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