direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×Q8⋊C4, Q8⋊3(C2×C28), (C2×Q8)⋊5C28, (Q8×C14)⋊15C4, C4.52(D4×C14), C2.1(C14×Q16), C28.459(C2×D4), (C2×C28).415D4, (C22×C8).5C14, C4.2(C22×C28), C14.48(C2×Q16), (C2×C14).20Q16, C2.2(C14×SD16), C23.55(C7×D4), C22.5(C7×Q16), (C22×C56).11C2, (C2×C14).45SD16, C14.82(C2×SD16), C22.42(D4×C14), (C22×Q8).4C14, C28.81(C22⋊C4), (C2×C28).891C23, C28.147(C22×C4), (C2×C56).357C22, (C22×C14).216D4, C22.11(C7×SD16), (Q8×C14).250C22, (C22×C28).582C22, (C7×Q8)⋊21(C2×C4), (C2×C4).69(C7×D4), (C2×C4⋊C4).12C14, (C14×C4⋊C4).41C2, (Q8×C2×C14).14C2, C4⋊C4.37(C2×C14), (C2×C4).48(C2×C28), (C2×C8).60(C2×C14), C4.13(C7×C22⋊C4), (C2×C28).269(C2×C4), (C2×C14).618(C2×D4), C2.18(C14×C22⋊C4), (C2×Q8).35(C2×C14), (C7×C4⋊C4).358C22, C14.106(C2×C22⋊C4), (C2×C4).66(C22×C14), C22.34(C7×C22⋊C4), (C22×C4).111(C2×C14), (C2×C14).139(C22⋊C4), SmallGroup(448,823)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×Q8⋊C4
G = < a,b,c,d | a14=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 242 in 162 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C28, C28, C28, C2×C14, C2×C14, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C56, C2×C28, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×C14, C2×Q8⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C22×C28, C22×C28, Q8×C14, Q8×C14, C7×Q8⋊C4, C14×C4⋊C4, C22×C56, Q8×C2×C14, C14×Q8⋊C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C28, C2×C14, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C2×C28, C7×D4, C22×C14, C2×Q8⋊C4, C7×C22⋊C4, C7×SD16, C7×Q16, C22×C28, D4×C14, C7×Q8⋊C4, C14×C22⋊C4, C14×SD16, C14×Q16, C14×Q8⋊C4
►Regular action on 448 points
Generators in S448
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154)(155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224)(225 226 227 228 229 230 231 232 233 234 235 236 237 238)(239 240 241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264 265 266)(267 268 269 270 271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290 291 292 293 294)(295 296 297 298 299 300 301 302 303 304 305 306 307 308)(309 310 311 312 313 314 315 316 317 318 319 320 321 322)(323 324 325 326 327 328 329 330 331 332 333 334 335 336)(337 338 339 340 341 342 343 344 345 346 347 348 349 350)(351 352 353 354 355 356 357 358 359 360 361 362 363 364)(365 366 367 368 369 370 371 372 373 374 375 376 377 378)(379 380 381 382 383 384 385 386 387 388 389 390 391 392)(393 394 395 396 397 398 399 400 401 402 403 404 405 406)(407 408 409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432 433 434)(435 436 437 438 439 440 441 442 443 444 445 446 447 448)
(1 73 151 101)(2 74 152 102)(3 75 153 103)(4 76 154 104)(5 77 141 105)(6 78 142 106)(7 79 143 107)(8 80 144 108)(9 81 145 109)(10 82 146 110)(11 83 147 111)(12 84 148 112)(13 71 149 99)(14 72 150 100)(15 411 227 127)(16 412 228 128)(17 413 229 129)(18 414 230 130)(19 415 231 131)(20 416 232 132)(21 417 233 133)(22 418 234 134)(23 419 235 135)(24 420 236 136)(25 407 237 137)(26 408 238 138)(27 409 225 139)(28 410 226 140)(29 312 213 160)(30 313 214 161)(31 314 215 162)(32 315 216 163)(33 316 217 164)(34 317 218 165)(35 318 219 166)(36 319 220 167)(37 320 221 168)(38 321 222 155)(39 322 223 156)(40 309 224 157)(41 310 211 158)(42 311 212 159)(43 174 331 380)(44 175 332 381)(45 176 333 382)(46 177 334 383)(47 178 335 384)(48 179 336 385)(49 180 323 386)(50 181 324 387)(51 182 325 388)(52 169 326 389)(53 170 327 390)(54 171 328 391)(55 172 329 392)(56 173 330 379)(57 294 117 442)(58 281 118 443)(59 282 119 444)(60 283 120 445)(61 284 121 446)(62 285 122 447)(63 286 123 448)(64 287 124 435)(65 288 125 436)(66 289 126 437)(67 290 113 438)(68 291 114 439)(69 292 115 440)(70 293 116 441)(85 378 269 206)(86 365 270 207)(87 366 271 208)(88 367 272 209)(89 368 273 210)(90 369 274 197)(91 370 275 198)(92 371 276 199)(93 372 277 200)(94 373 278 201)(95 374 279 202)(96 375 280 203)(97 376 267 204)(98 377 268 205)(183 433 342 248)(184 434 343 249)(185 421 344 250)(186 422 345 251)(187 423 346 252)(188 424 347 239)(189 425 348 240)(190 426 349 241)(191 427 350 242)(192 428 337 243)(193 429 338 244)(194 430 339 245)(195 431 340 246)(196 432 341 247)(253 299 359 398)(254 300 360 399)(255 301 361 400)(256 302 362 401)(257 303 363 402)(258 304 364 403)(259 305 351 404)(260 306 352 405)(261 307 353 406)(262 308 354 393)(263 295 355 394)(264 296 356 395)(265 297 357 396)(266 298 358 397)
(1 330 151 56)(2 331 152 43)(3 332 153 44)(4 333 154 45)(5 334 141 46)(6 335 142 47)(7 336 143 48)(8 323 144 49)(9 324 145 50)(10 325 146 51)(11 326 147 52)(12 327 148 53)(13 328 149 54)(14 329 150 55)(15 354 227 262)(16 355 228 263)(17 356 229 264)(18 357 230 265)(19 358 231 266)(20 359 232 253)(21 360 233 254)(22 361 234 255)(23 362 235 256)(24 363 236 257)(25 364 237 258)(26 351 238 259)(27 352 225 260)(28 353 226 261)(29 63 213 123)(30 64 214 124)(31 65 215 125)(32 66 216 126)(33 67 217 113)(34 68 218 114)(35 69 219 115)(36 70 220 116)(37 57 221 117)(38 58 222 118)(39 59 223 119)(40 60 224 120)(41 61 211 121)(42 62 212 122)(71 171 99 391)(72 172 100 392)(73 173 101 379)(74 174 102 380)(75 175 103 381)(76 176 104 382)(77 177 105 383)(78 178 106 384)(79 179 107 385)(80 180 108 386)(81 181 109 387)(82 182 110 388)(83 169 111 389)(84 170 112 390)(85 191 269 350)(86 192 270 337)(87 193 271 338)(88 194 272 339)(89 195 273 340)(90 196 274 341)(91 183 275 342)(92 184 276 343)(93 185 277 344)(94 186 278 345)(95 187 279 346)(96 188 280 347)(97 189 267 348)(98 190 268 349)(127 393 411 308)(128 394 412 295)(129 395 413 296)(130 396 414 297)(131 397 415 298)(132 398 416 299)(133 399 417 300)(134 400 418 301)(135 401 419 302)(136 402 420 303)(137 403 407 304)(138 404 408 305)(139 405 409 306)(140 406 410 307)(155 281 321 443)(156 282 322 444)(157 283 309 445)(158 284 310 446)(159 285 311 447)(160 286 312 448)(161 287 313 435)(162 288 314 436)(163 289 315 437)(164 290 316 438)(165 291 317 439)(166 292 318 440)(167 293 319 441)(168 294 320 442)(197 432 369 247)(198 433 370 248)(199 434 371 249)(200 421 372 250)(201 422 373 251)(202 423 374 252)(203 424 375 239)(204 425 376 240)(205 426 377 241)(206 427 378 242)(207 428 365 243)(208 429 366 244)(209 430 367 245)(210 431 368 246)
(1 393 287 375)(2 394 288 376)(3 395 289 377)(4 396 290 378)(5 397 291 365)(6 398 292 366)(7 399 293 367)(8 400 294 368)(9 401 281 369)(10 402 282 370)(11 403 283 371)(12 404 284 372)(13 405 285 373)(14 406 286 374)(15 161 347 56)(16 162 348 43)(17 163 349 44)(18 164 350 45)(19 165 337 46)(20 166 338 47)(21 167 339 48)(22 168 340 49)(23 155 341 50)(24 156 342 51)(25 157 343 52)(26 158 344 53)(27 159 345 54)(28 160 346 55)(29 423 172 140)(30 424 173 127)(31 425 174 128)(32 426 175 129)(33 427 176 130)(34 428 177 131)(35 429 178 132)(36 430 179 133)(37 431 180 134)(38 432 181 135)(39 433 182 136)(40 434 169 137)(41 421 170 138)(42 422 171 139)(57 273 108 255)(58 274 109 256)(59 275 110 257)(60 276 111 258)(61 277 112 259)(62 278 99 260)(63 279 100 261)(64 280 101 262)(65 