346,250,333)(107,347,251,334)(108,348,252,335)(109,349,239,336)(110,350,240,323)(111,337,241,324)(112,338,242,325)(183,392,222,225)(184,379,223,226)(185,380,224,227)(186,381,211,228)(187,382,212,229)(188,383,213,230)(189,384,214,231)(190,385,215,232)(191,386,216,233)(192,387,217,234)(193,388,218,235)(194,389,219,236)(195,390,220,237)(196,391,221,238), (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), (1,436,8,443)(2,435,9,442)(3,448,10,441)(4,447,11,440)(5,446,12,439)(6,445,13,438)(7,444,14,437)(15,197,22,204)(16,210,23,203)(17,209,24,202)(18,208,25,201)(19,207,26,200)(20,206,27,199)(21,205,28,198)(29,234,36,227)(30,233,37,226)(31,232,38,225)(32,231,39,238)(33,230,40,237)(34,229,41,236)(35,228,42,235)(43,93,50,86)(44,92,51,85)(45,91,52,98)(46,90,53,97)(47,89,54,96)(48,88,55,95)(49,87,56,94)(57,279,64,272)(58,278,65,271)(59,277,66,270)(60,276,67,269)(61,275,68,268)(62,274,69,267)(63,273,70,280)(71,347,78,340)(72,346,79,339)(73,345,80,338)(74,344,81,337)(75,343,82,350)(76,342,83,349)(77,341,84,348)(99,359,106,352)(100,358,107,351)(101,357,108,364)(102,356,109,363)(103,355,110,362)(104,354,111,361)(105,353,112,360)(113,224,120,217)(114,223,121,216)(115,222,122,215)(116,221,123,214)(117,220,124,213)(118,219,125,212)(119,218,126,211)(127,158,134,165)(128,157,135,164)(129,156,136,163)(130,155,137,162)(131,168,138,161)(132,167,139,160)(133,166,140,159)(141,433,148,426)(142,432,149,425)(143,431,150,424)(144,430,151,423)(145,429,152,422)(146,428,153,421)(147,427,154,434)(169,249,176,242)(170,248,177,241)(171,247,178,240)(172,246,179,239)(173,245,180,252)(174,244,181,251)(175,243,182,250)(183,315,190,322)(184,314,191,321)(185,313,192,320)(186,312,193,319)(187,311,194,318)(188,310,195,317)(189,309,196,316)(253,332,260,325)(254,331,261,324)(255,330,262,323)(256,329,263,336)(257,328,264,335)(258,327,265,334)(259,326,266,333)(281,406,288,399)(282,405,289,398)(283,404,290,397)(284,403,291,396)(285,402,292,395)(286,401,293,394)(287,400,294,393)(295,386,302,379)(296,385,303,392)(297,384,304,391)(298,383,305,390)(299,382,306,389)(300,381,307,388)(301,380,308,387)(365,409,372,416)(366,408,373,415)(367,407,374,414)(368,420,375,413)(369,419,376,412)(370,418,377,411)(371,417,378,410) );
G=PermutationGroup([[(1,168,391,338),(2,155,392,339),(3,156,379,340),(4,157,380,341),(5,158,381,342),(6,159,382,343),(7,160,383,344),(8,161,384,345),(9,162,385,346),(10,163,386,347),(11,164,387,348),(12,165,388,349),(13,166,389,350),(14,167,390,337),(15,361,317,397),(16,362,318,398),(17,363,319,399),(18,364,320,400),(19,351,321,401),(20,352,322,402),(21,353,309,403),(22,354,310,404),(23,355,311,405),(24,356,312,406),(25,357,313,393),(26,358,314,394),(27,359,315,395),(28,360,316,396),(29,264,145,276),(30,265,146,277),(31,266,147,278),(32,253,148,279),(33,254,149,280),(34,255,150,267),(35,256,151,268),(36,257,152,269),(37,258,153,270),(38,259,154,271),(39,260,141,272),(40,261,142,273),(41,262,143,274),(42,263,144,275),(43,375,243,222),(44,376,244,223),(45,377,245,224),(46,378,246,211),(47,365,247,212),(48,366,248,213),(49,367,249,214),(50,368,250,215),(51,369,251,216),(52,370,252,217),(53,371,239,218),(54,372,240,219),(55,373,241,220),(56,374,242,221),(57,238,325,433),(58,225,326,434),(59,226,327,421),(60,227,328,422),(61,228,329,423),(62,229,330,424),(63,230,331,425),(64,231,332,426),(65,232,333,427),(66,233,334,428),(67,234,335,429),(68,235,336,430),(69,236,323,431),(70,237,324,432),(71,448,136,295),(72,435,137,296),(73,436,138,297),(74,437,139,298),(75,438,140,299),(76,439,127,300),(77,440,128,301),(78,441,129,302),(79,442,130,303),(80,443,131,304),(81,444,132,305),(82,445,133,306),(83,446,134,307),(84,447,135,308),(85,419,174,114),(86,420,175,115),(87,407,176,116),(88,408,177,117),(89,409,178,118),(90,410,179,119),(91,411,180,120),(92,412,181,121),(93,413,182,122),(94,414,169,123),(95,415,170,124),(96,416,171,125),(97,417,172,126),(98,418,173,113),(99,183,292,206),(100,184,293,207),(101,185,294,208),(102,186,281,209),(103,187,282,210),(104,188,283,197),(105,189,284,198),(106,190,285,199),(107,191,286,200),(108,192,287,201),(109,193,288,202),(110,194,289,203),(111,195,290,204),(112,196,291,205)], [(1,374,433,205),(2,375,434,206),(3,376,421,207),(4,377,422,208),(5,378,423,209),(6,365,424,210),(7,366,425,197),(8,367,426,198),(9,368,427,199),(10,369,428,200),(11,370,429,201),(12,371,430,202),(13,372,431,203),(14,373,432,204),(15,149,415,437),(16,150,416,438),(17,151,417,439),(18,152,418,440),(19,153,419,441),(20,154,420,442),(21,141,407,443),(22,142,408,444),(23,143,409,445),(24,144,410,446),(25,145,411,447),(26,146,412,448),(27,147,413,435),(28,148,414,436),(29,120,308,313),(30,121,295,314),(31,122,296,315),(32,123,297,316),(33,124,298,317),(34,125,299,318),(35,126,300,319),(36,113,301,320),(37,114,302,321),(38,115,303,322),(39,116,304,309),(40,117,305,310),(41,118,306,311),(42,119,307,312),(43,58,292,155),(44,59,293,156),(45,60,294,157),(46,61,281,158),(47,62,282,159),(48,63,283,160),(49,64,284,161),(50,65,285,162),(51,66,286,163),(52,67,287,164),(53,68,288,165),(54,69,289,166),(55,70,290,167),(56,57,291,168),(71,358,265,181),(72,359,266,182),(73,360,253,169),(74,361,254,170),(75,362,255,171),(76,363,256,172),(77,364,257,173),(78,351,258,174),(79,352,259,175),(80,353,260,176),(81,354,261,177),(82,355,262,178),(83,356,263,179),(84,357,264,180),(85,129,401,270),(86,130,402,271),(87,131,403,272),(88,132,404,273),(89,133,405,274),(90,134,406,275),(91,135,393,276),(92,136,394,277),(93,137,395,278),(94,138,396,279),(95,139,397,280),(96,140,398,267),(97,127,399,268),(98,128,400,269),(99,339,243,326),(100,340,244,327),(101,341,245,328),(102,342,246,329),(103,343,247,330),(104,344,248,331),(105,345,249,332),(106,346,250,333),(107,347,251,334),(108,348,252,335),(109,349,239,336),(110,350,240,323),(111,337,241,324),(112,338,242,325),(183,392,222,225),(184,379,223,226),(185,380,224,227),(186,381,211,228),(187,382,212,229),(188,383,213,230),(189,384,214,231),(190,385,215,232),(191,386,216,233),(192,387,217,234),(193,388,218,235),(194,389,219,236),(195,390,220,237),(196,391,221,238)], [(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)], [(1,436,8,443),(2,435,9,442),(3,448,10,441),(4,447,11,440),(5,446,12,439),(6,445,13,438),(7,444,14,437),(15,197,22,204),(16,210,23,203),(17,209,24,202),(18,208,25,201),(19,207,26,200),(20,206,27,199),(21,205,28,198),(29,234,36,227),(30,233,37,226),(31,232,38,225),(32,231,39,238),(33,230,40,237),(34,229,41,236),(35,228,42,235),(43,93,50,86),(44,92,51,85),(45,91,52,98),(46,90,53,97),(47,89,54,96),(48,88,55,95),(49,87,56,94),(57,279,64,272),(58,278,65,271),(59,277,66,270),(60,276,67,269),(61,275,68,268),(62,274,69,267),(63,273,70,280),(71,347,78,340),(72,346,79,339),(73,345,80,338),(74,344,81,337),(75,343,82,350),(76,342,83,349),(77,341,84,348),(99,359,106,352),(100,358,107,351),(101,357,108,364),(102,356,109,363),(103,355,110,362),(104,354,111,361),(105,353,112,360),(113,224,120,217),(114,223,121,216),(115,222,122,215),(116,221,123,214),(117,220,124,213),(118,219,125,212),(119,218,126,211),(127,158,134,165),(128,157,135,164),(129,156,136,163),(130,155,137,162),(131,168,138,161),(132,167,139,160),(133,166,140,159),(141,433,148,426),(142,432,149,425),(143,431,150,424),(144,430,151,423),(145,429,152,422),(146,428,153,421),(147,427,154,434),(169,249,176,242),(170,248,177,241),(171,247,178,240),(172,246,179,239),(173,245,180,252),(174,244,181,251),(175,243,182,250),(183,315,190,322),(184,314,191,321),(185,313,192,320),(186,312,193,319),(187,311,194,318),(188,310,195,317),(189,309,196,316),(253,332,260,325),(254,331,261,324),(255,330,262,323),(256,329,263,336),(257,328,264,335),(258,327,265,334),(259,326,266,333),(281,406,288,399),(282,405,289,398),(283,404,290,397),(284,403,291,396),(285,402,292,395),(286,401,293,394),(287,400,294,393),(295,386,302,379),(296,385,303,392),(297,384,304,391),(298,383,305,390),(299,382,306,389),(300,381,307,388),(301,380,308,387),(365,409,372,416),(366,408,373,415),(367,407,374,414),(368,420,375,413),(369,419,376,412),(370,418,377,411),(371,417,378,410)]])
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 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | - | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | Q8 | D7 | C4○D4 | Dic7 | D14 | Dic14 | D28 | D4×D7 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 |
| kernel | C4⋊(C4⋊Dic7) | C14.C42 | C2×C4×Dic7 | C2×C4⋊Dic7 | C14×C4⋊C4 | C7×C4⋊C4 | C2×Dic7 | C2×Dic7 | C2×C28 | C2×C28 | C2×C4⋊C4 | C2×C14 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 8 | 2 | 2 | 2 | 2 | 3 | 4 | 12 | 9 | 12 | 12 | 3 | 3 | 3 | 3 |
Matrix representation of C4⋊(C4⋊Dic7) ►in GL6(𝔽29)
| 13 | 27 | 0 | 0 | 0 | 0 |
| 27 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 20 | 0 | 0 |
| 0 | 0 | 9 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 5 |
| 0 | 0 | 0 | 0 | 28 | 27 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 23 | 8 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 27 | 12 |
G:=sub<GL(6,GF(29))| [13,27,0,0,0,0,27,16,0,0,0,0,0,0,14,9,0,0,0,0,20,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,28,0,0,0,0,5,27],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,3,23,0,0,0,0,1,8],[16,2,0,0,0,0,2,13,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,17,27,0,0,0,0,0,12] >;
C4⋊(C4⋊Dic7) in GAP, Magma, Sage, TeX
C_4\rtimes (C_4\rtimes {\rm Dic}_7) % in TeX
G:=Group("C4:(C4:Dic7)"); // GroupNames label
G:=SmallGroup(448,519);
// by ID
G=gap.SmallGroup(448,519);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,422,387,100,18822]);
// 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,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations