direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C4⋊Q8, C28⋊7(C2×Q8), C4⋊1(Q8×C14), (C2×C28)⋊15Q8, C4.14(D4×C14), C28.321(C2×D4), (C2×C28).431D4, (C2×C42).19C14, C42.89(C2×C14), C22.64(D4×C14), (C22×Q8).8C14, C14.59(C22×Q8), C22.19(Q8×C14), (C4×C28).374C22, (C2×C28).962C23, (C2×C14).351C24, C14.187(C22×D4), C22.25(C23×C14), C23.73(C22×C14), (Q8×C14).267C22, (C22×C14).470C23, (C22×C28).445C22, (C2×C4)⋊4(C7×Q8), C2.5(Q8×C2×C14), (C2×C4×C28).42C2, C2.11(D4×C2×C14), (C2×C4).87(C7×D4), (C2×C4⋊C4).19C14, (C14×C4⋊C4).48C2, (Q8×C2×C14).18C2, C4⋊C4.65(C2×C14), (C2×C14).685(C2×D4), (C2×Q8).54(C2×C14), (C2×C14).117(C2×Q8), (C7×C4⋊C4).388C22, (C2×C4).18(C22×C14), (C22×C4).55(C2×C14), SmallGroup(448,1314)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C4⋊Q8
G = < a,b,c,d | a14=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 370 in 290 conjugacy classes, 210 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, C2×C28, C2×C28, C7×Q8, C22×C14, C2×C4⋊Q8, C4×C28, C7×C4⋊C4, C22×C28, C22×C28, Q8×C14, Q8×C14, C2×C4×C28, C14×C4⋊C4, C7×C4⋊Q8, Q8×C2×C14, C14×C4⋊Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C24, C2×C14, C4⋊Q8, C22×D4, C22×Q8, C7×D4, C7×Q8, C22×C14, C2×C4⋊Q8, D4×C14, Q8×C14, C23×C14, C7×C4⋊Q8, D4×C2×C14, Q8×C2×C14, C14×C4⋊Q8
►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 96 216 365)(2 97 217 366)(3 98 218 367)(4 85 219 368)(5 86 220 369)(6 87 221 370)(7 88 222 371)(8 89 223 372)(9 90 224 373)(10 91 211 374)(11 92 212 375)(12 93 213 376)(13 94 214 377)(14 95 215 378)(15 184 131 156)(16 185 132 157)(17 186 133 158)(18 187 134 159)(19 188 135 160)(20 189 136 161)(21 190 137 162)(22 191 138 163)(23 192 139 164)(24 193 140 165)(25 194 127 166)(26 195 128 167)(27 196 129 168)(28 183 130 155)(29 244 397 58)(30 245 398 59)(31 246 399 60)(32 247 400 61)(33 248 401 62)(34 249 402 63)(35 250 403 64)(36 251 404 65)(37 252 405 66)(38 239 406 67)(39 240 393 68)(40 241 394 69)(41 242 395 70)(42 243 396 57)(43 232 392 77)(44 233 379 78)(45 234 380 79)(46 235 381 80)(47 236 382 81)(48 237 383 82)(49 238 384 83)(50 225 385 84)(51 226 386 71)(52 227 387 72)(53 228 388 73)(54 229 389 74)(55 230 390 75)(56 231 391 76)(99 171 324 273)(100 172 325 274)(101 173 326 275)(102 174 327 276)(103 175 328 277)(104 176 329 278)(105 177 330 279)(106 178 331 280)(107 179 332 267)(108 180 333 268)(109 181 334 269)(110 182 335 270)(111 169 336 271)(112 170 323 272)(113 282 302 342)(114 283 303 343)(115 284 304 344)(116 285 305 345)(117 286 306 346)(118 287 307 347)(119 288 308 348)(120 289 295 349)(121 290 296 350)(122 291 297 337)(123 292 298 338)(124 293 299 339)(125 294 300 340)(126 281 301 341)(141 363 443 412)(142 364 444 413)(143 351 445 414)(144 352 446 415)(145 353 447 416)(146 354 448 417)(147 355 435 418)(148 356 436 419)(149 357 437 420)(150 358 438 407)(151 359 439 408)(152 360 440 409)(153 361 441 410)(154 362 442 411)(197 430 309 258)(198 431 310 259)(199 432 311 260)(200 433 312 261)(201 434 313 262)(202 421 314 263)(203 422 315 264)(204 423 316 265)(205 424 317 266)(206 425 318 253)(207 426 319 254)(208 427 320 255)(209 428 321 256)(210 429 322 257)
(1 406 348 233)(2 393 349 234)(3 394 350 235)(4 395 337 236)(5 396 338 237)(6 397 339 238)(7 398 340 225)(8 399 341 226)(9 400 342 227)(10 401 343 228)(11 402 344 229)(12 403 345 230)(13 404 346 231)(14 405 347 232)(15 446 427 278)(16 447 428 279)(17 448 429 280)(18 435 430 267)(19 436 431 268)(20 437 432 269)(21 438 433 270)(22 439 434 271)(23 440 421 272)(24 441 422 273)(25 442 423 274)(26 443 424 275)(27 444 425 276)(28 445 426 277)(29 293 83 221)(30 294 84 222)(31 281 71 223)(32 282 72 224)(33 283 73 211)(34 284 74 212)(35 285 75 213)(36 286 76 214)(37 287 77 215)(38 288 78 216)(39 289 79 217)(40 290 80 218)(41 291 81 219)(42 292 82 220)(43 378 252 307)(44 365 239 308)(45 366 240 295)(46 367 241 296)(47 368 242 297)(48 369 243 298)(49 370 244 299)(50 371 245 300)(51 372 246 301)(52 373 247 302)(53 374 248 303)(54 375 249 304)(55 376 250 305)(56 377 251 306)(57 123 383 86)(58 124 384 87)(59 125 385 88)(60 126 386 89)(61 113 387 90)(62 114 388 91)(63 115 389 92)(64 116 390 93)(65 117 391 94)(66 118 392 95)(67 119 379 96)(68 120 380 97)(69 121 381 98)(70 122 382 85)(99 193 410 315)(100 194 411 316)(101 195 412 317)(102 196 413 318)(103 183 414 319)(104 184 415 320)(105 185 416 321)(106 186 417 322)(107 187 418 309)(108 188 419 310)(109 189 420 311)(110 190 407 312)(111 191 408 313)(112 192 409 314)(127 154 265 172)(128 141 266 173)(129 142 253 174)(130 143 254 175)(131 144 255 176)(132 145 256 177)(133 146 257 178)(134 147 258 179)(135 148 259 180)(136 149 260 181)(137 150 261 182)(138 151 262 169)(139 152 263 170)(140 153 264 171)(155 351 207 328)(156 352 208 329)(157 353 209 330)(158 354 210 331)(159 355 197 332)(160 356 198 333)(161 357 199 334)(162 358 200 335)(163 359 201 336)(164 360 202 323)(165 361 203 324)(166 362 204 325)(167 363 205 326)(168 364 206 327)
(1 158 348 210)(2 159 349 197)(3 160 350 198)(4 161 337 199)(5 162 338 200)(6 163 339 201)(7 164 340 202)(8 165 341 203)(9 166 342 204)(10 167 343 205)(11 168 344 206)(12 155 345 207)(13 156 346 208)(14 157 347 209)(15 306 427 377)(16 307 428 378)(17 308 429 365)(18 295 430 366)(19 296 431 367)(20 297 432 368)(21 298 433 369)(22 299 434 370)(23 300 421 371)(24 301 422 372)(25 302 423 373)(26 303 424 374)(27 304 425 375)(28 305 426 376)(29 111 83 408)(30 112 84 409)(31 99 71 410)(32 100 72 411)(33 101 73 412)(34 102 74 413)(35 103 75 414)(36 104 76 415)(37 105 77 416)(38 106 78 417)(39 107 79 418)(40 108 80 419)(41 109 81 420)(42 110 82 407)(43 447 252 279)(44 448 239 280)(45 435 240 267)(46 436 241 268)(47 437 242 269)(48 438 243 270)(49 439 244 271)(50 440 245 272)(51 441 246 273)(52 442 247 274)(53 443 248 275)(54 444 249 276)(55 445 250 277)(56 446 251 278)(57 182 383 150)(58 169 384 151)(59 170 385 152)(60 171 386 153)(61 172 387 154)(62 173 388 141)(63 174 389 142)(64 175 390 143)(65 176 391 144)(66 177 392 145)(67 178 379 146)(68 179 380 147)(69 180 381 148)(70 181 382 149)(85 136 122 260)(86 137 123 261)(87 138 124 262)(88 139 125 263)(89 140 126 264)(90 127 113 265)(91 128 114 266)(92 129 115 253)(93 130 116 254)(94 131 117 255)(95 132 118 256)(96 133 119 257)(97 134 120 258)(98 135 121 259)(183 285 319 213)(184 286 320 214)(185 287 321 215)(186 288 322 216)(187 289 309 217)(188 290 310 218)(189 291 311 219)(190 292 312 220)(191 293 313 221)(192 294 314 222)(193 281 315 223)(194 282 316 224)(195 283 317 211)(196 284 318 212)(225 360 398 323)(226 361 399 324)(227 362 400 325)(228 363 401 326)(229 364 402 327)(230 351 403 328)(231 352 404 329)(232 353 405 330)(233 354 406 331)(234 355 393 332)(235 356 394 333)(236 357 395 334)(237 358 396 335)(238 359 397 336)
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), (1,96,216,365)(2,97,217,366)(3,98,218,367)(4,85,219,368)(5,86,220,369)(6,87,221,370)(7,88,222,371)(8,89,223,372)(9,90,224,373)(10,91,211,374)(11,92,212,375)(12,93,213,376)(13,94,214,377)(14,95,215,378)(15,184,131,156)(16,185,132,157)(17,186,133,158)(18,187,134,159)(19,188,135,160)(20,189,136,161)(21,190,137,162)(22,191,138,163)(23,192,139,164)(24,193,140,165)(25,194,127,166)(26,195,128,167)(27,196,129,168)(28,183,130,155)(29,244,397,58)(30,245,398,59)(31,246,399,60)(32,247,400,61)(33,248,401,62)(34,249,402,63)(35,250,403,64)(36,251,404,65)(37,252,405,66)(38,239,406,67)(39,240,393,68)(40,241,394,69)(41,242,395,70)(42,243,396,57)(43,232,392,77)(44,233,379,78)(45,234,380,79)(46,235,381,80)(47,236,382,81)(48,237,383,82)(49,238,384,83)(50,225,385,84)(51,226,386,71)(52,227,387,72)(53,228,388,73)(54,229,389,74)(55,230,390,75)(56,231,391,76)(99,171,324,273)(100,172,325,274)(101,173,326,275)(102,174,327,276)(103,175,328,277)(104,176,329,278)(105,177,330,279)(106,178,331,280)(107,179,332,267)(108,180,333,268)(109,181,334,269)(110,182,335,270)(111,169,336,271)(112,170,323,272)(113,282,302,342)(114,283,303,343)(115,284,304,344)(116,285,305,345)(117,286,306,346)(118,287,307,347)(119,288,308,348)(120,289,295,349)(121,290,296,350)(122,291,297,337)(123,292,298,338)(124,293,299,339)(125,294,300,340)(126,281,301,341)(141,363,443,412)(142,364,444,413)(143,351,445,414)(144,352,446,415)(145,353,447,416)(146,354,448,417)(147,355,435,418)(148,356,436,419)(149,357,437,420)(150,358,438,407)(151,359,439,408)(152,360,440,409)(153,361,441,410)(154,362,442,411)(197,430,309,258)(198,431,310,259)(199,432,311,260)(200,433,312,261)(201,434,313,262)(202,421,314,263)(203,422,315,264)(204,423,316,265)(205,424,317,266)(206,425,318,253)(207,426,319,254)(208,427,320,255)(209,428,321,256)(210,429,322,257), (1,406,348,233)(2,393,349,234)(3,394,350,235)(4,395,337,236)(5,396,338,237)(6,397,339,238)(7,398,340,225)(8,399,341,226)(9,400,342,227)(10,401,343,228)(11,402,344,229)(12,403,345,230)(13,404,346,231)(14,405,347,232)(15,446,427,278)(16,447,428,279)(17,448,429,280)(18,435,430,267)(19,436,431,268)(20,437,432,269)(21,438,433,270)(22,439,434,271)(23,440,421,272)(24,441,422,273)(25,442,423,274)(26,443,424,275)(27,444,425,276)(28,445,426,277)(29,293,83,221)(30,294,84,222)(31,281,71,223)(32,282,72,224)(33,283,73,211)(34,284,74,212)(35,285,75,213)(36,286,76,214)(37,287,77,215)(38,288,78,216)(39,289,79,217)(40,290,80,218)(41,291,81,219)(42,292,82,220)(43,378,252,307)(44,365,239,308)(45,366,240,295)(46,367,241,296)(47,368,242,297)(48,369,243,298)(49,370,244,299)(50,371,245,300)(51,372,246,301)(52,373,247,302)(53,374,248,303)(54,375,249,304)(55,376,250,305)(56,377,251,306)(57,123,383,86)(58,124,384,87)(59,125,385,88)(60,126,386,89)(61,113,387,90)(62,114,388,91)(63,115,389,92)(64,116,390,93)(65,117,391,94)(66,118,392,95)(67,119,379,96)(68,120,380,97)(69,121,381,98)(70,122,382,85)(99,193,410,315)(100,194,411,316)(101,195,412,317)(102,196,413,318)(103,183,414,319)(104,184,415,320)(105,185,416,321)(106,186,417,322)(107,187,418,309)(108,188,419,310)(109,189,420,311)(110,190,407,312)(111,191,408,313)(112,192,409,314)(127,154,265,172)(128,141,266,173)(129,142,253,174)(130,143,254,175)(131,144,255,176)(132,145,256,177)(133,146,257,178)(134,147,258,179)(135,148,259,180)(136,149,260,181)(137,150,261,182)(138,151,262,169)(139,152,263,170)(140,153,264,171)(155,351,207,328)(156,352,208,329)(157,353,209,330)(158,354,210,331)(159,355,197,332)(160,356,198,333)(161,357,199,334)(162,358,200,335)(163,359,201,336)(164,360,202,323)(165,361,203,324)(166,362,204,325)(167,363,205,326)(168,364,206,327), (1,158,348,210)(2,159,349,197)(3,160,350,198)(4,161,337,199)(5,162,338,200)(6,163,339,201)(7,164,340,202)(8,165,341,203)(9,166,342,204)(10,167,343,205)(11,168,344,206)(12,155,345,207)(13,156,346,208)(14,157,347,209)(15,306,427,377)(16,307,428,378)(17,308,429,365)(18,295,430,366)(19,296,431,367)(20,297,432,368)(21,298,433,369)(22,299,434,370)(23,300,421,371)(24,301,422,372)(25,302,423,373)(26,303,424,374)(27,304,425,375)(28,305,426,376)(29,111,83,408)(30,112,84,409)(31,99,71,410)(32,100,72,411)(33,101,73,412)(34,102,74,413)(35,103,75,414)(36,104,76,415)(37,105,77,416)(38,106,78,417)(39,107,79,418)(40,108,80,419)(41,109,81,420)(42,110,82,407)(43,447,252,279)(44,448,239,280)(45,435,240,267)(46,436,241,268)(47,437,242,269)(48,438,243,270)(49,439,244,271)(50,440,245,272)(51,441,246,273)(52,442,247,274)(53,443,248,275)(54,444,249,276)(55,445,250,277)(56,446,251,278)(57,182,383,150)(58,169,384,151)(59,170,385,152)(60,171,386,153)(61,172,387,154)(62,173,388,141)(63,174,389,142)(64,175,390,143)(65,176,391,144)(66,177,392,145)(67,178,379,146)(68,179,380,147)(69,180,381,148)(70,181,382,149)(85,136,122,260)(86,137,123,261)(87,138,124,262)(88,139,125,263)(89,140,126,264)(90,127,113,265)(91,128,114,266)(92,129,115,253)(93,130,116,254)(94,131,117,255)(95,132,118,256)(96,133,119,257)(97,134,120,258)(98,135,121,259)(183,285,319,213)(184,286,320,214)(185,287,321,215)(186,288,322,216)(187,289,309,217)(188,290,310,218)(189,291,311,219)(190,292,312,220)(191,293,313,221)(192,294,314,222)(193,281,315,223)(194,282,316,224)(195,283,317,211)(196,284,318,212)(225,360,398,323)(226,361,399,324)(227,362,400,325)(228,363,401,326)(229,364,402,327)(230,351,403,328)(231,352,404,329)(232,353,405,330)(233,354,406,331)(234,355,393,332)(235,356,394,333)(236,357,395,334)(237,358,396,335)(238,359,397,336)>;
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), (1,96,216,365)(2,97,217,366)(3,98,218,367)(4,85,219,368)(5,86,220,369)(6,87,221,370)(7,88,222,371)(8,89,223,372)(9,90,224,373)(10,91,211,374)(11,92,212,375)(12,93,213,376)(13,94,214,377)(14,95,215,378)(15,184,131,156)(16,185,132,157)(17,186,133,158)(18,187,134,159)(19,188,135,160)(20,189,136,161)(21,190,137,162)(22,191,138,163)(23,192,139,164)(24,193,140,165)(25,194,127,166)(26,195,128,167)(27,196,129,168)(28,183,130,155)(29,244,397,58)(30,245,398,59)(31,246,399,60)(32,247,400,61)(33,248,401,62)(34,249,402,63)(35,250,403,64)(36,251,404,65)(37,252,405,66)(38,239,406,67)(39,240,393,68)(40,241,394,69)(41,242,395,70)(42,243,396,57)(43,232,392,77)(44,233,379,78)(45,234,380,79)(46,235,381,80)(47,236,382,81)(48,237,383,82)(49,238,384,83)(50,225,385,84)(51,226,386,71)(52,227,387,72)(53,228,388,73)(54,229,389,74)(55,230,390,75)(56,231,391,76)(99,171,324,273)(100,172,325,274)(101,173,326,275)(102,174,327,276)(103,175,328,277)(104,176,329,278)(105,177,330,279)(106,178,331,280)(107,179,332,267)(108,180,333,268)(109,181,334,269)(110,182,335,270)(111,169,336,271)(112,170,323,272)(113,282,302,342)(114,283,303,343)(115,284,304,344)(116,285,305,345)(117,286,306,346)(118,287,307,347)(119,288,308,348)(120,289,295,349)(121,290,296,350)(122,291,297,337)(123,292,298,338)(124,293,299,339)(125,294,300,340)(126,281,301,341)(141,363,443,412)(142,364,444,413)(143,351,445,414)(144,352,446,415)(145,353,447,416)(146,354,448,417)(147,355,435,418)(148,356,436,419)(149,357,437,420)(150,358,438,407)(151,359,439,408)(152,360,440,409)(153,361,441,410)(154,362,442,411)(197,430,309,258)(198,431,310,259)(199,432,311,260)(200,433,312,261)(201,434,313,262)(202,421,314,263)(203,422,315,264)(204,423,316,265)(205,424,317,266)(206,425,318,253)(207,426,319,254)(208,427,320,255)(209,428,321,256)(210,429,322,257), (1,406,348,233)(2,393,349,234)(3,394,350,235)(4,395,337,236)(5,396,338,237)(6,397,339,238)(7,398,340,225)(8,399,341,226)(9,400,342,227)(10,401,343,228)(11,402,344,229)(12,403,345,230)(13,404,346,231)(14,405,347,232)(15,446,427,278)(16,447,428,279)(17,448,429,280)(18,435,430,267)(19,436,431,268)(20,437,432,269)(21,438,433,270)(22,439,434,271)(23,440,421,272)(24,441,422,273)(25,442,423,274)(26,443,424,275)(27,444,425,276)(28,445,426,277)(29,293,83,221)(30,294,84,222)(31,281,71,223)(32,282,72,224)(33,283,73,211)(34,284,74,212)(35,285,75,213)(36,286,76,214)(37,287,77,215)(38,288,78,216)(39,289,79,217)(40,290,80,218)(41,291,81,219)(42,292,82,220)(43,378,252,307)(44,365,239,308)(45,366,240,295)(46,367,241,296)(47,368,242,297)(48,369,243,298)(49,370,244,299)(50,371,245,300)(51,372,246,301)(52,373,247,302)(53,374,248,303)(54,375,249,304)(55,376,250,305)(56,377,251,306)(57,123,383,86)(58,124,384,87)(59,125,385,88)(60,126,386,89)(61,113,387,90)(62,114,388,91)(63,115,389,92)(64,116,390,93)(65,117,391,94)(66,118,392,95)(67,119,379,96)(68,120,380,97)(69,121,381,98)(70,122,382,85)(99,193,410,315)(100,194,411,316)(101,195,412,317)(102,196,413,318)(103,183,414,319)(104,184,415,320)(105,185,416,321)(106,186,417,322)(107,187,418,309)(108,188,419,310)(109,189,420,311)(110,190,407,312)(111,191,408,313)(112,192,409,314)(127,154,265,172)(128,141,266,173)(129,142,253,174)(130,143,254,175)(131,144,255,176)(132,145,256,177)(133,146,257,178)(134,147,258,179)(135,148,259,180)(136,149,260,181)(137,150,261,182)(138,151,262,169)(139,152,263,170)(140,153,264,171)(155,351,207,328)(156,352,208,329)(157,353,209,330)(158,354,210,331)(159,355,197,332)(160,356,198,333)(161,357,199,334)(162,358,200,335)(163,359,201,336)(164,360,202,323)(165,361,203,324)(166,362,204,325)(167,363,205,326)(168,364,206,327), (1,158,348,210)(2,159,349,197)(3,160,350,198)(4,161,337,199)(5,162,338,200)(6,163,339,201)(7,164,340,202)(8,165,341,203)(9,166,342,204)(10,167,343,205)(11,168,344,206)(12,155,345,207)(13,156,346,208)(14,157,347,209)(15,306,427,377)(16,307,428,378)(17,308,429,365)(18,295,430,366)(19,296,431,367)(20,297,432,368)(21,298,433,369)(22,299,434,370)(23,300,421,371)(24,301,422,372)(25,302,423,373)(26,303,424,374)(27,304,425,375)(28,305,426,376)(29,111,83,408)(30,112,84,409)(31,99,71,410)(32,100,72,411)(33,101,73,412)(34,102,74,413)(35,103,75,414)(36,104,76,415)(37,105,77,416)(38,106,78,417)(39,107,79,418)(40,108,80,419)(41,109,81,420)(42,110,82,407)(43,447,252,279)(44,448,239,280)(45,435,240,267)(46,436,241,268)(47,437,242,269)(48,438,243,270)(49,439,244,271)(50,440,245,272)(51,441,246,273)(52,442,247,274)(53,443,248,275)(54,444,249,276)(55,445,250,277)(56,446,251,278)(57,182,383,150)(58,169,384,151)(59,170,385,152)(60,171,386,153)(61,172,387,154)(62,173,388,141)(63,174,389,142)(64,175,390,143)(65,176,391,144)(66,177,392,145)(67,178,379,146)(68,179,380,147)(69,180,381,148)(70,181,382,149)(85,136,122,260)(86,137,123,261)(87,138,124,262)(88,139,125,263)(89,140,126,264)(90,127,113,265)(91,128,114,266)(92,129,115,253)(93,130,116,254)(94,131,117,255)(95,132,118,256)(96,133,119,257)(97,134,120,258)(98,135,121,259)(183,285,319,213)(184,286,320,214)(185,287,321,215)(186,288,322,216)(187,289,309,217)(188,290,310,218)(189,291,311,219)(190,292,312,220)(191,293,313,221)(192,294,314,222)(193,281,315,223)(194,282,316,224)(195,283,317,211)(196,284,318,212)(225,360,398,323)(226,361,399,324)(227,362,400,325)(228,363,401,326)(229,364,402,327)(230,351,403,328)(231,352,404,329)(232,353,405,330)(233,354,406,331)(234,355,393,332)(235,356,394,333)(236,357,395,334)(237,358,396,335)(238,359,397,336) );
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)], [(1,96,216,365),(2,97,217,366),(3,98,218,367),(4,85,219,368),(5,86,220,369),(6,87,221,370),(7,88,222,371),(8,89,223,372),(9,90,224,373),(10,91,211,374),(11,92,212,375),(12,93,213,376),(13,94,214,377),(14,95,215,378),(15,184,131,156),(16,185,132,157),(17,186,133,158),(18,187,134,159),(19,188,135,160),(20,189,136,161),(21,190,137,162),(22,191,138,163),(23,192,139,164),(24,193,140,165),(25,194,127,166),(26,195,128,167),(27,196,129,168),(28,183,130,155),(29,244,397,58),(30,245,398,59),(31,246,399,60),(32,247,400,61),(33,248,401,62),(34,249,402,63),(35,250,403,64),(36,251,404,65),(37,252,405,66),(38,239,406,67),(39,240,393,68),(40,241,394,69),(41,242,395,70),(42,243,396,57),(43,232,392,77),(44,233,379,78),(45,234,380,79),(46,235,381,80),(47,236,382,81),(48,237,383,82),(49,238,384,83),(50,225,385,84),(51,226,386,71),(52,227,387,72),(53,228,388,73),(54,229,389,74),(55,230,390,75),(56,231,391,76),(99,171,324,273),(100,172,325,274),(101,173,326,275),(102,174,327,276),(103,175,328,277),(104,176,329,278),(105,177,330,279),(106,178,331,280),(107,179,332,267),(108,180,333,268),(109,181,334,269),(110,182,335,270),(111,169,336,271),(112,170,323,272),(113,282,302,342),(114,283,303,343),(115,284,304,344),(116,285,305,345),(117,286,306,346),(118,287,307,347),(119,288,308,348),(120,289,295,349),(121,290,296,350),(122,291,297,337),(123,292,298,338),(124,293,299,339),(125,294,300,340),(126,281,301,341),(141,363,443,412),(142,364,444,413),(143,351,445,414),(144,352,446,415),(145,353,447,416),(146,354,448,417),(147,355,435,418),(148,356,436,419),(149,357,437,420),(150,358,438,407),(151,359,439,408),(152,360,440,409),(153,361,441,410),(154,362,442,411),(197,430,309,258),(198,431,310,259),(199,432,311,260),(200,433,312,261),(201,434,313,262),(202,421,314,263),(203,422,315,264),(204,423,316,265),(205,424,317,266),(206,425,318,253),(207,426,319,254),(208,427,320,255),(209,428,321,256),(210,429,322,257)], [(1,406,348,233),(2,393,349,234),(3,394,350,235),(4,395,337,236),(5,396,338,237),(6,397,339,238),(7,398,340,225),(8,399,341,226),(9,400,342,227),(10,401,343,228),(11,402,344,229),(12,403,345,230),(13,404,346,231),(14,405,347,232),(15,446,427,278),(16,447,428,279),(17,448,429,280),(18,435,430,267),(19,436,431,268),(20,437,432,269),(21,438,433,270),(22,439,434,271),(23,440,421,272),(24,441,422,273),(25,442,423,274),(26,443,424,275),(27,444,425,276),(28,445,426,277),(29,293,83,221),(30,294,84,222),(31,281,71,223),(32,282,72,224),(33,283,73,211),(34,284,74,212),(35,285,75,213),(36,286,76,214),(37,287,77,215),(38,288,78,216),(39,289,79,217),(40,290,80,218),(41,291,81,219),(42,292,82,220),(43,378,252,307),(44,365,239,308),(45,366,240,295),(46,367,241,296),(47,368,242,297),(48,369,243,298),(49,370,244,299),(50,371,245,300),(51,372,246,301),(52,373,247,302),(53,374,248,303),(54,375,249,304),(55,376,250,305),(56,377,251,306),(57,123,383,86),(58,124,384,87),(59,125,385,88),(60,126,386,89),(61,113,387,90),(62,114,388,91),(63,115,389,92),(64,116,390,93),(65,117,391,94),(66,118,392,95),(67,119,379,96),(68,120,380,97),(69,121,381,98),(70,122,382,85),(99,193,410,315),(100,194,411,316),(101,195,412,317),(102,196,413,318),(103,183,414,319),(104,184,415,320),(105,185,416,321),(106,186,417,322),(107,187,418,309),(108,188,419,310),(109,189,420,311),(110,190,407,312),(111,191,408,313),(112,192,409,314),(127,154,265,172),(128,141,266,173),(129,142,253,174),(130,143,254,175),(131,144,255,176),(132,145,256,177),(133,146,257,178),(134,147,258,179),(135,148,259,180),(136,149,260,181),(137,150,261,182),(138,151,262,169),(139,152,263,170),(140,153,264,171),(155,351,207,328),(156,352,208,329),(157,353,209,330),(158,354,210,331),(159,355,197,332),(160,356,198,333),(161,357,199,334),(162,358,200,335),(163,359,201,336),(164,360,202,323),(165,361,203,324),(166,362,204,325),(167,363,205,326),(168,364,206,327)], [(1,158,348,210),(2,159,349,197),(3,160,350,198),(4,161,337,199),(5,162,338,200),(6,163,339,201),(7,164,340,202),(8,165,341,203),(9,166,342,204),(10,167,343,205),(11,168,344,206),(12,155,345,207),(13,156,346,208),(14,157,347,209),(15,306,427,377),(16,307,428,378),(17,308,429,365),(18,295,430,366),(19,296,431,367),(20,297,432,368),(21,298,433,369),(22,299,434,370),(23,300,421,371),(24,301,422,372),(25,302,423,373),(26,303,424,374),(27,304,425,375),(28,305,426,376),(29,111,83,408),(30,112,84,409),(31,99,71,410),(32,100,72,411),(33,101,73,412),(34,102,74,413),(35,103,75,414),(36,104,76,415),(37,105,77,416),(38,106,78,417),(39,107,79,418),(40,108,80,419),(41,109,81,420),(42,110,82,407),(43,447,252,279),(44,448,239,280),(45,435,240,267),(46,436,241,268),(47,437,242,269),(48,438,243,270),(49,439,244,271),(50,440,245,272),(51,441,246,273),(52,442,247,274),(53,443,248,275),(54,444,249,276),(55,445,250,277),(56,446,251,278),(57,182,383,150),(58,169,384,151),(59,170,385,152),(60,171,386,153),(61,172,387,154),(62,173,388,141),(63,174,389,142),(64,175,390,143),(65,176,391,144),(66,177,392,145),(67,178,379,146),(68,179,380,147),(69,180,381,148),(70,181,382,149),(85,136,122,260),(86,137,123,261),(87,138,124,262),(88,139,125,263),(89,140,126,264),(90,127,113,265),(91,128,114,266),(92,129,115,253),(93,130,116,254),(94,131,117,255),(95,132,118,256),(96,133,119,257),(97,134,120,258),(98,135,121,259),(183,285,319,213),(184,286,320,214),(185,287,321,215),(186,288,322,216),(187,289,309,217),(188,290,310,218),(189,291,311,219),(190,292,312,220),(191,293,313,221),(192,294,314,222),(193,281,315,223),(194,282,316,224),(195,283,317,211),(196,284,318,212),(225,360,398,323),(226,361,399,324),(227,362,400,325),(228,363,401,326),(229,364,402,327),(230,351,403,328),(231,352,404,329),(232,353,405,330),(233,354,406,331),(234,355,393,332),(235,356,394,333),(236,357,395,334),(237,358,396,335),(238,359,397,336)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 7A | ··· | 7F | 14A | ··· | 14AP | 28A | ··· | 28BT | 28BU | ··· | 28DP |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | Q8 | C7×D4 | C7×Q8 |
| kernel | C14×C4⋊Q8 | C2×C4×C28 | C14×C4⋊C4 | C7×C4⋊Q8 | Q8×C2×C14 | C2×C4⋊Q8 | C2×C42 | C2×C4⋊C4 | C4⋊Q8 | C22×Q8 | C2×C28 | C2×C28 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 2 | 6 | 6 | 24 | 48 | 12 | 4 | 8 | 24 | 48 |
Matrix representation of C14×C4⋊Q8 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 0 | 25 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 13 | 0 | 0 |
| 0 | 13 | 27 | 0 | 0 |
| 0 | 0 | 0 | 20 | 23 |
| 0 | 0 | 0 | 23 | 9 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,25],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,2,13,0,0,0,13,27,0,0,0,0,0,20,23,0,0,0,23,9] >;
C14×C4⋊Q8 in GAP, Magma, Sage, TeX
C_{14}\times C_4\rtimes Q_8 % in TeX
G:=Group("C14xC4:Q8"); // GroupNames label
G:=SmallGroup(448,1314);
// by ID
G=gap.SmallGroup(448,1314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790,1192]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=c^4=1,d^2=c^2,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