267 102 263)(66 268 103 264)(67 269 104 265)(68 270 105 266)(69 271 106 253)(70 272 107 254)(71 352 122 94)(72 353 123 95)(73 354 124 96)(74 355 125 97)(75 356 126 98)(76 357 113 85)(77 358 114 86)(78 359 115 87)(79 360 116 88)(80 361 117 89)(81 362 118 90)(82 363 119 91)(83 364 120 92)(84 351 121 93)(141 298 439 207)(142 299 440 208)(143 300 441 209)(144 301 442 210)(145 302 443 197)(146 303 444 198)(147 304 445 199)(148 305 446 200)(149 306 447 201)(150 307 448 202)(151 308 435 203)(152 295 436 204)(153 296 437 205)(154 297 438 206)(183 325 236 322)(184 326 237 309)(185 327 238 310)(186 328 225 311)(187 329 226 312)(188 330 227 313)(189 331 228 314)(190 332 229 315)(191 333 230 316)(192 334 231 317)(193 335 232 318)(194 336 233 319)(195 323 234 320)(196 324 235 321)(211 250 390 408)(212 251 391 409)(213 252 392 410)(214 239 379 411)(215 240 380 412)(216 241 381 413)(217 242 382 414)(218 243 383 415)(219 244 384 416)(220 245 385 417)(221 246 386 418)(222 247 387 419)(223 248 388 420)(224 249 389 407)
G:=sub<Sym(448)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232,233,234,235,236,237,238)(239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266)(267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294)(295,296,297,298,299,300,301,302,303,304,305,306,307,308)(309,310,311,312,313,314,315,316,317,318,319,320,321,322)(323,324,325,326,327,328,329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344,345,346,347,348,349,350)(351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400,401,402,403,404,405,406)(407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434)(435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,73,151,101)(2,74,152,102)(3,75,153,103)(4,76,154,104)(5,77,141,105)(6,78,142,106)(7,79,143,107)(8,80,144,108)(9,81,145,109)(10,82,146,110)(11,83,147,111)(12,84,148,112)(13,71,149,99)(14,72,150,100)(15,411,227,127)(16,412,228,128)(17,413,229,129)(18,414,230,130)(19,415,231,131)(20,416,232,132)(21,417,233,133)(22,418,234,134)(23,419,235,135)(24,420,236,136)(25,407,237,137)(26,408,238,138)(27,409,225,139)(28,410,226,140)(29,312,213,160)(30,313,214,161)(31,314,215,162)(32,315,216,163)(33,316,217,164)(34,317,218,165)(35,318,219,166)(36,319,220,167)(37,320,221,168)(38,321,222,155)(39,322,223,156)(40,309,224,157)(41,310,211,158)(42,311,212,159)(43,174,331,380)(44,175,332,381)(45,176,333,382)(46,177,334,383)(47,178,335,384)(48,179,336,385)(49,180,323,386)(50,181,324,387)(51,182,325,388)(52,169,326,389)(53,170,327,390)(54,171,328,391)(55,172,329,392)(56,173,330,379)(57,294,117,442)(58,281,118,443)(59,282,119,444)(60,283,120,445)(61,284,121,446)(62,285,122,447)(63,286,123,448)(64,287,124,435)(65,288,125,436)(66,289,126,437)(67,290,113,438)(68,291,114,439)(69,292,115,440)(70,293,116,441)(85,378,269,206)(86,365,270,207)(87,366,271,208)(88,367,272,209)(89,368,273,210)(90,369,274,197)(91,370,275,198)(92,371,276,199)(93,372,277,200)(94,373,278,201)(95,374,279,202)(96,375,280,203)(97,376,267,204)(98,377,268,205)(183,433,342,248)(184,434,343,249)(185,421,344,250)(186,422,345,251)(187,423,346,252)(188,424,347,239)(189,425,348,240)(190,426,349,241)(191,427,350,242)(192,428,337,243)(193,429,338,244)(194,430,339,245)(195,431,340,246)(196,432,341,247)(253,299,359,398)(254,300,360,399)(255,301,361,400)(256,302,362,401)(257,303,363,402)(258,304,364,403)(259,305,351,404)(260,306,352,405)(261,307,353,406)(262,308,354,393)(263,295,355,394)(264,296,356,395)(265,297,357,396)(266,298,358,397), (1,330,151,56)(2,331,152,43)(3,332,153,44)(4,333,154,45)(5,334,141,46)(6,335,142,47)(7,336,143,48)(8,323,144,49)(9,324,145,50)(10,325,146,51)(11,326,147,52)(12,327,148,53)(13,328,149,54)(14,329,150,55)(15,354,227,262)(16,355,228,263)(17,356,229,264)(18,357,230,265)(19,358,231,266)(20,359,232,253)(21,360,233,254)(22,361,234,255)(23,362,235,256)(24,363,236,257)(25,364,237,258)(26,351,238,259)(27,352,225,260)(28,353,226,261)(29,63,213,123)(30,64,214,124)(31,65,215,125)(32,66,216,126)(33,67,217,113)(34,68,218,114)(35,69,219,115)(36,70,220,116)(37,57,221,117)(38,58,222,118)(39,59,223,119)(40,60,224,120)(41,61,211,121)(42,62,212,122)(71,171,99,391)(72,172,100,392)(73,173,101,379)(74,174,102,380)(75,175,103,381)(76,176,104,382)(77,177,105,383)(78,178,106,384)(79,179,107,385)(80,180,108,386)(81,181,109,387)(82,182,110,388)(83,169,111,389)(84,170,112,390)(85,191,269,350)(86,192,270,337)(87,193,271,338)(88,194,272,339)(89,195,273,340)(90,196,274,341)(91,183,275,342)(92,184,276,343)(93,185,277,344)(94,186,278,345)(95,187,279,346)(96,188,280,347)(97,189,267,348)(98,190,268,349)(127,393,411,308)(128,394,412,295)(129,395,413,296)(130,396,414,297)(131,397,415,298)(132,398,416,299)(133,399,417,300)(134,400,418,301)(135,401,419,302)(136,402,420,303)(137,403,407,304)(138,404,408,305)(139,405,409,306)(140,406,410,307)(155,281,321,443)(156,282,322,444)(157,283,309,445)(158,284,310,446)(159,285,311,447)(160,286,312,448)(161,287,313,435)(162,288,314,436)(163,289,315,437)(164,290,316,438)(165,291,317,439)(166,292,318,440)(167,293,319,441)(168,294,320,442)(197,432,369,247)(198,433,370,248)(199,434,371,249)(200,421,372,250)(201,422,373,251)(202,423,374,252)(203,424,375,239)(204,425,376,240)(205,426,377,241)(206,427,378,242)(207,428,365,243)(208,429,366,244)(209,430,367,245)(210,431,368,246), (1,393,287,375)(2,394,288,376)(3,395,289,377)(4,396,290,378)(5,397,291,365)(6,398,292,366)(7,399,293,367)(8,400,294,368)(9,401,281,369)(10,402,282,370)(11,403,283,371)(12,404,284,372)(13,405,285,373)(14,406,286,374)(15,161,347,56)(16,162,348,43)(17,163,349,44)(18,164,350,45)(19,165,337,46)(20,166,338,47)(21,167,339,48)(22,168,340,49)(23,155,341,50)(24,156,342,51)(25,157,343,52)(26,158,344,53)(27,159,345,54)(28,160,346,55)(29,423,172,140)(30,424,173,127)(31,425,174,128)(32,426,175,129)(33,427,176,130)(34,428,177,131)(35,429,178,132)(36,430,179,133)(37,431,180,134)(38,432,181,135)(39,433,182,136)(40,434,169,137)(41,421,170,138)(42,422,171,139)(57,273,108,255)(58,274,109,256)(59,275,110,257)(60,276,111,258)(61,277,112,259)(62,278,99,260)(63,279,100,261)(64,280,101,262)(65,267,102,263)(66,268,103,264)(67,269,104,265)(68,270,105,266)(69,271,106,253)(70,272,107,254)(71,352,122,94)(72,353,123,95)(73,354,124,96)(74,355,125,97)(75,356,126,98)(76,357,113,85)(77,358,114,86)(78,359,115,87)(79,360,116,88)(80,361,117,89)(81,362,118,90)(82,363,119,91)(83,364,120,92)(84,351,121,93)(141,298,439,207)(142,299,440,208)(143,300,441,209)(144,301,442,210)(145,302,443,197)(146,303,444,198)(147,304,445,199)(148,305,446,200)(149,306,447,201)(150,307,448,202)(151,308,435,203)(152,295,436,204)(153,296,437,205)(154,297,438,206)(183,325,236,322)(184,326,237,309)(185,327,238,310)(186,328,225,311)(187,329,226,312)(188,330,227,313)(189,331,228,314)(190,332,229,315)(191,333,230,316)(192,334,231,317)(193,335,232,318)(194,336,233,319)(195,323,234,320)(196,324,235,321)(211,250,390,408)(212,251,391,409)(213,252,392,410)(214,239,379,411)(215,240,380,412)(216,241,381,413)(217,242,382,414)(218,243,383,415)(219,244,384,416)(220,245,385,417)(221,246,386,418)(222,247,387,419)(223,248,388,420)(224,249,389,407)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232,233,234,235,236,237,238)(239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266)(267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294)(295,296,297,298,299,300,301,302,303,304,305,306,307,308)(309,310,311,312,313,314,315,316,317,318,319,320,321,322)(323,324,325,326,327,328,329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344,345,346,347,348,349,350)(351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400,401,402,403,404,405,406)(407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434)(435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,73,151,101)(2,74,152,102)(3,75,153,103)(4,76,154,104)(5,77,141,105)(6,78,142,106)(7,79,143,107)(8,80,144,108)(9,81,145,109)(10,82,146,110)(11,83,147,111)(12,84,148,112)(13,71,149,99)(14,72,150,100)(15,411,227,127)(16,412,228,128)(17,413,229,129)(18,414,230,130)(19,415,231,131)(20,416,232,132)(21,417,233,133)(22,418,234,134)(23,419,235,135)(24,420,236,136)(25,407,237,137)(26,408,238,138)(27,409,225,139)(28,410,226,140)(29,312,213,160)(30,313,214,161)(31,314,215,162)(32,315,216,163)(33,316,217,164)(34,317,218,165)(35,318,219,166)(36,319,220,167)(37,320,221,168)(38,321,222,155)(39,322,223,156)(40,309,224,157)(41,310,211,158)(42,311,212,159)(43,174,331,380)(44,175,332,381)(45,176,333,382)(46,177,334,383)(47,178,335,384)(48,179,336,385)(49,180,323,386)(50,181,324,387)(51,182,325,388)(52,169,326,389)(53,170,327,390)(54,171,328,391)(55,172,329,392)(56,173,330,379)(57,294,117,442)(58,281,118,443)(59,282,119,444)(60,283,120,445)(61,284,121,446)(62,285,122,447)(63,286,123,448)(64,287,124,435)(65,288,125,436)(66,289,126,437)(67,290,113,438)(68,291,114,439)(69,292,115,440)(70,293,116,441)(85,378,269,206)(86,365,270,207)(87,366,271,208)(88,367,272,209)(89,368,273,210)(90,369,274,197)(91,370,275,198)(92,371,276,199)(93,372,277,200)(94,373,278,201)(95,374,279,202)(96,375,280,203)(97,376,267,204)(98,377,268,205)(183,433,342,248)(184,434,343,249)(185,421,344,250)(186,422,345,251)(187,423,346,252)(188,424,347,239)(189,425,348,240)(190,426,349,241)(191,427,350,242)(192,428,337,243)(193,429,338,244)(194,430,339,245)(195,431,340,246)(196,432,341,247)(253,299,359,398)(254,300,360,399)(255,301,361,400)(256,302,362,401)(257,303,363,402)(258,304,364,403)(259,305,351,404)(260,306,352,405)(261,307,353,406)(262,308,354,393)(263,295,355,394)(264,296,356,395)(265,297,357,396)(266,298,358,397), (1,330,151,56)(2,331,152,43)(3,332,153,44)(4,333,154,45)(5,334,141,46)(6,335,142,47)(7,336,143,48)(8,323,144,49)(9,324,145,50)(10,325,146,51)(11,326,147,52)(12,327,148,53)(13,328,149,54)(14,329,150,55)(15,354,227,262)(16,355,228,263)(17,356,229,264)(18,357,230,265)(19,358,231,266)(20,359,232,253)(21,360,233,254)(22,361,234,255)(23,362,235,256)(24,363,236,257)(25,364,237,258)(26,351,238,259)(27,352,225,260)(28,353,226,261)(29,63,213,123)(30,64,214,124)(31,65,215,125)(32,66,216,126)(33,67,217,113)(34,68,218,114)(35,69,219,115)(36,70,220,116)(37,57,221,117)(38,58,222,118)(39,59,223,119)(40,60,224,120)(41,61,211,121)(42,62,212,122)(71,171,99,391)(72,172,100,392)(73,173,101,379)(74,174,102,380)(75,175,103,381)(76,176,104,382)(77,177,105,383)(78,178,106,384)(79,179,107,385)(80,180,108,386)(81,181,109,387)(82,182,110,388)(83,169,111,389)(84,170,112,390)(85,191,269,350)(86,192,270,337)(87,193,271,338)(88,194,272,339)(89,195,273,340)(90,196,274,341)(91,183,275,342)(92,184,276,343)(93,185,277,344)(94,186,278,345)(95,187,279,346)(96,188,280,347)(97,189,267,348)(98,190,268,349)(127,393,411,308)(128,394,412,295)(129,395,413,296)(130,396,414,297)(131,397,415,298)(132,398,416,299)(133,399,417,300)(134,400,418,301)(135,401,419,302)(136,402,420,303)(137,403,407,304)(138,404,408,305)(139,405,409,306)(140,406,410,307)(155,281,321,443)(156,282,322,444)(157,283,309,445)(158,284,310,446)(159,285,311,447)(160,286,312,448)(161,287,313,435)(162,288,314,436)(163,289,315,437)(164,290,316,438)(165,291,317,439)(166,292,318,440)(167,293,319,441)(168,294,320,442)(197,432,369,247)(198,433,370,248)(199,434,371,249)(200,421,372,250)(201,422,373,251)(202,423,374,252)(203,424,375,239)(204,425,376,240)(205,426,377,241)(206,427,378,242)(207,428,365,243)(208,429,366,244)(209,430,367,245)(210,431,368,246), (1,393,287,375)(2,394,288,376)(3,395,289,377)(4,396,290,378)(5,397,291,365)(6,398,292,366)(7,399,293,367)(8,400,294,368)(9,401,281,369)(10,402,282,370)(11,403,283,371)(12,404,284,372)(13,405,285,373)(14,406,286,374)(15,161,347,56)(16,162,348,43)(17,163,349,44)(18,164,350,45)(19,165,337,46)(20,166,338,47)(21,167,339,48)(22,168,340,49)(23,155,341,50)(24,156,342,51)(25,157,343,52)(26,158,344,53)(27,159,345,54)(28,160,346,55)(29,423,172,140)(30,424,173,127)(31,425,174,128)(32,426,175,129)(33,427,176,130)(34,428,177,131)(35,429,178,132)(36,430,179,133)(37,431,180,134)(38,432,181,135)(39,433,182,136)(40,434,169,137)(41,421,170,138)(42,422,171,139)(57,273,108,255)(58,274,109,256)(59,275,110,257)(60,276,111,258)(61,277,112,259)(62,278,99,260)(63,279,100,261)(64,280,101,262)(65,267,102,263)(66,268,103,264)(67,269,104,265)(68,270,105,266)(69,271,106,253)(70,272,107,254)(71,352,122,94)(72,353,123,95)(73,354,124,96)(74,355,125,97)(75,356,126,98)(76,357,113,85)(77,358,114,86)(78,359,115,87)(79,360,116,88)(80,361,117,89)(81,362,118,90)(82,363,119,91)(83,364,120,92)(84,351,121,93)(141,298,439,207)(142,299,440,208)(143,300,441,209)(144,301,442,210)(145,302,443,197)(146,303,444,198)(147,304,445,199)(148,305,446,200)(149,306,447,201)(150,307,448,202)(151,308,435,203)(152,295,436,204)(153,296,437,205)(154,297,438,206)(183,325,236,322)(184,326,237,309)(185,327,238,310)(186,328,225,311)(187,329,226,312)(188,330,227,313)(189,331,228,314)(190,332,229,315)(191,333,230,316)(192,334,231,317)(193,335,232,318)(194,336,233,319)(195,323,234,320)(196,324,235,321)(211,250,390,408)(212,251,391,409)(213,252,392,410)(214,239,379,411)(215,240,380,412)(216,241,381,413)(217,242,382,414)(218,243,383,415)(219,244,384,416)(220,245,385,417)(221,246,386,418)(222,247,387,419)(223,248,388,420)(224,249,389,407) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154),(155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224),(225,226,227,228,229,230,231,232,233,234,235,236,237,238),(239,240,241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264,265,266),(267,268,269,270,271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290,291,292,293,294),(295,296,297,298,299,300,301,302,303,304,305,306,307,308),(309,310,311,312,313,314,315,316,317,318,319,320,321,322),(323,324,325,326,327,328,329,330,331,332,333,334,335,336),(337,338,339,340,341,342,343,344,345,346,347,348,349,350),(351,352,353,354,355,356,357,358,359,360,361,362,363,364),(365,366,367,368,369,370,371,372,373,374,375,376,377,378),(379,380,381,382,383,384,385,386,387,388,389,390,391,392),(393,394,395,396,397,398,399,400,401,402,403,404,405,406),(407,408,409,410,411,412,413,414,415,416,417,418,419,420),(421,422,423,424,425,426,427,428,429,430,431,432,433,434),(435,436,437,438,439,440,441,442,443,444,445,446,447,448)], [(1,73,151,101),(2,74,152,102),(3,75,153,103),(4,76,154,104),(5,77,141,105),(6,78,142,106),(7,79,143,107),(8,80,144,108),(9,81,145,109),(10,82,146,110),(11,83,147,111),(12,84,148,112),(13,71,149,99),(14,72,150,100),(15,411,227,127),(16,412,228,128),(17,413,229,129),(18,414,230,130),(19,415,231,131),(20,416,232,132),(21,417,233,133),(22,418,234,134),(23,419,235,135),(24,420,236,136),(25,407,237,137),(26,408,238,138),(27,409,225,139),(28,410,226,140),(29,312,213,160),(30,313,214,161),(31,314,215,162),(32,315,216,163),(33,316,217,164),(34,317,218,165),(35,318,219,166),(36,319,220,167),(37,320,221,168),(38,321,222,155),(39,322,223,156),(40,309,224,157),(41,310,211,158),(42,311,212,159),(43,174,331,380),(44,175,332,381),(45,176,333,382),(46,177,334,383),(47,178,335,384),(48,179,336,385),(49,180,323,386),(50,181,324,387),(51,182,325,388),(52,169,326,389),(53,170,327,390),(54,171,328,391),(55,172,329,392),(56,173,330,379),(57,294,117,442),(58,281,118,443),(59,282,119,444),(60,283,120,445),(61,284,121,446),(62,285,122,447),(63,286,123,448),(64,287,124,435),(65,288,125,436),(66,289,126,437),(67,290,113,438),(68,291,114,439),(69,292,115,440),(70,293,116,441),(85,378,269,206),(86,365,270,207),(87,366,271,208),(88,367,272,209),(89,368,273,210),(90,369,274,197),(91,370,275,198),(92,371,276,199),(93,372,277,200),(94,373,278,201),(95,374,279,202),(96,375,280,203),(97,376,267,204),(98,377,268,205),(183,433,342,248),(184,434,343,249),(185,421,344,250),(186,422,345,251),(187,423,346,252),(188,424,347,239),(189,425,348,240),(190,426,349,241),(191,427,350,242),(192,428,337,243),(193,429,338,244),(194,430,339,245),(195,431,340,246),(196,432,341,247),(253,299,359,398),(254,300,360,399),(255,301,361,400),(256,302,362,401),(257,303,363,402),(258,304,364,403),(259,305,351,404),(260,306,352,405),(261,307,353,406),(262,308,354,393),(263,295,355,394),(264,296,356,395),(265,297,357,396),(266,298,358,397)], [(1,330,151,56),(2,331,152,43),(3,332,153,44),(4,333,154,45),(5,334,141,46),(6,335,142,47),(7,336,143,48),(8,323,144,49),(9,324,145,50),(10,325,146,51),(11,326,147,52),(12,327,148,53),(13,328,149,54),(14,329,150,55),(15,354,227,262),(16,355,228,263),(17,356,229,264),(18,357,230,265),(19,358,231,266),(20,359,232,253),(21,360,233,254),(22,361,234,255),(23,362,235,256),(24,363,236,257),(25,364,237,258),(26,351,238,259),(27,352,225,260),(28,353,226,261),(29,63,213,123),(30,64,214,124),(31,65,215,125),(32,66,216,126),(33,67,217,113),(34,68,218,114),(35,69,219,115),(36,70,220,116),(37,57,221,117),(38,58,222,118),(39,59,223,119),(40,60,224,120),(41,61,211,121),(42,62,212,122),(71,171,99,391),(72,172,100,392),(73,173,101,379),(74,174,102,380),(75,175,103,381),(76,176,104,382),(77,177,105,383),(78,178,106,384),(79,179,107,385),(80,180,108,386),(81,181,109,387),(82,182,110,388),(83,169,111,389),(84,170,112,390),(85,191,269,350),(86,192,270,337),(87,193,271,338),(88,194,272,339),(89,195,273,340),(90,196,274,341),(91,183,275,342),(92,184,276,343),(93,185,277,344),(94,186,278,345),(95,187,279,346),(96,188,280,347),(97,189,267,348),(98,190,268,349),(127,393,411,308),(128,394,412,295),(129,395,413,296),(130,396,414,297),(131,397,415,298),(132,398,416,299),(133,399,417,300),(134,400,418,301),(135,401,419,302),(136,402,420,303),(137,403,407,304),(138,404,408,305),(139,405,409,306),(140,406,410,307),(155,281,321,443),(156,282,322,444),(157,283,309,445),(158,284,310,446),(159,285,311,447),(160,286,312,448),(161,287,313,435),(162,288,314,436),(163,289,315,437),(164,290,316,438),(165,291,317,439),(166,292,318,440),(167,293,319,441),(168,294,320,442),(197,432,369,247),(198,433,370,248),(199,434,371,249),(200,421,372,250),(201,422,373,251),(202,423,374,252),(203,424,375,239),(204,425,376,240),(205,426,377,241),(206,427,378,242),(207,428,365,243),(208,429,366,244),(209,430,367,245),(210,431,368,246)], [(1,393,287,375),(2,394,288,376),(3,395,289,377),(4,396,290,378),(5,397,291,365),(6,398,292,366),(7,399,293,367),(8,400,294,368),(9,401,281,369),(10,402,282,370),(11,403,283,371),(12,404,284,372),(13,405,285,373),(14,406,286,374),(15,161,347,56),(16,162,348,43),(17,163,349,44),(18,164,350,45),(19,165,337,46),(20,166,338,47),(21,167,339,48),(22,168,340,49),(23,155,341,50),(24,156,342,51),(25,157,343,52),(26,158,344,53),(27,159,345,54),(28,160,346,55),(29,423,172,140),(30,424,173,127),(31,425,174,128),(32,426,175,129),(33,427,176,130),(34,428,177,131),(35,429,178,132),(36,430,179,133),(37,431,180,134),(38,432,181,135),(39,433,182,136),(40,434,169,137),(41,421,170,138),(42,422,171,139),(57,273,108,255),(58,274,109,256),(59,275,110,257),(60,276,111,258),(61,277,112,259),(62,278,99,260),(63,279,100,261),(64,280,101,262),(65,267,102,263),(66,268,103,264),(67,269,104,265),(68,270,105,266),(69,271,106,253),(70,272,107,254),(71,352,122,94),(72,353,123,95),(73,354,124,96),(74,355,125,97),(75,356,126,98),(76,357,113,85),(77,358,114,86),(78,359,115,87),(79,360,116,88),(80,361,117,89),(81,362,118,90),(82,363,119,91),(83,364,120,92),(84,351,121,93),(141,298,439,207),(142,299,440,208),(143,300,441,209),(144,301,442,210),(145,302,443,197),(146,303,444,198),(147,304,445,199),(148,305,446,200),(149,306,447,201),(150,307,448,202),(151,308,435,203),(152,295,436,204),(153,296,437,205),(154,297,438,206),(183,325,236,322),(184,326,237,309),(185,327,238,310),(186,328,225,311),(187,329,226,312),(188,330,227,313),(189,331,228,314),(190,332,229,315),(191,333,230,316),(192,334,231,317),(193,335,232,318),(194,336,233,319),(195,323,234,320),(196,324,235,321),(211,250,390,408),(212,251,391,409),(213,252,392,410),(214,239,379,411),(215,240,380,412),(216,241,381,413),(217,242,382,414),(218,243,383,415),(219,244,384,416),(220,245,385,417),(221,246,386,418),(222,247,387,419),(223,248,388,420),(224,249,389,407)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28BT | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | D4 | D4 | SD16 | Q16 | C7×D4 | C7×D4 | C7×SD16 | C7×Q16 |
| kernel | C14×Q8⋊C4 | C7×Q8⋊C4 | C14×C4⋊C4 | C22×C56 | Q8×C2×C14 | Q8×C14 | C2×Q8⋊C4 | Q8⋊C4 | C2×C4⋊C4 | C22×C8 | C22×Q8 | C2×Q8 | C2×C28 | C22×C14 | C2×C14 | C2×C14 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 6 | 24 | 6 | 6 | 6 | 48 | 3 | 1 | 4 | 4 | 18 | 6 | 24 | 24 |
Matrix representation of C14×Q8⋊C4 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 97 | 0 |
| 0 | 0 | 0 | 97 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 45 | 11 |
| 0 | 0 | 11 | 68 |
| 15 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 50 | 10 |
| 0 | 0 | 10 | 63 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,97,0,0,0,0,97],[1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,45,11,0,0,11,68],[15,0,0,0,0,1,0,0,0,0,50,10,0,0,10,63] >;
C14×Q8⋊C4 in GAP, Magma, Sage, TeX
C_{14}\times Q_8\rtimes C_4 % in TeX
G:=Group("C14xQ8:C4"); // GroupNames label
G:=SmallGroup(448,823);
// by ID
G=gap.SmallGroup(448,823);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations