direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4×Dic7, C28⋊C42, C4⋊1(C4×Dic7), C4⋊Dic7⋊13C4, (C4×Dic7)⋊7C4, C14.58(C4×D4), C14.22(C4×Q8), C2.1(Q8×Dic7), C2.2(D4×Dic7), C22.21(Q8×D7), C14.20(C2×C42), (C2×Dic7).22Q8, C22.106(D4×D7), C2.3(D28⋊C4), (C2×Dic7).211D4, (C22×C4).333D14, C2.3(Dic7⋊3Q8), C23.289(C22×D7), C14.34(C42⋊C2), C22.54(D4⋊2D7), C14.C42.34C2, (C22×C14).339C23, (C22×C28).346C22, C22.22(Q8⋊2D7), C22.20(C22×Dic7), (C22×Dic7).187C22, C7⋊4(C4×C4⋊C4), (C7×C4⋊C4)⋊7C4, C2.4(D7×C4⋊C4), C14.19(C2×C4⋊C4), C2.9(C2×C4×Dic7), (C2×C4⋊C4).28D7, (C14×C4⋊C4).11C2, (C2×C4×Dic7).5C2, C22.55(C2×C4×D7), (C2×C28).82(C2×C4), (C2×C4).151(C4×D7), (C2×C14).75(C2×Q8), C2.4(C4⋊C4⋊7D7), (C2×C14).328(C2×D4), (C2×C4⋊Dic7).32C2, (C2×C4).32(C2×Dic7), (C2×Dic7).62(C2×C4), (C2×C14).151(C4○D4), (C2×C14).115(C22×C4), SmallGroup(448,509)
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, bd=db, dcd-1=c-1 >
Subgroups: 644 in 194 conjugacy classes, 111 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, 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, C4×C4⋊C4, C4×Dic7, 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, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, D14, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×D7, C2×Dic7, C22×D7, C4×C4⋊C4, C4×Dic7, C2×C4×D7, D4×D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C22×Dic7, Dic7⋊3Q8, D7×C4⋊C4, C4⋊C4⋊7D7, D28⋊C4, C2×C4×Dic7, D4×Dic7, Q8×Dic7, C4⋊C4×Dic7
►Regular action on 448 points
Generators in S448
(1 394 61 83)(2 395 62 84)(3 396 63 71)(4 397 64 72)(5 398 65 73)(6 399 66 74)(7 400 67 75)(8 401 68 76)(9 402 69 77)(10 403 70 78)(11 404 57 79)(12 405 58 80)(13 406 59 81)(14 393 60 82)(15 345 209 100)(16 346 210 101)(17 347 197 102)(18 348 198 103)(19 349 199 104)(20 350 200 105)(21 337 201 106)(22 338 202 107)(23 339 203 108)(24 340 204 109)(25 341 205 110)(26 342 206 111)(27 343 207 112)(28 344 208 99)(29 333 181 296)(30 334 182 297)(31 335 169 298)(32 336 170 299)(33 323 171 300)(34 324 172 301)(35 325 173 302)(36 326 174 303)(37 327 175 304)(38 328 176 305)(39 329 177 306)(40 330 178 307)(41 331 179 308)(42 332 180 295)(43 157 390 119)(44 158 391 120)(45 159 392 121)(46 160 379 122)(47 161 380 123)(48 162 381 124)(49 163 382 125)(50 164 383 126)(51 165 384 113)(52 166 385 114)(53 167 386 115)(54 168 387 116)(55 155 388 117)(56 156 389 118)(85 368 148 225)(86 369 149 226)(87 370 150 227)(88 371 151 228)(89 372 152 229)(90 373 153 230)(91 374 154 231)(92 375 141 232)(93 376 142 233)(94 377 143 234)(95 378 144 235)(96 365 145 236)(97 366 146 237)(98 367 147 238)(127 280 313 185)(128 267 314 186)(129 268 315 187)(130 269 316 188)(131 270 317 189)(132 271 318 190)(133 272 319 191)(134 273 320 192)(135 274 321 193)(136 275 322 194)(137 276 309 195)(138 277 310 196)(139 278 311 183)(140 279 312 184)(211 429 447 411)(212 430 448 412)(213 431 435 413)(214 432 436 414)(215 433 437 415)(216 434 438 416)(217 421 439 417)(218 422 440 418)(219 423 441 419)(220 424 442 420)(221 425 443 407)(222 426 444 408)(223 427 445 409)(224 428 446 410)(239 353 262 293)(240 354 263 294)(241 355 264 281)(242 356 265 282)(243 357 266 283)(244 358 253 284)(245 359 254 285)(246 360 255 286)(247 361 256 287)(248 362 257 288)(249 363 258 289)(250 364 259 290)(251 351 260 291)(252 352 261 292)
(1 155 312 283)(2 156 313 284)(3 157 314 285)(4 158 315 286)(5 159 316 287)(6 160 317 288)(7 161 318 289)(8 162 319 290)(9 163 320 291)(10 164 321 292)(11 165 322 293)(12 166 309 294)(13 167 310 281)(14 168 311 282)(15 212 142 39)(16 213 143 40)(17 214 144 41)(18 215 145 42)(19 216 146 29)(20 217 147 30)(21 218 148 31)(22 219 149 32)(23 220 150 33)(24 221 151 34)(25 222 152 35)(26 223 153 36)(27 224 154 37)(28 211 141 38)(43 186 254 396)(44 187 255 397)(45 188 256 398)(46 189 257 399)(47 190 258 400)(48 191 259 401)(49 192 260 402)(50 193 261 403)(51 194 262 404)(52 195 263 405)(53 196 264 406)(54 183 265 393)(55 184 266 394)(56 185 253 395)(57 113 136 353)(58 114 137 354)(59 115 138 355)(60 116 139 356)(61 117 140 357)(62 118 127 358)(63 119 128 359)(64 120 129 360)(65 121 130 361)(66 122 131 362)(67 123 132 363)(68 124 133 364)(69 125 134 351)(70 126 135 352)(71 390 267 245)(72 391 268 246)(73 392 269 247)(74 379 270 248)(75 380 271 249)(76 381 272 250)(77 382 273 251)(78 383 274 252)(79 384 275 239)(80 385 276 240)(81 386 277 241)(82 387 278 242)(83 388 279 243)(84 389 280 244)(85 169 201 440)(86 170 202 441)(87 171 203 442)(88 172 204 443)(89 173 205 444)(90 174 206 445)(91 175 207 446)(92 176 208 447)(93 177 209 448)(94 178 210 435)(95 179 197 436)(96 180 198 437)(97 181 199 438)(98 182 200 439)(99 429 375 328)(100 430 376 329)(101 431 377 330)(102 432 378 331)(103 433 365 332)(104 434 366 333)(105 421 367 334)(106 422 368 335)(107 423 369 336)(108 424 370 323)(109 425 371 324)(110 426 372 325)(111 427 373 326)(112 428 374 327)(225 298 337 418)(226 299 338 419)(227 300 339 420)(228 301 340 407)(229 302 341 408)(230 303 342 409)(231 304 343 410)(232 305 344 411)(233 306 345 412)(234 307 346 413)(235 308 347 414)(236 295 348 415)(237 296 349 416)(238 297 350 417)
(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)
(1 227 8 234)(2 226 9 233)(3 225 10 232)(4 238 11 231)(5 237 12 230)(6 236 13 229)(7 235 14 228)(15 280 22 273)(16 279 23 272)(17 278 24 271)(18 277 25 270)(19 276 26 269)(20 275 27 268)(21 274 28 267)(29 385 36 392)(30 384 37 391)(31 383 38 390)(32 382 39 389)(33 381 40 388)(34 380 41 387)(35 379 42 386)(43 169 50 176)(44 182 51 175)(45 181 52 174)(46 180 53 173)(47 179 54 172)(48 178 55 171)(49 177 56 170)(57 374 64 367)(58 373 65 366)(59 372 66 365)(60 371 67 378)(61 370 68 377)(62 369 69 376)(63 368 70 375)(71 148 78 141)(72 147 79 154)(73 146 80 153)(74 145 81 152)(75 144 82 151)(76 143 83 150)(77 142 84 149)(85 403 92 396)(86 402 93 395)(87 401 94 394)(88 400 95 393)(89 399 96 406)(90 398 97 405)(91 397 98 404)(99 128 106 135)(100 127 107 134)(101 140 108 133)(102 139 109 132)(103 138 110 131)(104 137 111 130)(105 136 112 129)(113 327 120 334)(114 326 121 333)(115 325 122 332)(116 324 123 331)(117 323 124 330)(118 336 125 329)(119 335 126 328)(155 300 162 307)(156 299 163 306)(157 298 164 305)(158 297 165 304)(159 296 166 303)(160 295 167 302)(161 308 168 301)(183 204 190 197)(184 203 191 210)(185 202 192 209)(186 201 193 208)(187 200 194 207)(188 199 195 206)(189 198 196 205)(211 245 218 252)(212 244 219 251)(213 243 220 250)(214 242 221 249)(215 241 222 248)(216 240 223 247)(217 239 224 246)(253 441 260 448)(254 440 261 447)(255 439 262 446)(256 438 263 445)(257 437 264 444)(258 436 265 443)(259 435 266 442)(281 408 288 415)(282 407 289 414)(283 420 290 413)(284 419 291 412)(285 418 292 411)(286 417 293 410)(287 416 294 409)(309 342 316 349)(310 341 317 348)(311 340 318 347)(312 339 319 346)(313 338 320 345)(314 337 321 344)(315 350 322 343)(351 430 358 423)(352 429 359 422)(353 428 360 421)(354 427 361 434)(355 426 362 433)(356 425 363 432)(357 424 364 431)
G:=sub<Sym(448)| (1,394,61,83)(2,395,62,84)(3,396,63,71)(4,397,64,72)(5,398,65,73)(6,399,66,74)(7,400,67,75)(8,401,68,76)(9,402,69,77)(10,403,70,78)(11,404,57,79)(12,405,58,80)(13,406,59,81)(14,393,60,82)(15,345,209,100)(16,346,210,101)(17,347,197,102)(18,348,198,103)(19,349,199,104)(20,350,200,105)(21,337,201,106)(22,338,202,107)(23,339,203,108)(24,340,204,109)(25,341,205,110)(26,342,206,111)(27,343,207,112)(28,344,208,99)(29,333,181,296)(30,334,182,297)(31,335,169,298)(32,336,170,299)(33,323,171,300)(34,324,172,301)(35,325,173,302)(36,326,174,303)(37,327,175,304)(38,328,176,305)(39,329,177,306)(40,330,178,307)(41,331,179,308)(42,332,180,295)(43,157,390,119)(44,158,391,120)(45,159,392,121)(46,160,379,122)(47,161,380,123)(48,162,381,124)(49,163,382,125)(50,164,383,126)(51,165,384,113)(52,166,385,114)(53,167,386,115)(54,168,387,116)(55,155,388,117)(56,156,389,118)(85,368,148,225)(86,369,149,226)(87,370,150,227)(88,371,151,228)(89,372,152,229)(90,373,153,230)(91,374,154,231)(92,375,141,232)(93,376,142,233)(94,377,143,234)(95,378,144,235)(96,365,145,236)(97,366,146,237)(98,367,147,238)(127,280,313,185)(128,267,314,186)(129,268,315,187)(130,269,316,188)(131,270,317,189)(132,271,318,190)(133,272,319,191)(134,273,320,192)(135,274,321,193)(136,275,322,194)(137,276,309,195)(138,277,310,196)(139,278,311,183)(140,279,312,184)(211,429,447,411)(212,430,448,412)(213,431,435,413)(214,432,436,414)(215,433,437,415)(216,434,438,416)(217,421,439,417)(218,422,440,418)(219,423,441,419)(220,424,442,420)(221,425,443,407)(222,426,444,408)(223,427,445,409)(224,428,446,410)(239,353,262,293)(240,354,263,294)(241,355,264,281)(242,356,265,282)(243,357,266,283)(244,358,253,284)(245,359,254,285)(246,360,255,286)(247,361,256,287)(248,362,257,288)(249,363,258,289)(250,364,259,290)(251,351,260,291)(252,352,261,292), (1,155,312,283)(2,156,313,284)(3,157,314,285)(4,158,315,286)(5,159,316,287)(6,160,317,288)(7,161,318,289)(8,162,319,290)(9,163,320,291)(10,164,321,292)(11,165,322,293)(12,166,309,294)(13,167,310,281)(14,168,311,282)(15,212,142,39)(16,213,143,40)(17,214,144,41)(18,215,145,42)(19,216,146,29)(20,217,147,30)(21,218,148,31)(22,219,149,32)(23,220,150,33)(24,221,151,34)(25,222,152,35)(26,223,153,36)(27,224,154,37)(28,211,141,38)(43,186,254,396)(44,187,255,397)(45,188,256,398)(46,189,257,399)(47,190,258,400)(48,191,259,401)(49,192,260,402)(50,193,261,403)(51,194,262,404)(52,195,263,405)(53,196,264,406)(54,183,265,393)(55,184,266,394)(56,185,253,395)(57,113,136,353)(58,114,137,354)(59,115,138,355)(60,116,139,356)(61,117,140,357)(62,118,127,358)(63,119,128,359)(64,120,129,360)(65,121,130,361)(66,122,131,362)(67,123,132,363)(68,124,133,364)(69,125,134,351)(70,126,135,352)(71,390,267,245)(72,391,268,246)(73,392,269,247)(74,379,270,248)(75,380,271,249)(76,381,272,250)(77,382,273,251)(78,383,274,252)(79,384,275,239)(80,385,276,240)(81,386,277,241)(82,387,278,242)(83,388,279,243)(84,389,280,244)(85,169,201,440)(86,170,202,441)(87,171,203,442)(88,172,204,443)(89,173,205,444)(90,174,206,445)(91,175,207,446)(92,176,208,447)(93,177,209,448)(94,178,210,435)(95,179,197,436)(96,180,198,437)(97,181,199,438)(98,182,200,439)(99,429,375,328)(100,430,376,329)(101,431,377,330)(102,432,378,331)(103,433,365,332)(104,434,366,333)(105,421,367,334)(106,422,368,335)(107,423,369,336)(108,424,370,323)(109,425,371,324)(110,426,372,325)(111,427,373,326)(112,428,374,327)(225,298,337,418)(226,299,338,419)(227,300,339,420)(228,301,340,407)(229,302,341,408)(230,303,342,409)(231,304,343,410)(232,305,344,411)(233,306,345,412)(234,307,346,413)(235,308,347,414)(236,295,348,415)(237,296,349,416)(238,297,350,417), (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), (1,227,8,234)(2,226,9,233)(3,225,10,232)(4,238,11,231)(5,237,12,230)(6,236,13,229)(7,235,14,228)(15,280,22,273)(16,279,23,272)(17,278,24,271)(18,277,25,270)(19,276,26,269)(20,275,27,268)(21,274,28,267)(29,385,36,392)(30,384,37,391)(31,383,38,390)(32,382,39,389)(33,381,40,388)(34,380,41,387)(35,379,42,386)(43,169,50,176)(44,182,51,175)(45,181,52,174)(46,180,53,173)(47,179,54,172)(48,178,55,171)(49,177,56,170)(57,374,64,367)(58,373,65,366)(59,372,66,365)(60,371,67,378)(61,370,68,377)(62,369,69,376)(63,368,70,375)(71,148,78,141)(72,147,79,154)(73,146,80,153)(74,145,81,152)(75,144,82,151)(76,143,83,150)(77,142,84,149)(85,403,92,396)(86,402,93,395)(87,401,94,394)(88,400,95,393)(89,399,96,406)(90,398,97,405)(91,397,98,404)(99,128,106,135)(100,127,107,134)(101,140,108,133)(102,139,109,132)(103,138,110,131)(104,137,111,130)(105,136,112,129)(113,327,120,334)(114,326,121,333)(115,325,122,332)(116,324,123,331)(117,323,124,330)(118,336,125,329)(119,335,126,328)(155,300,162,307)(156,299,163,306)(157,298,164,305)(158,297,165,304)(159,296,166,303)(160,295,167,302)(161,308,168,301)(183,204,190,197)(184,203,191,210)(185,202,192,209)(186,201,193,208)(187,200,194,207)(188,199,195,206)(189,198,196,205)(211,245,218,252)(212,244,219,251)(213,243,220,250)(214,242,221,249)(215,241,222,248)(216,240,223,247)(217,239,224,246)(253,441,260,448)(254,440,261,447)(255,439,262,446)(256,438,263,445)(257,437,264,444)(258,436,265,443)(259,435,266,442)(281,408,288,415)(282,407,289,414)(283,420,290,413)(284,419,291,412)(285,418,292,411)(286,417,293,410)(287,416,294,409)(309,342,316,349)(310,341,317,348)(311,340,318,347)(312,339,319,346)(313,338,320,345)(314,337,321,344)(315,350,322,343)(351,430,358,423)(352,429,359,422)(353,428,360,421)(354,427,361,434)(355,426,362,433)(356,425,363,432)(357,424,364,431)>;
G:=Group( (1,394,61,83)(2,395,62,84)(3,396,63,71)(4,397,64,72)(5,398,65,73)(6,399,66,74)(7,400,67,75)(8,401,68,76)(9,402,69,77)(10,403,70,78)(11,404,57,79)(12,405,58,80)(13,406,59,81)(14,393,60,82)(15,345,209,100)(16,346,210,101)(17,347,197,102)(18,348,198,103)(19,349,199,104)(20,350,200,105)(21,337,201,106)(22,338,202,107)(23,339,203,108)(24,340,204,109)(25,341,205,110)(26,342,206,111)(27,343,207,112)(28,344,208,99)(29,333,181,296)(30,334,182,297)(31,335,169,298)(32,336,170,299)(33,323,171,300)(34,324,172,301)(35,325,173,302)(36,326,174,303)(37,327,175,304)(38,328,176,305)(39,329,177,306)(40,330,178,307)(41,331,179,308)(42,332,180,295)(43,157,390,119)(44,158,391,120)(45,159,392,121)(46,160,379,122)(47,161,380,123)(48,162,381,124)(49,163,382,125)(50,164,383,126)(51,165,384,113)(52,166,385,114)(53,167,386,115)(54,168,387,116)(55,155,388,117)(56,156,389,118)(85,368,148,225)(86,369,149,226)(87,370,150,227)(88,371,151,228)(89,372,152,229)(90,373,153,230)(91,374,154,231)(92,375,141,232)(93,376,142,233)(94,377,143,234)(95,378,144,235)(96,365,145,236)(97,366,146,237)(98,367,147,238)(127,280,313,185)(128,267,314,186)(129,268,315,187)(130,269,316,188)(131,270,317,189)(132,271,318,190)(133,272,319,191)(134,273,320,192)(135,274,321,193)(136,275,322,194)(137,276,309,195)(138,277,310,196)(139,278,311,183)(140,279,312,184)(211,429,447,411)(212,430,448,412)(213,431,435,413)(214,432,436,414)(215,433,437,415)(216,434,438,416)(217,421,439,417)(218,422,440,418)(219,423,441,419)(220,424,442,420)(221,425,443,407)(222,426,444,408)(223,427,445,409)(224,428,446,410)(239,353,262,293)(240,354,263,294)(241,355,264,281)(242,356,265,282)(243,357,266,283)(244,358,253,284)(245,359,254,285)(246,360,255,286)(247,361,256,287)(248,362,257,288)(249,363,258,289)(250,364,259,290)(251,351,260,291)(252,352,261,292), (1,155,312,283)(2,156,313,284)(3,157,314,285)(4,158,315,286)(5,159,316,287)(6,160,317,288)(7,161,318,289)(8,162,319,290)(9,163,320,291)(10,164,321,292)(11,165,322,293)(12,166,309,294)(13,167,310,281)(14,168,311,282)(15,212,142,39)(16,213,143,40)(17,214,144,41)(18,215,145,42)(19,216,146,29)(20,217,147,30)(21,218,148,31)(22,219,149,32)(23,220,150,33)(24,221,151,34)(25,222,152,35)(26,223,153,36)(27,224,154,37)(28,211,141,38)(43,186,254,396)(44,187,255,397)(45,188,256,398)(46,189,257,399)(47,190,258,400)(48,191,259,401)(49,192,260,402)(50,193,261,403)(51,194,262,404)(52,195,263,405)(53,196,264,406)(54,183,265,393)(55,184,266,394)(56,185,253,395)(57,113,136,353)(58,114,137,354)(59,115,138,355)(60,116,139,356)(61,117,140,357)(62,118,127,358)(63,119,128,359)(64,120,129,360)(65,121,130,361)(66,122,131,362)(67,123,132,363)(68,124,133,364)(69,125,134,351)(70,126,135,352)(71,390,267,245)(72,391,268,246)(73,392,269,247)(74,379,270,248)(75,380,271,249)(76,381,272,250)(77,382,273,251)(78,383,274,252)(79,384,275,239)(80,385,276,240)(81,386,277,241)(82,387,278,242)(83,388,279,243)(84,389,280,244)(85,169,201,440)(86,170,202,441)(87,171,203,442)(88,172,204,443)(89,173,205,444)(90,174,206,445)(91,175,207,446)(92,176,208,447)(93,177,209,448)(94,178,210,435)(95,179,197,436)(96,180,198,437)(97,181,199,438)(98,182,200,439)(99,429,375,328)(100,430,376,329)(101,431,377,330)(102,432,378,331)(103,433,365,332)(104,434,366,333)(105,421,367,334)(106,422,368,335)(107,423,369,336)(108,424,370,323)(109,425,371,324)(110,426,372,325)(111,427,373,326)(112,428,374,327)(225,298,337,418)(226,299,338,419)(227,300,339,420)(228,301,340,407)(229,302,341,408)(230,303,342,409)(231,304,343,410)(232,305,344,411)(233,306,345,412)(234,307,346,413)(235,308,347,414)(236,295,348,415)(237,296,349,416)(238,297,350,417), (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), (1,227,8,234)(2,226,9,233)(3,225,10,232)(4,238,11,231)(5,237,12,230)(6,236,13,229)(7,235,14,228)(15,280,22,273)(16,279,23,272)(17,278,24,271)(18,277,25,270)(19,276,26,269)(20,275,27,268)(21,274,28,267)(29,385,36,392)(30,384,37,391)(31,383,38,390)(32,382,39,389)(33,381,40,388)(34,380,41,387)(35,379,42,386)(43,169,50,176)(44,182,51,175)(45,181,52,174)(46,180,53,173)(47,179,54,172)(48,178,55,171)(49,177,56,170)(57,374,64,367)(58,373,65,366)(59,372,66,365)(60,371,67,378)(61,370,68,377)(62,369,69,376)(63,368,70,375)(71,148,78,141)(72,147,79,154)(73,146,80,153)(74,145,81,152)(75,144,82,151)(76,143,83,150)(77,142,84,149)(85,403,92,396)(86,402,93,395)(87,401,94,394)(88,400,95,393)(89,399,96,406)(90,398,97,405)(91,397,98,404)(99,128,106,135)(100,127,107,134)(101,140,108,133)(102,139,109,132)(103,138,110,131)(104,137,111,130)(105,136,112,129)(113,327,120,334)(114,326,121,333)(115,325,122,332)(116,324,123,331)(117,323,124,330)(118,336,125,329)(119,335,126,328)(155,300,162,307)(156,299,163,306)(157,298,164,305)(158,297,165,304)(159,296,166,303)(160,295,167,302)(161,308,168,301)(183,204,190,197)(184,203,191,210)(185,202,192,209)(186,201,193,208)(187,200,194,207)(188,199,195,206)(189,198,196,205)(211,245,218,252)(212,244,219,251)(213,243,220,250)(214,242,221,249)(215,241,222,248)(216,240,223,247)(217,239,224,246)(253,441,260,448)(254,440,261,447)(255,439,262,446)(256,438,263,445)(257,437,264,444)(258,436,265,443)(259,435,266,442)(281,408,288,415)(282,407,289,414)(283,420,290,413)(284,419,291,412)(285,418,292,411)(286,417,293,410)(287,416,294,409)(309,342,316,349)(310,341,317,348)(311,340,318,347)(312,339,319,346)(313,338,320,345)(314,337,321,344)(315,350,322,343)(351,430,358,423)(352,429,359,422)(353,428,360,421)(354,427,361,434)(355,426,362,433)(356,425,363,432)(357,424,364,431) );
G=PermutationGroup([[(1,394,61,83),(2,395,62,84),(3,396,63,71),(4,397,64,72),(5,398,65,73),(6,399,66,74),(7,400,67,75),(8,401,68,76),(9,402,69,77),(10,403,70,78),(11,404,57,79),(12,405,58,80),(13,406,59,81),(14,393,60,82),(15,345,209,100),(16,346,210,101),(17,347,197,102),(18,348,198,103),(19,349,199,104),(20,350,200,105),(21,337,201,106),(22,338,202,107),(23,339,203,108),(24,340,204,109),(25,341,205,110),(26,342,206,111),(27,343,207,112),(28,344,208,99),(29,333,181,296),(30,334,182,297),(31,335,169,298),(32,336,170,299),(33,323,171,300),(34,324,172,301),(35,325,173,302),(36,326,174,303),(37,327,175,304),(38,328,176,305),(39,329,177,306),(40,330,178,307),(41,331,179,308),(42,332,180,295),(43,157,390,119),(44,158,391,120),(45,159,392,121),(46,160,379,122),(47,161,380,123),(48,162,381,124),(49,163,382,125),(50,164,383,126),(51,165,384,113),(52,166,385,114),(53,167,386,115),(54,168,387,116),(55,155,388,117),(56,156,389,118),(85,368,148,225),(86,369,149,226),(87,370,150,227),(88,371,151,228),(89,372,152,229),(90,373,153,230),(91,374,154,231),(92,375,141,232),(93,376,142,233),(94,377,143,234),(95,378,144,235),(96,365,145,236),(97,366,146,237),(98,367,147,238),(127,280,313,185),(128,267,314,186),(129,268,315,187),(130,269,316,188),(131,270,317,189),(132,271,318,190),(133,272,319,191),(134,273,320,192),(135,274,321,193),(136,275,322,194),(137,276,309,195),(138,277,310,196),(139,278,311,183),(140,279,312,184),(211,429,447,411),(212,430,448,412),(213,431,435,413),(214,432,436,414),(215,433,437,415),(216,434,438,416),(217,421,439,417),(218,422,440,418),(219,423,441,419),(220,424,442,420),(221,425,443,407),(222,426,444,408),(223,427,445,409),(224,428,446,410),(239,353,262,293),(240,354,263,294),(241,355,264,281),(242,356,265,282),(243,357,266,283),(244,358,253,284),(245,359,254,285),(246,360,255,286),(247,361,256,287),(248,362,257,288),(249,363,258,289),(250,364,259,290),(251,351,260,291),(252,352,261,292)], [(1,155,312,283),(2,156,313,284),(3,157,314,285),(4,158,315,286),(5,159,316,287),(6,160,317,288),(7,161,318,289),(8,162,319,290),(9,163,320,291),(10,164,321,292),(11,165,322,293),(12,166,309,294),(13,167,310,281),(14,168,311,282),(15,212,142,39),(16,213,143,40),(17,214,144,41),(18,215,145,42),(19,216,146,29),(20,217,147,30),(21,218,148,31),(22,219,149,32),(23,220,150,33),(24,221,151,34),(25,222,152,35),(26,223,153,36),(27,224,154,37),(28,211,141,38),(43,186,254,396),(44,187,255,397),(45,188,256,398),(46,189,257,399),(47,190,258,400),(48,191,259,401),(49,192,260,402),(50,193,261,403),(51,194,262,404),(52,195,263,405),(53,196,264,406),(54,183,265,393),(55,184,266,394),(56,185,253,395),(57,113,136,353),(58,114,137,354),(59,115,138,355),(60,116,139,356),(61,117,140,357),(62,118,127,358),(63,119,128,359),(64,120,129,360),(65,121,130,361),(66,122,131,362),(67,123,132,363),(68,124,133,364),(69,125,134,351),(70,126,135,352),(71,390,267,245),(72,391,268,246),(73,392,269,247),(74,379,270,248),(75,380,271,249),(76,381,272,250),(77,382,273,251),(78,383,274,252),(79,384,275,239),(80,385,276,240),(81,386,277,241),(82,387,278,242),(83,388,279,243),(84,389,280,244),(85,169,201,440),(86,170,202,441),(87,171,203,442),(88,172,204,443),(89,173,205,444),(90,174,206,445),(91,175,207,446),(92,176,208,447),(93,177,209,448),(94,178,210,435),(95,179,197,436),(96,180,198,437),(97,181,199,438),(98,182,200,439),(99,429,375,328),(100,430,376,329),(101,431,377,330),(102,432,378,331),(103,433,365,332),(104,434,366,333),(105,421,367,334),(106,422,368,335),(107,423,369,336),(108,424,370,323),(109,425,371,324),(110,426,372,325),(111,427,373,326),(112,428,374,327),(225,298,337,418),(226,299,338,419),(227,300,339,420),(228,301,340,407),(229,302,341,408),(230,303,342,409),(231,304,343,410),(232,305,344,411),(233,306,345,412),(234,307,346,413),(235,308,347,414),(236,295,348,415),(237,296,349,416),(238,297,350,417)], [(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)], [(1,227,8,234),(2,226,9,233),(3,225,10,232),(4,238,11,231),(5,237,12,230),(6,236,13,229),(7,235,14,228),(15,280,22,273),(16,279,23,272),(17,278,24,271),(18,277,25,270),(19,276,26,269),(20,275,27,268),(21,274,28,267),(29,385,36,392),(30,384,37,391),(31,383,38,390),(32,382,39,389),(33,381,40,388),(34,380,41,387),(35,379,42,386),(43,169,50,176),(44,182,51,175),(45,181,52,174),(46,180,53,173),(47,179,54,172),(48,178,55,171),(49,177,56,170),(57,374,64,367),(58,373,65,366),(59,372,66,365),(60,371,67,378),(61,370,68,377),(62,369,69,376),(63,368,70,375),(71,148,78,141),(72,147,79,154),(73,146,80,153),(74,145,81,152),(75,144,82,151),(76,143,83,150),(77,142,84,149),(85,403,92,396),(86,402,93,395),(87,401,94,394),(88,400,95,393),(89,399,96,406),(90,398,97,405),(91,397,98,404),(99,128,106,135),(100,127,107,134),(101,140,108,133),(102,139,109,132),(103,138,110,131),(104,137,111,130),(105,136,112,129),(113,327,120,334),(114,326,121,333),(115,325,122,332),(116,324,123,331),(117,323,124,330),(118,336,125,329),(119,335,126,328),(155,300,162,307),(156,299,163,306),(157,298,164,305),(158,297,165,304),(159,296,166,303),(160,295,167,302),(161,308,168,301),(183,204,190,197),(184,203,191,210),(185,202,192,209),(186,201,193,208),(187,200,194,207),(188,199,195,206),(189,198,196,205),(211,245,218,252),(212,244,219,251),(213,243,220,250),(214,242,221,249),(215,241,222,248),(216,240,223,247),(217,239,224,246),(253,441,260,448),(254,440,261,447),(255,439,262,446),(256,438,263,445),(257,437,264,444),(258,436,265,443),(259,435,266,442),(281,408,288,415),(282,407,289,414),(283,420,290,413),(284,419,291,412),(285,418,292,411),(286,417,293,410),(287,416,294,409),(309,342,316,349),(310,341,317,348),(311,340,318,347),(312,339,319,346),(313,338,320,345),(314,337,321,344),(315,350,322,343),(351,430,358,423),(352,429,359,422),(353,428,360,421),(354,427,361,434),(355,426,362,433),(356,425,363,432),(357,424,364,431)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AF | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D7 | C4○D4 | Dic7 | D14 | C4×D7 | D4×D7 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 |
| kernel | C4⋊C4×Dic7 | C14.C42 | C2×C4×Dic7 | C2×C4⋊Dic7 | C14×C4⋊C4 | C4×Dic7 | C4⋊Dic7 | C7×C4⋊C4 | C2×Dic7 | C2×Dic7 | C2×C4⋊C4 | C2×C14 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 3 | 1 | 1 | 8 | 8 | 8 | 2 | 2 | 3 | 4 | 12 | 9 | 24 | 3 | 3 | 3 | 3 |
Matrix representation of C4⋊C4×Dic7 ►in GL6(𝔽29)
| 16 | 27 | 0 | 0 | 0 | 0 |
| 27 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 28 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 26 |
| 0 | 0 | 0 | 0 | 13 | 26 |
G:=sub<GL(6,GF(29))| [16,27,0,0,0,0,27,13,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,2,0,0,0,0,28,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,13,0,0,0,0,26,26] >;
C4⋊C4×Dic7 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times {\rm Dic}_7 % in TeX
G:=Group("C4:C4xDic7"); // GroupNames label
G:=SmallGroup(448,509);
// by ID
G=gap.SmallGroup(448,509);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,387,100,18822]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations