metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).3Q8, (C2×C4).18D28, (C2×C28).29D4, C14.1(C4⋊Q8), C2.6(C4⋊D28), (C2×Dic7).11D4, C2.5(C28⋊2Q8), (C2×C4).10Dic14, C22.154(D4×D7), (C22×C4).69D14, C22.79(C2×D28), C14.6(C22⋊Q8), C14.33(C4⋊D4), C2.7(C28.3Q8), (C22×C28).45C22, C14.14(C42.C2), C22.43(C2×Dic14), C2.C42.16D7, C23.357(C22×D7), C22.86(D4⋊2D7), C14.C42.26C2, (C22×C14).286C23, C7⋊2(C23.81C23), C22.42(Q8⋊2D7), C14.7(C22.D4), C2.7(C22.D28), C2.11(C22⋊Dic14), (C22×Dic7).11C22, (C2×C14).95(C2×D4), (C2×C14).22(C2×Q8), (C2×C4⋊Dic7).10C2, (C2×Dic7⋊C4).20C2, (C2×C14).181(C4○D4), (C7×C2.C42).12C2, SmallGroup(448,195)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.(C4⋊Q8)
G = < a,b,c,d | a14=b4=c4=1, d2=a7c2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a7b, dbd-1=b-1, dcd-1=a7c-1 >
Subgroups: 636 in 150 conjugacy classes, 65 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2.C42, C2.C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.81C23, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C7×C2.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×C4⋊Dic7, C14.(C4⋊Q8)
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, Dic14, D28, C22×D7, C23.81C23, C2×Dic14, C2×D28, D4×D7, D4⋊2D7, Q8⋊2D7, C28⋊2Q8, C22⋊Dic14, C22.D28, C28.3Q8, C4⋊D28, 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)
(1 187 49 206)(2 188 50 207)(3 189 51 208)(4 190 52 209)(5 191 53 210)(6 192 54 197)(7 193 55 198)(8 194 56 199)(9 195 43 200)(10 196 44 201)(11 183 45 202)(12 184 46 203)(13 185 47 204)(14 186 48 205)(15 171 37 211)(16 172 38 212)(17 173 39 213)(18 174 40 214)(19 175 41 215)(20 176 42 216)(21 177 29 217)(22 178 30 218)(23 179 31 219)(24 180 32 220)(25 181 33 221)(26 182 34 222)(27 169 35 223)(28 170 36 224)(57 161 102 122)(58 162 103 123)(59 163 104 124)(60 164 105 125)(61 165 106 126)(62 166 107 113)(63 167 108 114)(64 168 109 115)(65 155 110 116)(66 156 111 117)(67 157 112 118)(68 158 99 119)(69 159 100 120)(70 160 101 121)(71 146 94 136)(72 147 95 137)(73 148 96 138)(74 149 97 139)(75 150 98 140)(76 151 85 127)(77 152 86 128)(78 153 87 129)(79 154 88 130)(80 141 89 131)(81 142 90 132)(82 143 91 133)(83 144 92 134)(84 145 93 135)(225 434 277 409)(226 421 278 410)(227 422 279 411)(228 423 280 412)(229 424 267 413)(230 425 268 414)(231 426 269 415)(232 427 270 416)(233 428 271 417)(234 429 272 418)(235 430 273 419)(236 431 274 420)(237 432 275 407)(238 433 276 408)(239 436 257 399)(240 437 258 400)(241 438 259 401)(242 439 260 402)(243 440 261 403)(244 441 262 404)(245 442 263 405)(246 443 264 406)(247 444 265 393)(248 445 266 394)(249 446 253 395)(250 447 254 396)(251 448 255 397)(252 435 256 398)(281 348 323 391)(282 349 324 392)(283 350 325 379)(284 337 326 380)(285 338 327 381)(286 339 328 382)(287 340 329 383)(288 341 330 384)(289 342 331 385)(290 343 332 386)(291 344 333 387)(292 345 334 388)(293 346 335 389)(294 347 336 390)(295 361 321 376)(296 362 322 377)(297 363 309 378)(298 364 310 365)(299 351 311 366)(300 352 312 367)(301 353 313 368)(302 354 314 369)(303 355 315 370)(304 356 316 371)(305 357 317 372)(306 358 318 373)(307 359 319 374)(308 360 320 375)
(1 138 23 117)(2 139 24 118)(3 140 25 119)(4 127 26 120)(5 128 27 121)(6 129 28 122)(7 130 15 123)(8 131 16 124)(9 132 17 125)(10 133 18 126)(11 134 19 113)(12 135 20 114)(13 136 21 115)(14 137 22 116)(29 168 47 146)(30 155 48 147)(31 156 49 148)(32 157 50 149)(33 158 51 150)(34 159 52 151)(35 160 53 152)(36 161 54 153)(37 162 55 154)(38 163 56 141)(39 164 43 142)(40 165 44 143)(41 166 45 144)(42 167 46 145)(57 185 78 177)(58 186 79 178)(59 187 80 179)(60 188 81 180)(61 189 82 181)(62 190 83 182)(63 191 84 169)(64 192 71 170)(65 193 72 171)(66 194 73 172)(67 195 74 173)(68 196 75 174)(69 183 76 175)(70 184 77 176)(85 215 100 202)(86 216 101 203)(87 217 102 204)(88 218 103 205)(89 219 104 206)(90 220 105 207)(91 221 106 208)(92 222 107 209)(93 223 108 210)(94 224 109 197)(95 211 110 198)(96 212 111 199)(97 213 112 200)(98 214 99 201)(225 342 242 361)(226 343 243 362)(227 344 244 363)(228 345 245 364)(229 346 246 351)(230 347 247 352)(231 348 248 353)(232 349 249 354)(233 350 250 355)(234 337 251 356)(235 338 252 357)(236 339 239 358)(237 340 240 359)(238 341 241 360)(253 369 270 392)(254 370 271 379)(255 371 272 380)(256 372 273 381)(257 373 274 382)(258 374 275 383)(259 375 276 384)(260 376 277 385)(261 377 278 386)(262 378 279 387)(263 365 280 388)(264 366 267 389)(265 367 268 390)(266 368 269 391)(281 401 301 408)(282 402 302 409)(283 403 303 410)(284 404 304 411)(285 405 305 412)(286 406 306 413)(287 393 307 414)(288 394 308 415)(289 395 295 416)(290 396 296 417)(291 397 297 418)(292 398 298 419)(293 399 299 420)(294 400 300 407)(309 429 333 448)(310 430 334 435)(311 431 335 436)(312 432 336 437)(313 433 323 438)(314 434 324 439)(315 421 325 440)(316 422 326 441)(317 423 327 442)(318 424 328 443)(319 425 329 444)(320 426 330 445)(321 427 331 446)(322 428 332 447)
(1 247 16 237)(2 246 17 236)(3 245 18 235)(4 244 19 234)(5 243 20 233)(6 242 21 232)(7 241 22 231)(8 240 23 230)(9 239 24 229)(10 252 25 228)(11 251 26 227)(12 250 27 226)(13 249 28 225)(14 248 15 238)(29 270 54 260)(30 269 55 259)(31 268 56 258)(32 267 43 257)(33 280 44 256)(34 279 45 255)(35 278 46 254)(36 277 47 253)(37 276 48 266)(38 275 49 265)(39 274 50 264)(40 273 51 263)(41 272 52 262)(42 271 53 261)(57 302 71 289)(58 301 72 288)(59 300 73 287)(60 299 74 286)(61 298 75 285)(62 297 76 284)(63 296 77 283)(64 295 78 282)(65 308 79 281)(66 307 80 294)(67 306 81 293)(68 305 82 292)(69 304 83 291)(70 303 84 290)(85 326 107 309)(86 325 108 322)(87 324 109 321)(88 323 110 320)(89 336 111 319)(90 335 112 318)(91 334 99 317)(92 333 100 316)(93 332 101 315)(94 331 102 314)(95 330 103 313)(96 329 104 312)(97 328 105 311)(98 327 106 310)(113 363 127 337)(114 362 128 350)(115 361 129 349)(116 360 130 348)(117 359 131 347)(118 358 132 346)(119 357 133 345)(120 356 134 344)(121 355 135 343)(122 354 136 342)(123 353 137 341)(124 352 138 340)(125 351 139 339)(126 364 140 338)(141 390 156 374)(142 389 157 373)(143 388 158 372)(144 387 159 371)(145 386 160 370)(146 385 161 369)(147 384 162 368)(148 383 163 367)(149 382 164 366)(150 381 165 365)(151 380 166 378)(152 379 167 377)(153 392 168 376)(154 391 155 375)(169 410 184 396)(170 409 185 395)(171 408 186 394)(172 407 187 393)(173 420 188 406)(174 419 189 405)(175 418 190 404)(176 417 191 403)(177 416 192 402)(178 415 193 401)(179 414 194 400)(180 413 195 399)(181 412 196 398)(182 411 183 397)(197 439 217 427)(198 438 218 426)(199 437 219 425)(200 436 220 424)(201 435 221 423)(202 448 222 422)(203 447 223 421)(204 446 224 434)(205 445 211 433)(206 444 212 432)(207 443 213 431)(208 442 214 430)(209 441 215 429)(210 440 216 428)
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), (1,187,49,206)(2,188,50,207)(3,189,51,208)(4,190,52,209)(5,191,53,210)(6,192,54,197)(7,193,55,198)(8,194,56,199)(9,195,43,200)(10,196,44,201)(11,183,45,202)(12,184,46,203)(13,185,47,204)(14,186,48,205)(15,171,37,211)(16,172,38,212)(17,173,39,213)(18,174,40,214)(19,175,41,215)(20,176,42,216)(21,177,29,217)(22,178,30,218)(23,179,31,219)(24,180,32,220)(25,181,33,221)(26,182,34,222)(27,169,35,223)(28,170,36,224)(57,161,102,122)(58,162,103,123)(59,163,104,124)(60,164,105,125)(61,165,106,126)(62,166,107,113)(63,167,108,114)(64,168,109,115)(65,155,110,116)(66,156,111,117)(67,157,112,118)(68,158,99,119)(69,159,100,120)(70,160,101,121)(71,146,94,136)(72,147,95,137)(73,148,96,138)(74,149,97,139)(75,150,98,140)(76,151,85,127)(77,152,86,128)(78,153,87,129)(79,154,88,130)(80,141,89,131)(81,142,90,132)(82,143,91,133)(83,144,92,134)(84,145,93,135)(225,434,277,409)(226,421,278,410)(227,422,279,411)(228,423,280,412)(229,424,267,413)(230,425,268,414)(231,426,269,415)(232,427,270,416)(233,428,271,417)(234,429,272,418)(235,430,273,419)(236,431,274,420)(237,432,275,407)(238,433,276,408)(239,436,257,399)(240,437,258,400)(241,438,259,401)(242,439,260,402)(243,440,261,403)(244,441,262,404)(245,442,263,405)(246,443,264,406)(247,444,265,393)(248,445,266,394)(249,446,253,395)(250,447,254,396)(251,448,255,397)(252,435,256,398)(281,348,323,391)(282,349,324,392)(283,350,325,379)(284,337,326,380)(285,338,327,381)(286,339,328,382)(287,340,329,383)(288,341,330,384)(289,342,331,385)(290,343,332,386)(291,344,333,387)(292,345,334,388)(293,346,335,389)(294,347,336,390)(295,361,321,376)(296,362,322,377)(297,363,309,378)(298,364,310,365)(299,351,311,366)(300,352,312,367)(301,353,313,368)(302,354,314,369)(303,355,315,370)(304,356,316,371)(305,357,317,372)(306,358,318,373)(307,359,319,374)(308,360,320,375), (1,138,23,117)(2,139,24,118)(3,140,25,119)(4,127,26,120)(5,128,27,121)(6,129,28,122)(7,130,15,123)(8,131,16,124)(9,132,17,125)(10,133,18,126)(11,134,19,113)(12,135,20,114)(13,136,21,115)(14,137,22,116)(29,168,47,146)(30,155,48,147)(31,156,49,148)(32,157,50,149)(33,158,51,150)(34,159,52,151)(35,160,53,152)(36,161,54,153)(37,162,55,154)(38,163,56,141)(39,164,43,142)(40,165,44,143)(41,166,45,144)(42,167,46,145)(57,185,78,177)(58,186,79,178)(59,187,80,179)(60,188,81,180)(61,189,82,181)(62,190,83,182)(63,191,84,169)(64,192,71,170)(65,193,72,171)(66,194,73,172)(67,195,74,173)(68,196,75,174)(69,183,76,175)(70,184,77,176)(85,215,100,202)(86,216,101,203)(87,217,102,204)(88,218,103,205)(89,219,104,206)(90,220,105,207)(91,221,106,208)(92,222,107,209)(93,223,108,210)(94,224,109,197)(95,211,110,198)(96,212,111,199)(97,213,112,200)(98,214,99,201)(225,342,242,361)(226,343,243,362)(227,344,244,363)(228,345,245,364)(229,346,246,351)(230,347,247,352)(231,348,248,353)(232,349,249,354)(233,350,250,355)(234,337,251,356)(235,338,252,357)(236,339,239,358)(237,340,240,359)(238,341,241,360)(253,369,270,392)(254,370,271,379)(255,371,272,380)(256,372,273,381)(257,373,274,382)(258,374,275,383)(259,375,276,384)(260,376,277,385)(261,377,278,386)(262,378,279,387)(263,365,280,388)(264,366,267,389)(265,367,268,390)(266,368,269,391)(281,401,301,408)(282,402,302,409)(283,403,303,410)(284,404,304,411)(285,405,305,412)(286,406,306,413)(287,393,307,414)(288,394,308,415)(289,395,295,416)(290,396,296,417)(291,397,297,418)(292,398,298,419)(293,399,299,420)(294,400,300,407)(309,429,333,448)(310,430,334,435)(311,431,335,436)(312,432,336,437)(313,433,323,438)(314,434,324,439)(315,421,325,440)(316,422,326,441)(317,423,327,442)(318,424,328,443)(319,425,329,444)(320,426,330,445)(321,427,331,446)(322,428,332,447), (1,247,16,237)(2,246,17,236)(3,245,18,235)(4,244,19,234)(5,243,20,233)(6,242,21,232)(7,241,22,231)(8,240,23,230)(9,239,24,229)(10,252,25,228)(11,251,26,227)(12,250,27,226)(13,249,28,225)(14,248,15,238)(29,270,54,260)(30,269,55,259)(31,268,56,258)(32,267,43,257)(33,280,44,256)(34,279,45,255)(35,278,46,254)(36,277,47,253)(37,276,48,266)(38,275,49,265)(39,274,50,264)(40,273,51,263)(41,272,52,262)(42,271,53,261)(57,302,71,289)(58,301,72,288)(59,300,73,287)(60,299,74,286)(61,298,75,285)(62,297,76,284)(63,296,77,283)(64,295,78,282)(65,308,79,281)(66,307,80,294)(67,306,81,293)(68,305,82,292)(69,304,83,291)(70,303,84,290)(85,326,107,309)(86,325,108,322)(87,324,109,321)(88,323,110,320)(89,336,111,319)(90,335,112,318)(91,334,99,317)(92,333,100,316)(93,332,101,315)(94,331,102,314)(95,330,103,313)(96,329,104,312)(97,328,105,311)(98,327,106,310)(113,363,127,337)(114,362,128,350)(115,361,129,349)(116,360,130,348)(117,359,131,347)(118,358,132,346)(119,357,133,345)(120,356,134,344)(121,355,135,343)(122,354,136,342)(123,353,137,341)(124,352,138,340)(125,351,139,339)(126,364,140,338)(141,390,156,374)(142,389,157,373)(143,388,158,372)(144,387,159,371)(145,386,160,370)(146,385,161,369)(147,384,162,368)(148,383,163,367)(149,382,164,366)(150,381,165,365)(151,380,166,378)(152,379,167,377)(153,392,168,376)(154,391,155,375)(169,410,184,396)(170,409,185,395)(171,408,186,394)(172,407,187,393)(173,420,188,406)(174,419,189,405)(175,418,190,404)(176,417,191,403)(177,416,192,402)(178,415,193,401)(179,414,194,400)(180,413,195,399)(181,412,196,398)(182,411,183,397)(197,439,217,427)(198,438,218,426)(199,437,219,425)(200,436,220,424)(201,435,221,423)(202,448,222,422)(203,447,223,421)(204,446,224,434)(205,445,211,433)(206,444,212,432)(207,443,213,431)(208,442,214,430)(209,441,215,429)(210,440,216,428)>;
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), (1,187,49,206)(2,188,50,207)(3,189,51,208)(4,190,52,209)(5,191,53,210)(6,192,54,197)(7,193,55,198)(8,194,56,199)(9,195,43,200)(10,196,44,201)(11,183,45,202)(12,184,46,203)(13,185,47,204)(14,186,48,205)(15,171,37,211)(16,172,38,212)(17,173,39,213)(18,174,40,214)(19,175,41,215)(20,176,42,216)(21,177,29,217)(22,178,30,218)(23,179,31,219)(24,180,32,220)(25,181,33,221)(26,182,34,222)(27,169,35,223)(28,170,36,224)(57,161,102,122)(58,162,103,123)(59,163,104,124)(60,164,105,125)(61,165,106,126)(62,166,107,113)(63,167,108,114)(64,168,109,115)(65,155,110,116)(66,156,111,117)(67,157,112,118)(68,158,99,119)(69,159,100,120)(70,160,101,121)(71,146,94,136)(72,147,95,137)(73,148,96,138)(74,149,97,139)(75,150,98,140)(76,151,85,127)(77,152,86,128)(78,153,87,129)(79,154,88,130)(80,141,89,131)(81,142,90,132)(82,143,91,133)(83,144,92,134)(84,145,93,135)(225,434,277,409)(226,421,278,410)(227,422,279,411)(228,423,280,412)(229,424,267,413)(230,425,268,414)(231,426,269,415)(232,427,270,416)(233,428,271,417)(234,429,272,418)(235,430,273,419)(236,431,274,420)(237,432,275,407)(238,433,276,408)(239,436,257,399)(240,437,258,400)(241,438,259,401)(242,439,260,402)(243,440,261,403)(244,441,262,404)(245,442,263,405)(246,443,264,406)(247,444,265,393)(248,445,266,394)(249,446,253,395)(250,447,254,396)(251,448,255,397)(252,435,256,398)(281,348,323,391)(282,349,324,392)(283,350,325,379)(284,337,326,380)(285,338,327,381)(286,339,328,382)(287,340,329,383)(288,341,330,384)(289,342,331,385)(290,343,332,386)(291,344,333,387)(292,345,334,388)(293,346,335,389)(294,347,336,390)(295,361,321,376)(296,362,322,377)(297,363,309,378)(298,364,310,365)(299,351,311,366)(300,352,312,367)(301,353,313,368)(302,354,314,369)(303,355,315,370)(304,356,316,371)(305,357,317,372)(306,358,318,373)(307,359,319,374)(308,360,320,375), (1,138,23,117)(2,139,24,118)(3,140,25,119)(4,127,26,120)(5,128,27,121)(6,129,28,122)(7,130,15,123)(8,131,16,124)(9,132,17,125)(10,133,18,126)(11,134,19,113)(12,135,20,114)(13,136,21,115)(14,137,22,116)(29,168,47,146)(30,155,48,147)(31,156,49,148)(32,157,50,149)(33,158,51,150)(34,159,52,151)(35,160,53,152)(36,161,54,153)(37,162,55,154)(38,163,56,141)(39,164,43,142)(40,165,44,143)(41,166,45,144)(42,167,46,145)(57,185,78,177)(58,186,79,178)(59,187,80,179)(60,188,81,180)(61,189,82,181)(62,190,83,182)(63,191,84,169)(64,192,71,170)(65,193,72,171)(66,194,73,172)(67,195,74,173)(68,196,75,174)(69,183,76,175)(70,184,77,176)(85,215,100,202)(86,216,101,203)(87,217,102,204)(88,218,103,205)(89,219,104,206)(90,220,105,207)(91,221,106,208)(92,222,107,209)(93,223,108,210)(94,224,109,197)(95,211,110,198)(96,212,111,199)(97,213,112,200)(98,214,99,201)(225,342,242,361)(226,343,243,362)(227,344,244,363)(228,345,245,364)(229,346,246,351)(230,347,247,352)(231,348,248,353)(232,349,249,354)(233,350,250,355)(234,337,251,356)(235,338,252,357)(236,339,239,358)(237,340,240,359)(238,341,241,360)(253,369,270,392)(254,370,271,379)(255,371,272,380)(256,372,273,381)(257,373,274,382)(258,374,275,383)(259,375,276,384)(260,376,277,385)(261,377,278,386)(262,378,279,387)(263,365,280,388)(264,366,267,389)(265,367,268,390)(266,368,269,391)(281,401,301,408)(282,402,302,409)(283,403,303,410)(284,404,304,411)(285,405,305,412)(286,406,306,413)(287,393,307,414)(288,394,308,415)(289,395,295,416)(290,396,296,417)(291,397,297,418)(292,398,298,419)(293,399,299,420)(294,400,300,407)(309,429,333,448)(310,430,334,435)(311,431,335,436)(312,432,336,437)(313,433,323,438)(314,434,324,439)(315,421,325,440)(316,422,326,441)(317,423,327,442)(318,424,328,443)(319,425,329,444)(320,426,330,445)(321,427,331,446)(322,428,332,447), (1,247,16,237)(2,246,17,236)(3,245,18,235)(4,244,19,234)(5,243,20,233)(6,242,21,232)(7,241,22,231)(8,240,23,230)(9,239,24,229)(10,252,25,228)(11,251,26,227)(12,250,27,226)(13,249,28,225)(14,248,15,238)(29,270,54,260)(30,269,55,259)(31,268,56,258)(32,267,43,257)(33,280,44,256)(34,279,45,255)(35,278,46,254)(36,277,47,253)(37,276,48,266)(38,275,49,265)(39,274,50,264)(40,273,51,263)(41,272,52,262)(42,271,53,261)(57,302,71,289)(58,301,72,288)(59,300,73,287)(60,299,74,286)(61,298,75,285)(62,297,76,284)(63,296,77,283)(64,295,78,282)(65,308,79,281)(66,307,80,294)(67,306,81,293)(68,305,82,292)(69,304,83,291)(70,303,84,290)(85,326,107,309)(86,325,108,322)(87,324,109,321)(88,323,110,320)(89,336,111,319)(90,335,112,318)(91,334,99,317)(92,333,100,316)(93,332,101,315)(94,331,102,314)(95,330,103,313)(96,329,104,312)(97,328,105,311)(98,327,106,310)(113,363,127,337)(114,362,128,350)(115,361,129,349)(116,360,130,348)(117,359,131,347)(118,358,132,346)(119,357,133,345)(120,356,134,344)(121,355,135,343)(122,354,136,342)(123,353,137,341)(124,352,138,340)(125,351,139,339)(126,364,140,338)(141,390,156,374)(142,389,157,373)(143,388,158,372)(144,387,159,371)(145,386,160,370)(146,385,161,369)(147,384,162,368)(148,383,163,367)(149,382,164,366)(150,381,165,365)(151,380,166,378)(152,379,167,377)(153,392,168,376)(154,391,155,375)(169,410,184,396)(170,409,185,395)(171,408,186,394)(172,407,187,393)(173,420,188,406)(174,419,189,405)(175,418,190,404)(176,417,191,403)(177,416,192,402)(178,415,193,401)(179,414,194,400)(180,413,195,399)(181,412,196,398)(182,411,183,397)(197,439,217,427)(198,438,218,426)(199,437,219,425)(200,436,220,424)(201,435,221,423)(202,448,222,422)(203,447,223,421)(204,446,224,434)(205,445,211,433)(206,444,212,432)(207,443,213,431)(208,442,214,430)(209,441,215,429)(210,440,216,428) );
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)], [(1,187,49,206),(2,188,50,207),(3,189,51,208),(4,190,52,209),(5,191,53,210),(6,192,54,197),(7,193,55,198),(8,194,56,199),(9,195,43,200),(10,196,44,201),(11,183,45,202),(12,184,46,203),(13,185,47,204),(14,186,48,205),(15,171,37,211),(16,172,38,212),(17,173,39,213),(18,174,40,214),(19,175,41,215),(20,176,42,216),(21,177,29,217),(22,178,30,218),(23,179,31,219),(24,180,32,220),(25,181,33,221),(26,182,34,222),(27,169,35,223),(28,170,36,224),(57,161,102,122),(58,162,103,123),(59,163,104,124),(60,164,105,125),(61,165,106,126),(62,166,107,113),(63,167,108,114),(64,168,109,115),(65,155,110,116),(66,156,111,117),(67,157,112,118),(68,158,99,119),(69,159,100,120),(70,160,101,121),(71,146,94,136),(72,147,95,137),(73,148,96,138),(74,149,97,139),(75,150,98,140),(76,151,85,127),(77,152,86,128),(78,153,87,129),(79,154,88,130),(80,141,89,131),(81,142,90,132),(82,143,91,133),(83,144,92,134),(84,145,93,135),(225,434,277,409),(226,421,278,410),(227,422,279,411),(228,423,280,412),(229,424,267,413),(230,425,268,414),(231,426,269,415),(232,427,270,416),(233,428,271,417),(234,429,272,418),(235,430,273,419),(236,431,274,420),(237,432,275,407),(238,433,276,408),(239,436,257,399),(240,437,258,400),(241,438,259,401),(242,439,260,402),(243,440,261,403),(244,441,262,404),(245,442,263,405),(246,443,264,406),(247,444,265,393),(248,445,266,394),(249,446,253,395),(250,447,254,396),(251,448,255,397),(252,435,256,398),(281,348,323,391),(282,349,324,392),(283,350,325,379),(284,337,326,380),(285,338,327,381),(286,339,328,382),(287,340,329,383),(288,341,330,384),(289,342,331,385),(290,343,332,386),(291,344,333,387),(292,345,334,388),(293,346,335,389),(294,347,336,390),(295,361,321,376),(296,362,322,377),(297,363,309,378),(298,364,310,365),(299,351,311,366),(300,352,312,367),(301,353,313,368),(302,354,314,369),(303,355,315,370),(304,356,316,371),(305,357,317,372),(306,358,318,373),(307,359,319,374),(308,360,320,375)], [(1,138,23,117),(2,139,24,118),(3,140,25,119),(4,127,26,120),(5,128,27,121),(6,129,28,122),(7,130,15,123),(8,131,16,124),(9,132,17,125),(10,133,18,126),(11,134,19,113),(12,135,20,114),(13,136,21,115),(14,137,22,116),(29,168,47,146),(30,155,48,147),(31,156,49,148),(32,157,50,149),(33,158,51,150),(34,159,52,151),(35,160,53,152),(36,161,54,153),(37,162,55,154),(38,163,56,141),(39,164,43,142),(40,165,44,143),(41,166,45,144),(42,167,46,145),(57,185,78,177),(58,186,79,178),(59,187,80,179),(60,188,81,180),(61,189,82,181),(62,190,83,182),(63,191,84,169),(64,192,71,170),(65,193,72,171),(66,194,73,172),(67,195,74,173),(68,196,75,174),(69,183,76,175),(70,184,77,176),(85,215,100,202),(86,216,101,203),(87,217,102,204),(88,218,103,205),(89,219,104,206),(90,220,105,207),(91,221,106,208),(92,222,107,209),(93,223,108,210),(94,224,109,197),(95,211,110,198),(96,212,111,199),(97,213,112,200),(98,214,99,201),(225,342,242,361),(226,343,243,362),(227,344,244,363),(228,345,245,364),(229,346,246,351),(230,347,247,352),(231,348,248,353),(232,349,249,354),(233,350,250,355),(234,337,251,356),(235,338,252,357),(236,339,239,358),(237,340,240,359),(238,341,241,360),(253,369,270,392),(254,370,271,379),(255,371,272,380),(256,372,273,381),(257,373,274,382),(258,374,275,383),(259,375,276,384),(260,376,277,385),(261,377,278,386),(262,378,279,387),(263,365,280,388),(264,366,267,389),(265,367,268,390),(266,368,269,391),(281,401,301,408),(282,402,302,409),(283,403,303,410),(284,404,304,411),(285,405,305,412),(286,406,306,413),(287,393,307,414),(288,394,308,415),(289,395,295,416),(290,396,296,417),(291,397,297,418),(292,398,298,419),(293,399,299,420),(294,400,300,407),(309,429,333,448),(310,430,334,435),(311,431,335,436),(312,432,336,437),(313,433,323,438),(314,434,324,439),(315,421,325,440),(316,422,326,441),(317,423,327,442),(318,424,328,443),(319,425,329,444),(320,426,330,445),(321,427,331,446),(322,428,332,447)], [(1,247,16,237),(2,246,17,236),(3,245,18,235),(4,244,19,234),(5,243,20,233),(6,242,21,232),(7,241,22,231),(8,240,23,230),(9,239,24,229),(10,252,25,228),(11,251,26,227),(12,250,27,226),(13,249,28,225),(14,248,15,238),(29,270,54,260),(30,269,55,259),(31,268,56,258),(32,267,43,257),(33,280,44,256),(34,279,45,255),(35,278,46,254),(36,277,47,253),(37,276,48,266),(38,275,49,265),(39,274,50,264),(40,273,51,263),(41,272,52,262),(42,271,53,261),(57,302,71,289),(58,301,72,288),(59,300,73,287),(60,299,74,286),(61,298,75,285),(62,297,76,284),(63,296,77,283),(64,295,78,282),(65,308,79,281),(66,307,80,294),(67,306,81,293),(68,305,82,292),(69,304,83,291),(70,303,84,290),(85,326,107,309),(86,325,108,322),(87,324,109,321),(88,323,110,320),(89,336,111,319),(90,335,112,318),(91,334,99,317),(92,333,100,316),(93,332,101,315),(94,331,102,314),(95,330,103,313),(96,329,104,312),(97,328,105,311),(98,327,106,310),(113,363,127,337),(114,362,128,350),(115,361,129,349),(116,360,130,348),(117,359,131,347),(118,358,132,346),(119,357,133,345),(120,356,134,344),(121,355,135,343),(122,354,136,342),(123,353,137,341),(124,352,138,340),(125,351,139,339),(126,364,140,338),(141,390,156,374),(142,389,157,373),(143,388,158,372),(144,387,159,371),(145,386,160,370),(146,385,161,369),(147,384,162,368),(148,383,163,367),(149,382,164,366),(150,381,165,365),(151,380,166,378),(152,379,167,377),(153,392,168,376),(154,391,155,375),(169,410,184,396),(170,409,185,395),(171,408,186,394),(172,407,187,393),(173,420,188,406),(174,419,189,405),(175,418,190,404),(176,417,191,403),(177,416,192,402),(178,415,193,401),(179,414,194,400),(180,413,195,399),(181,412,196,398),(182,411,183,397),(197,439,217,427),(198,438,218,426),(199,437,219,425),(200,436,220,424),(201,435,221,423),(202,448,222,422),(203,447,223,421),(204,446,224,434),(205,445,211,433),(206,444,212,432),(207,443,213,431),(208,442,214,430),(209,441,215,429),(210,440,216,428)]])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | D28 | D4×D7 | D4⋊2D7 | Q8⋊2D7 |
| kernel | C14.(C4⋊Q8) | C14.C42 | C7×C2.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×Dic7 | C2×C28 | C2×C28 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 4 | 3 | 6 | 9 | 24 | 12 | 3 | 6 | 3 |
Matrix representation of C14.(C4⋊Q8) ►in GL8(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 21 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 18 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 24 | 17 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 5 |
G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,21,18],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,27,13,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,21,6,0,0,0,0,0,0,23,8,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,6,21,0,0,0,0,0,0,8,23,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,11,24,0,0,0,0,0,0,24,18,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,24,2,0,0,0,0,0,0,17,5] >;
C14.(C4⋊Q8) in GAP, Magma, Sage, TeX
C_{14}.(C_4\rtimes Q_8) % in TeX
G:=Group("C14.(C4:Q8)"); // GroupNames label
G:=SmallGroup(448,195);
// by ID
G=gap.SmallGroup(448,195);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,344,254,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=c^4=1,d^2=a^7*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^7*b,d*b*d^-1=b^-1,d*c*d^-1=a^7*c^-1>;
// generators/relations