direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C4⋊Dic7, C28⋊2C42, C42⋊7Dic7, C28⋊7(C4⋊C4), (C4×C28)⋊13C4, C4⋊2(C4×Dic7), C2.2(C4×D28), C14.16(C4×D4), C14.12(C4×Q8), (C2×C28).67Q8, (C2×C28).400D4, (C2×C4).167D28, C2.3(C4×Dic14), (C2×C42).15D7, C14.18(C2×C42), (C2×C4).57Dic14, C22.35(C2×D28), (C22×C4).418D14, C22.44(C4○D28), C22.20(C2×Dic14), C23.267(C22×D7), C14.41(C42⋊C2), C14.C42.43C2, (C22×C28).474C22, (C22×C14).309C23, C22.18(C22×Dic7), C2.3(C23.21D14), (C22×Dic7).180C22, C7⋊3(C4×C4⋊C4), (C2×C4×C28).21C2, C14.26(C2×C4⋊C4), C2.7(C2×C4×Dic7), C2.2(C2×C4⋊Dic7), C22.52(C2×C4×D7), (C2×C4).110(C4×D7), (C2×C14).27(C2×Q8), (C2×C4×Dic7).33C2, (C2×C28).317(C2×C4), (C2×C14).145(C2×D4), (C2×C4⋊Dic7).46C2, (C2×C4).61(C2×Dic7), (C2×C14).69(C4○D4), (C2×C14).98(C22×C4), (C2×Dic7).57(C2×C4), SmallGroup(448,468)
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, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 644 in 194 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C42, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C4×C4⋊C4, C4×Dic7, C4⋊Dic7, C4×C28, C22×Dic7, C22×C28, C14.C42, C2×C4×Dic7, C2×C4⋊Dic7, C2×C4×C28, C4×C4⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, D14, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, Dic14, C4×D7, D28, C2×Dic7, C22×D7, C4×C4⋊C4, C4×Dic7, C4⋊Dic7, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C22×Dic7, C4×Dic14, C4×D28, C2×C4×Dic7, C2×C4⋊Dic7, C23.21D14, C4×C4⋊Dic7
►Regular action on 448 points
Generators in S448
(1 42 296 240)(2 29 297 241)(3 30 298 242)(4 31 299 243)(5 32 300 244)(6 33 301 245)(7 34 302 246)(8 35 303 247)(9 36 304 248)(10 37 305 249)(11 38 306 250)(12 39 307 251)(13 40 308 252)(14 41 295 239)(15 221 352 171)(16 222 353 172)(17 223 354 173)(18 224 355 174)(19 211 356 175)(20 212 357 176)(21 213 358 177)(22 214 359 178)(23 215 360 179)(24 216 361 180)(25 217 362 181)(26 218 363 182)(27 219 364 169)(28 220 351 170)(43 83 61 105)(44 84 62 106)(45 71 63 107)(46 72 64 108)(47 73 65 109)(48 74 66 110)(49 75 67 111)(50 76 68 112)(51 77 69 99)(52 78 70 100)(53 79 57 101)(54 80 58 102)(55 81 59 103)(56 82 60 104)(85 124 343 333)(86 125 344 334)(87 126 345 335)(88 113 346 336)(89 114 347 323)(90 115 348 324)(91 116 349 325)(92 117 350 326)(93 118 337 327)(94 119 338 328)(95 120 339 329)(96 121 340 330)(97 122 341 331)(98 123 342 332)(127 392 201 266)(128 379 202 253)(129 380 203 254)(130 381 204 255)(131 382 205 256)(132 383 206 257)(133 384 207 258)(134 385 208 259)(135 386 209 260)(136 387 210 261)(137 388 197 262)(138 389 198 263)(139 390 199 264)(140 391 200 265)(141 234 435 312)(142 235 436 313)(143 236 437 314)(144 237 438 315)(145 238 439 316)(146 225 440 317)(147 226 441 318)(148 227 442 319)(149 228 443 320)(150 229 444 321)(151 230 445 322)(152 231 446 309)(153 232 447 310)(154 233 448 311)(155 272 433 188)(156 273 434 189)(157 274 421 190)(158 275 422 191)(159 276 423 192)(160 277 424 193)(161 278 425 194)(162 279 426 195)(163 280 427 196)(164 267 428 183)(165 268 429 184)(166 269 430 185)(167 270 431 186)(168 271 432 187)(281 406 417 371)(282 393 418 372)(283 394 419 373)(284 395 420 374)(285 396 407 375)(286 397 408 376)(287 398 409 377)(288 399 410 378)(289 400 411 365)(290 401 412 366)(291 402 413 367)(292 403 414 368)(293 404 415 369)(294 405 416 370)
(1 328 287 205)(2 329 288 206)(3 330 289 207)(4 331 290 208)(5 332 291 209)(6 333 292 210)(7 334 293 197)(8 335 294 198)(9 336 281 199)(10 323 282 200)(11 324 283 201)(12 325 284 202)(13 326 285 203)(14 327 286 204)(15 196 48 236)(16 183 49 237)(17 184 50 238)(18 185 51 225)(19 186 52 226)(20 187 53 227)(21 188 54 228)(22 189 55 229)(23 190 56 230)(24 191 43 231)(25 192 44 232)(26 193 45 233)(27 194 46 234)(28 195 47 235)(29 95 399 257)(30 96 400 258)(31 97 401 259)(32 98 402 260)(33 85 403 261)(34 86 404 262)(35 87 405 263)(36 88 406 264)(37 89 393 265)(38 90 394 266)(39 91 395 253)(40 92 396 254)(41 93 397 255)(42 94 398 256)(57 319 357 271)(58 320 358 272)(59 321 359 273)(60 322 360 274)(61 309 361 275)(62 310 362 276)(63 311 363 277)(64 312 364 278)(65 313 351 279)(66 314 352 280)(67 315 353 267)(68 316 354 268)(69 317 355 269)(70 318 356 270)(71 448 218 160)(72 435 219 161)(73 436 220 162)(74 437 221 163)(75 438 222 164)(76 439 223 165)(77 440 224 166)(78 441 211 167)(79 442 212 168)(80 443 213 155)(81 444 214 156)(82 445 215 157)(83 446 216 158)(84 447 217 159)(99 146 174 430)(100 147 175 431)(101 148 176 432)(102 149 177 433)(103 150 178 434)(104 151 179 421)(105 152 180 422)(106 153 181 423)(107 154 182 424)(108 141 169 425)(109 142 170 426)(110 143 171 427)(111 144 172 428)(112 145 173 429)(113 417 139 304)(114 418 140 305)(115 419 127 306)(116 420 128 307)(117 407 129 308)(118 408 130 295)(119 409 131 296)(120 410 132 297)(121 411 133 298)(122 412 134 299)(123 413 135 300)(124 414 136 301)(125 415 137 302)(126 416 138 303)(239 337 376 381)(240 338 377 382)(241 339 378 383)(242 340 365 384)(243 341 366 385)(244 342 367 386)(245 343 368 387)(246 344 369 388)(247 345 370 389)(248 346 371 390)(249 347 372 391)(250 348 373 392)(251 349 374 379)(252 350 375 380)
(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 28 8 21)(2 27 9 20)(3 26 10 19)(4 25 11 18)(5 24 12 17)(6 23 13 16)(7 22 14 15)(29 219 36 212)(30 218 37 211)(31 217 38 224)(32 216 39 223)(33 215 40 222)(34 214 41 221)(35 213 42 220)(43 284 50 291)(44 283 51 290)(45 282 52 289)(46 281 53 288)(47 294 54 287)(48 293 55 286)(49 292 56 285)(57 410 64 417)(58 409 65 416)(59 408 66 415)(60 407 67 414)(61 420 68 413)(62 419 69 412)(63 418 70 411)(71 393 78 400)(72 406 79 399)(73 405 80 398)(74 404 81 397)(75 403 82 396)(76 402 83 395)(77 401 84 394)(85 445 92 438)(86 444 93 437)(87 443 94 436)(88 442 95 435)(89 441 96 448)(90 440 97 447)(91 439 98 446)(99 366 106 373)(100 365 107 372)(101 378 108 371)(102 377 109 370)(103 376 110 369)(104 375 111 368)(105 374 112 367)(113 319 120 312)(114 318 121 311)(115 317 122 310)(116 316 123 309)(117 315 124 322)(118 314 125 321)(119 313 126 320)(127 269 134 276)(128 268 135 275)(129 267 136 274)(130 280 137 273)(131 279 138 272)(132 278 139 271)(133 277 140 270)(141 346 148 339)(142 345 149 338)(143 344 150 337)(144 343 151 350)(145 342 152 349)(146 341 153 348)(147 340 154 347)(155 256 162 263)(156 255 163 262)(157 254 164 261)(158 253 165 260)(159 266 166 259)(160 265 167 258)(161 264 168 257)(169 248 176 241)(170 247 177 240)(171 246 178 239)(172 245 179 252)(173 244 180 251)(174 243 181 250)(175 242 182 249)(183 210 190 203)(184 209 191 202)(185 208 192 201)(186 207 193 200)(187 206 194 199)(188 205 195 198)(189 204 196 197)(225 331 232 324)(226 330 233 323)(227 329 234 336)(228 328 235 335)(229 327 236 334)(230 326 237 333)(231 325 238 332)(295 352 302 359)(296 351 303 358)(297 364 304 357)(298 363 305 356)(299 362 306 355)(300 361 307 354)(301 360 308 353)(379 429 386 422)(380 428 387 421)(381 427 388 434)(382 426 389 433)(383 425 390 432)(384 424 391 431)(385 423 392 430)
G:=sub<Sym(448)| (1,42,296,240)(2,29,297,241)(3,30,298,242)(4,31,299,243)(5,32,300,244)(6,33,301,245)(7,34,302,246)(8,35,303,247)(9,36,304,248)(10,37,305,249)(11,38,306,250)(12,39,307,251)(13,40,308,252)(14,41,295,239)(15,221,352,171)(16,222,353,172)(17,223,354,173)(18,224,355,174)(19,211,356,175)(20,212,357,176)(21,213,358,177)(22,214,359,178)(23,215,360,179)(24,216,361,180)(25,217,362,181)(26,218,363,182)(27,219,364,169)(28,220,351,170)(43,83,61,105)(44,84,62,106)(45,71,63,107)(46,72,64,108)(47,73,65,109)(48,74,66,110)(49,75,67,111)(50,76,68,112)(51,77,69,99)(52,78,70,100)(53,79,57,101)(54,80,58,102)(55,81,59,103)(56,82,60,104)(85,124,343,333)(86,125,344,334)(87,126,345,335)(88,113,346,336)(89,114,347,323)(90,115,348,324)(91,116,349,325)(92,117,350,326)(93,118,337,327)(94,119,338,328)(95,120,339,329)(96,121,340,330)(97,122,341,331)(98,123,342,332)(127,392,201,266)(128,379,202,253)(129,380,203,254)(130,381,204,255)(131,382,205,256)(132,383,206,257)(133,384,207,258)(134,385,208,259)(135,386,209,260)(136,387,210,261)(137,388,197,262)(138,389,198,263)(139,390,199,264)(140,391,200,265)(141,234,435,312)(142,235,436,313)(143,236,437,314)(144,237,438,315)(145,238,439,316)(146,225,440,317)(147,226,441,318)(148,227,442,319)(149,228,443,320)(150,229,444,321)(151,230,445,322)(152,231,446,309)(153,232,447,310)(154,233,448,311)(155,272,433,188)(156,273,434,189)(157,274,421,190)(158,275,422,191)(159,276,423,192)(160,277,424,193)(161,278,425,194)(162,279,426,195)(163,280,427,196)(164,267,428,183)(165,268,429,184)(166,269,430,185)(167,270,431,186)(168,271,432,187)(281,406,417,371)(282,393,418,372)(283,394,419,373)(284,395,420,374)(285,396,407,375)(286,397,408,376)(287,398,409,377)(288,399,410,378)(289,400,411,365)(290,401,412,366)(291,402,413,367)(292,403,414,368)(293,404,415,369)(294,405,416,370), (1,328,287,205)(2,329,288,206)(3,330,289,207)(4,331,290,208)(5,332,291,209)(6,333,292,210)(7,334,293,197)(8,335,294,198)(9,336,281,199)(10,323,282,200)(11,324,283,201)(12,325,284,202)(13,326,285,203)(14,327,286,204)(15,196,48,236)(16,183,49,237)(17,184,50,238)(18,185,51,225)(19,186,52,226)(20,187,53,227)(21,188,54,228)(22,189,55,229)(23,190,56,230)(24,191,43,231)(25,192,44,232)(26,193,45,233)(27,194,46,234)(28,195,47,235)(29,95,399,257)(30,96,400,258)(31,97,401,259)(32,98,402,260)(33,85,403,261)(34,86,404,262)(35,87,405,263)(36,88,406,264)(37,89,393,265)(38,90,394,266)(39,91,395,253)(40,92,396,254)(41,93,397,255)(42,94,398,256)(57,319,357,271)(58,320,358,272)(59,321,359,273)(60,322,360,274)(61,309,361,275)(62,310,362,276)(63,311,363,277)(64,312,364,278)(65,313,351,279)(66,314,352,280)(67,315,353,267)(68,316,354,268)(69,317,355,269)(70,318,356,270)(71,448,218,160)(72,435,219,161)(73,436,220,162)(74,437,221,163)(75,438,222,164)(76,439,223,165)(77,440,224,166)(78,441,211,167)(79,442,212,168)(80,443,213,155)(81,444,214,156)(82,445,215,157)(83,446,216,158)(84,447,217,159)(99,146,174,430)(100,147,175,431)(101,148,176,432)(102,149,177,433)(103,150,178,434)(104,151,179,421)(105,152,180,422)(106,153,181,423)(107,154,182,424)(108,141,169,425)(109,142,170,426)(110,143,171,427)(111,144,172,428)(112,145,173,429)(113,417,139,304)(114,418,140,305)(115,419,127,306)(116,420,128,307)(117,407,129,308)(118,408,130,295)(119,409,131,296)(120,410,132,297)(121,411,133,298)(122,412,134,299)(123,413,135,300)(124,414,136,301)(125,415,137,302)(126,416,138,303)(239,337,376,381)(240,338,377,382)(241,339,378,383)(242,340,365,384)(243,341,366,385)(244,342,367,386)(245,343,368,387)(246,344,369,388)(247,345,370,389)(248,346,371,390)(249,347,372,391)(250,348,373,392)(251,349,374,379)(252,350,375,380), (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), (1,28,8,21)(2,27,9,20)(3,26,10,19)(4,25,11,18)(5,24,12,17)(6,23,13,16)(7,22,14,15)(29,219,36,212)(30,218,37,211)(31,217,38,224)(32,216,39,223)(33,215,40,222)(34,214,41,221)(35,213,42,220)(43,284,50,291)(44,283,51,290)(45,282,52,289)(46,281,53,288)(47,294,54,287)(48,293,55,286)(49,292,56,285)(57,410,64,417)(58,409,65,416)(59,408,66,415)(60,407,67,414)(61,420,68,413)(62,419,69,412)(63,418,70,411)(71,393,78,400)(72,406,79,399)(73,405,80,398)(74,404,81,397)(75,403,82,396)(76,402,83,395)(77,401,84,394)(85,445,92,438)(86,444,93,437)(87,443,94,436)(88,442,95,435)(89,441,96,448)(90,440,97,447)(91,439,98,446)(99,366,106,373)(100,365,107,372)(101,378,108,371)(102,377,109,370)(103,376,110,369)(104,375,111,368)(105,374,112,367)(113,319,120,312)(114,318,121,311)(115,317,122,310)(116,316,123,309)(117,315,124,322)(118,314,125,321)(119,313,126,320)(127,269,134,276)(128,268,135,275)(129,267,136,274)(130,280,137,273)(131,279,138,272)(132,278,139,271)(133,277,140,270)(141,346,148,339)(142,345,149,338)(143,344,150,337)(144,343,151,350)(145,342,152,349)(146,341,153,348)(147,340,154,347)(155,256,162,263)(156,255,163,262)(157,254,164,261)(158,253,165,260)(159,266,166,259)(160,265,167,258)(161,264,168,257)(169,248,176,241)(170,247,177,240)(171,246,178,239)(172,245,179,252)(173,244,180,251)(174,243,181,250)(175,242,182,249)(183,210,190,203)(184,209,191,202)(185,208,192,201)(186,207,193,200)(187,206,194,199)(188,205,195,198)(189,204,196,197)(225,331,232,324)(226,330,233,323)(227,329,234,336)(228,328,235,335)(229,327,236,334)(230,326,237,333)(231,325,238,332)(295,352,302,359)(296,351,303,358)(297,364,304,357)(298,363,305,356)(299,362,306,355)(300,361,307,354)(301,360,308,353)(379,429,386,422)(380,428,387,421)(381,427,388,434)(382,426,389,433)(383,425,390,432)(384,424,391,431)(385,423,392,430)>;
G:=Group( (1,42,296,240)(2,29,297,241)(3,30,298,242)(4,31,299,243)(5,32,300,244)(6,33,301,245)(7,34,302,246)(8,35,303,247)(9,36,304,248)(10,37,305,249)(11,38,306,250)(12,39,307,251)(13,40,308,252)(14,41,295,239)(15,221,352,171)(16,222,353,172)(17,223,354,173)(18,224,355,174)(19,211,356,175)(20,212,357,176)(21,213,358,177)(22,214,359,178)(23,215,360,179)(24,216,361,180)(25,217,362,181)(26,218,363,182)(27,219,364,169)(28,220,351,170)(43,83,61,105)(44,84,62,106)(45,71,63,107)(46,72,64,108)(47,73,65,109)(48,74,66,110)(49,75,67,111)(50,76,68,112)(51,77,69,99)(52,78,70,100)(53,79,57,101)(54,80,58,102)(55,81,59,103)(56,82,60,104)(85,124,343,333)(86,125,344,334)(87,126,345,335)(88,113,346,336)(89,114,347,323)(90,115,348,324)(91,116,349,325)(92,117,350,326)(93,118,337,327)(94,119,338,328)(95,120,339,329)(96,121,340,330)(97,122,341,331)(98,123,342,332)(127,392,201,266)(128,379,202,253)(129,380,203,254)(130,381,204,255)(131,382,205,256)(132,383,206,257)(133,384,207,258)(134,385,208,259)(135,386,209,260)(136,387,210,261)(137,388,197,262)(138,389,198,263)(139,390,199,264)(140,391,200,265)(141,234,435,312)(142,235,436,313)(143,236,437,314)(144,237,438,315)(145,238,439,316)(146,225,440,317)(147,226,441,318)(148,227,442,319)(149,228,443,320)(150,229,444,321)(151,230,445,322)(152,231,446,309)(153,232,447,310)(154,233,448,311)(155,272,433,188)(156,273,434,189)(157,274,421,190)(158,275,422,191)(159,276,423,192)(160,277,424,193)(161,278,425,194)(162,279,426,195)(163,280,427,196)(164,267,428,183)(165,268,429,184)(166,269,430,185)(167,270,431,186)(168,271,432,187)(281,406,417,371)(282,393,418,372)(283,394,419,373)(284,395,420,374)(285,396,407,375)(286,397,408,376)(287,398,409,377)(288,399,410,378)(289,400,411,365)(290,401,412,366)(291,402,413,367)(292,403,414,368)(293,404,415,369)(294,405,416,370), (1,328,287,205)(2,329,288,206)(3,330,289,207)(4,331,290,208)(5,332,291,209)(6,333,292,210)(7,334,293,197)(8,335,294,198)(9,336,281,199)(10,323,282,200)(11,324,283,201)(12,325,284,202)(13,326,285,203)(14,327,286,204)(15,196,48,236)(16,183,49,237)(17,184,50,238)(18,185,51,225)(19,186,52,226)(20,187,53,227)(21,188,54,228)(22,189,55,229)(23,190,56,230)(24,191,43,231)(25,192,44,232)(26,193,45,233)(27,194,46,234)(28,195,47,235)(29,95,399,257)(30,96,400,258)(31,97,401,259)(32,98,402,260)(33,85,403,261)(34,86,404,262)(35,87,405,263)(36,88,406,264)(37,89,393,265)(38,90,394,266)(39,91,395,253)(40,92,396,254)(41,93,397,255)(42,94,398,256)(57,319,357,271)(58,320,358,272)(59,321,359,273)(60,322,360,274)(61,309,361,275)(62,310,362,276)(63,311,363,277)(64,312,364,278)(65,313,351,279)(66,314,352,280)(67,315,353,267)(68,316,354,268)(69,317,355,269)(70,318,356,270)(71,448,218,160)(72,435,219,161)(73,436,220,162)(74,437,221,163)(75,438,222,164)(76,439,223,165)(77,440,224,166)(78,441,211,167)(79,442,212,168)(80,443,213,155)(81,444,214,156)(82,445,215,157)(83,446,216,158)(84,447,217,159)(99,146,174,430)(100,147,175,431)(101,148,176,432)(102,149,177,433)(103,150,178,434)(104,151,179,421)(105,152,180,422)(106,153,181,423)(107,154,182,424)(108,141,169,425)(109,142,170,426)(110,143,171,427)(111,144,172,428)(112,145,173,429)(113,417,139,304)(114,418,140,305)(115,419,127,306)(116,420,128,307)(117,407,129,308)(118,408,130,295)(119,409,131,296)(120,410,132,297)(121,411,133,298)(122,412,134,299)(123,413,135,300)(124,414,136,301)(125,415,137,302)(126,416,138,303)(239,337,376,381)(240,338,377,382)(241,339,378,383)(242,340,365,384)(243,341,366,385)(244,342,367,386)(245,343,368,387)(246,344,369,388)(247,345,370,389)(248,346,371,390)(249,347,372,391)(250,348,373,392)(251,349,374,379)(252,350,375,380), (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), (1,28,8,21)(2,27,9,20)(3,26,10,19)(4,25,11,18)(5,24,12,17)(6,23,13,16)(7,22,14,15)(29,219,36,212)(30,218,37,211)(31,217,38,224)(32,216,39,223)(33,215,40,222)(34,214,41,221)(35,213,42,220)(43,284,50,291)(44,283,51,290)(45,282,52,289)(46,281,53,288)(47,294,54,287)(48,293,55,286)(49,292,56,285)(57,410,64,417)(58,409,65,416)(59,408,66,415)(60,407,67,414)(61,420,68,413)(62,419,69,412)(63,418,70,411)(71,393,78,400)(72,406,79,399)(73,405,80,398)(74,404,81,397)(75,403,82,396)(76,402,83,395)(77,401,84,394)(85,445,92,438)(86,444,93,437)(87,443,94,436)(88,442,95,435)(89,441,96,448)(90,440,97,447)(91,439,98,446)(99,366,106,373)(100,365,107,372)(101,378,108,371)(102,377,109,370)(103,376,110,369)(104,375,111,368)(105,374,112,367)(113,319,120,312)(114,318,121,311)(115,317,122,310)(116,316,123,309)(117,315,124,322)(118,314,125,321)(119,313,126,320)(127,269,134,276)(128,268,135,275)(129,267,136,274)(130,280,137,273)(131,279,138,272)(132,278,139,271)(133,277,140,270)(141,346,148,339)(142,345,149,338)(143,344,150,337)(144,343,151,350)(145,342,152,349)(146,341,153,348)(147,340,154,347)(155,256,162,263)(156,255,163,262)(157,254,164,261)(158,253,165,260)(159,266,166,259)(160,265,167,258)(161,264,168,257)(169,248,176,241)(170,247,177,240)(171,246,178,239)(172,245,179,252)(173,244,180,251)(174,243,181,250)(175,242,182,249)(183,210,190,203)(184,209,191,202)(185,208,192,201)(186,207,193,200)(187,206,194,199)(188,205,195,198)(189,204,196,197)(225,331,232,324)(226,330,233,323)(227,329,234,336)(228,328,235,335)(229,327,236,334)(230,326,237,333)(231,325,238,332)(295,352,302,359)(296,351,303,358)(297,364,304,357)(298,363,305,356)(299,362,306,355)(300,361,307,354)(301,360,308,353)(379,429,386,422)(380,428,387,421)(381,427,388,434)(382,426,389,433)(383,425,390,432)(384,424,391,431)(385,423,392,430) );
G=PermutationGroup([[(1,42,296,240),(2,29,297,241),(3,30,298,242),(4,31,299,243),(5,32,300,244),(6,33,301,245),(7,34,302,246),(8,35,303,247),(9,36,304,248),(10,37,305,249),(11,38,306,250),(12,39,307,251),(13,40,308,252),(14,41,295,239),(15,221,352,171),(16,222,353,172),(17,223,354,173),(18,224,355,174),(19,211,356,175),(20,212,357,176),(21,213,358,177),(22,214,359,178),(23,215,360,179),(24,216,361,180),(25,217,362,181),(26,218,363,182),(27,219,364,169),(28,220,351,170),(43,83,61,105),(44,84,62,106),(45,71,63,107),(46,72,64,108),(47,73,65,109),(48,74,66,110),(49,75,67,111),(50,76,68,112),(51,77,69,99),(52,78,70,100),(53,79,57,101),(54,80,58,102),(55,81,59,103),(56,82,60,104),(85,124,343,333),(86,125,344,334),(87,126,345,335),(88,113,346,336),(89,114,347,323),(90,115,348,324),(91,116,349,325),(92,117,350,326),(93,118,337,327),(94,119,338,328),(95,120,339,329),(96,121,340,330),(97,122,341,331),(98,123,342,332),(127,392,201,266),(128,379,202,253),(129,380,203,254),(130,381,204,255),(131,382,205,256),(132,383,206,257),(133,384,207,258),(134,385,208,259),(135,386,209,260),(136,387,210,261),(137,388,197,262),(138,389,198,263),(139,390,199,264),(140,391,200,265),(141,234,435,312),(142,235,436,313),(143,236,437,314),(144,237,438,315),(145,238,439,316),(146,225,440,317),(147,226,441,318),(148,227,442,319),(149,228,443,320),(150,229,444,321),(151,230,445,322),(152,231,446,309),(153,232,447,310),(154,233,448,311),(155,272,433,188),(156,273,434,189),(157,274,421,190),(158,275,422,191),(159,276,423,192),(160,277,424,193),(161,278,425,194),(162,279,426,195),(163,280,427,196),(164,267,428,183),(165,268,429,184),(166,269,430,185),(167,270,431,186),(168,271,432,187),(281,406,417,371),(282,393,418,372),(283,394,419,373),(284,395,420,374),(285,396,407,375),(286,397,408,376),(287,398,409,377),(288,399,410,378),(289,400,411,365),(290,401,412,366),(291,402,413,367),(292,403,414,368),(293,404,415,369),(294,405,416,370)], [(1,328,287,205),(2,329,288,206),(3,330,289,207),(4,331,290,208),(5,332,291,209),(6,333,292,210),(7,334,293,197),(8,335,294,198),(9,336,281,199),(10,323,282,200),(11,324,283,201),(12,325,284,202),(13,326,285,203),(14,327,286,204),(15,196,48,236),(16,183,49,237),(17,184,50,238),(18,185,51,225),(19,186,52,226),(20,187,53,227),(21,188,54,228),(22,189,55,229),(23,190,56,230),(24,191,43,231),(25,192,44,232),(26,193,45,233),(27,194,46,234),(28,195,47,235),(29,95,399,257),(30,96,400,258),(31,97,401,259),(32,98,402,260),(33,85,403,261),(34,86,404,262),(35,87,405,263),(36,88,406,264),(37,89,393,265),(38,90,394,266),(39,91,395,253),(40,92,396,254),(41,93,397,255),(42,94,398,256),(57,319,357,271),(58,320,358,272),(59,321,359,273),(60,322,360,274),(61,309,361,275),(62,310,362,276),(63,311,363,277),(64,312,364,278),(65,313,351,279),(66,314,352,280),(67,315,353,267),(68,316,354,268),(69,317,355,269),(70,318,356,270),(71,448,218,160),(72,435,219,161),(73,436,220,162),(74,437,221,163),(75,438,222,164),(76,439,223,165),(77,440,224,166),(78,441,211,167),(79,442,212,168),(80,443,213,155),(81,444,214,156),(82,445,215,157),(83,446,216,158),(84,447,217,159),(99,146,174,430),(100,147,175,431),(101,148,176,432),(102,149,177,433),(103,150,178,434),(104,151,179,421),(105,152,180,422),(106,153,181,423),(107,154,182,424),(108,141,169,425),(109,142,170,426),(110,143,171,427),(111,144,172,428),(112,145,173,429),(113,417,139,304),(114,418,140,305),(115,419,127,306),(116,420,128,307),(117,407,129,308),(118,408,130,295),(119,409,131,296),(120,410,132,297),(121,411,133,298),(122,412,134,299),(123,413,135,300),(124,414,136,301),(125,415,137,302),(126,416,138,303),(239,337,376,381),(240,338,377,382),(241,339,378,383),(242,340,365,384),(243,341,366,385),(244,342,367,386),(245,343,368,387),(246,344,369,388),(247,345,370,389),(248,346,371,390),(249,347,372,391),(250,348,373,392),(251,349,374,379),(252,350,375,380)], [(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)], [(1,28,8,21),(2,27,9,20),(3,26,10,19),(4,25,11,18),(5,24,12,17),(6,23,13,16),(7,22,14,15),(29,219,36,212),(30,218,37,211),(31,217,38,224),(32,216,39,223),(33,215,40,222),(34,214,41,221),(35,213,42,220),(43,284,50,291),(44,283,51,290),(45,282,52,289),(46,281,53,288),(47,294,54,287),(48,293,55,286),(49,292,56,285),(57,410,64,417),(58,409,65,416),(59,408,66,415),(60,407,67,414),(61,420,68,413),(62,419,69,412),(63,418,70,411),(71,393,78,400),(72,406,79,399),(73,405,80,398),(74,404,81,397),(75,403,82,396),(76,402,83,395),(77,401,84,394),(85,445,92,438),(86,444,93,437),(87,443,94,436),(88,442,95,435),(89,441,96,448),(90,440,97,447),(91,439,98,446),(99,366,106,373),(100,365,107,372),(101,378,108,371),(102,377,109,370),(103,376,110,369),(104,375,111,368),(105,374,112,367),(113,319,120,312),(114,318,121,311),(115,317,122,310),(116,316,123,309),(117,315,124,322),(118,314,125,321),(119,313,126,320),(127,269,134,276),(128,268,135,275),(129,267,136,274),(130,280,137,273),(131,279,138,272),(132,278,139,271),(133,277,140,270),(141,346,148,339),(142,345,149,338),(143,344,150,337),(144,343,151,350),(145,342,152,349),(146,341,153,348),(147,340,154,347),(155,256,162,263),(156,255,163,262),(157,254,164,261),(158,253,165,260),(159,266,166,259),(160,265,167,258),(161,264,168,257),(169,248,176,241),(170,247,177,240),(171,246,178,239),(172,245,179,252),(173,244,180,251),(174,243,181,250),(175,242,182,249),(183,210,190,203),(184,209,191,202),(185,208,192,201),(186,207,193,200),(187,206,194,199),(188,205,195,198),(189,204,196,197),(225,331,232,324),(226,330,233,323),(227,329,234,336),(228,328,235,335),(229,327,236,334),(230,326,237,333),(231,325,238,332),(295,352,302,359),(296,351,303,358),(297,364,304,357),(298,363,305,356),(299,362,306,355),(300,361,307,354),(301,360,308,353),(379,429,386,422),(380,428,387,421),(381,427,388,434),(382,426,389,433),(383,425,390,432),(384,424,391,431),(385,423,392,430)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D7 | C4○D4 | Dic7 | D14 | Dic14 | C4×D7 | D28 | C4○D28 |
| kernel | C4×C4⋊Dic7 | C14.C42 | C2×C4×Dic7 | C2×C4⋊Dic7 | C2×C4×C28 | C4⋊Dic7 | C4×C28 | C2×C28 | C2×C28 | C2×C42 | C2×C14 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 8 | 2 | 2 | 3 | 4 | 12 | 9 | 12 | 24 | 12 | 24 |
Matrix representation of C4×C4⋊Dic7 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 24 |
| 0 | 0 | 0 | 13 | 8 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 19 | 28 |
| 0 | 0 | 0 | 20 | 28 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 9 | 6 | 0 | 0 |
| 0 | 6 | 20 | 0 | 0 |
| 0 | 0 | 0 | 20 | 22 |
| 0 | 0 | 0 | 28 | 9 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,1,0,0,0,0,0,0,21,13,0,0,0,24,8],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,19,20,0,0,0,28,28],[17,0,0,0,0,0,9,6,0,0,0,6,20,0,0,0,0,0,20,28,0,0,0,22,9] >;
C4×C4⋊Dic7 in GAP, Magma, Sage, TeX
C_4\times C_4\rtimes {\rm Dic}_7 % in TeX
G:=Group("C4xC4:Dic7"); // GroupNames label
G:=SmallGroup(448,468);
// by ID
G=gap.SmallGroup(448,468);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,232,100,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c^7,a*b=b*a,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