Copied to
clipboard

## G = C14×C4.Q8order 448 = 26·7

### Direct product of C14 and C4.Q8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C14×C4.Q8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — C7×C4⋊C4 — C7×C4.Q8 — C14×C4.Q8
 Lower central C1 — C2 — C4 — C14×C4.Q8
 Upper central C1 — C22×C14 — C22×C28 — C14×C4.Q8

Generators and relations for C14×C4.Q8
G = < a,b,c,d | a14=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 194 in 130 conjugacy classes, 98 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C28, C28, C28, C2×C14, C2×C14, C4.Q8, C2×C4⋊C4, C22×C8, C56, C2×C28, C2×C28, C2×C28, C22×C14, C2×C4.Q8, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C22×C28, C22×C28, C7×C4.Q8, C14×C4⋊C4, C22×C56, C14×C4.Q8
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C4.Q8, C2×C4⋊C4, C2×SD16, C2×C28, C7×D4, C7×Q8, C22×C14, C2×C4.Q8, C7×C4⋊C4, C7×SD16, C22×C28, D4×C14, Q8×C14, C7×C4.Q8, C14×C4⋊C4, C14×SD16, C14×C4.Q8

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

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

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

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

196 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 7A ··· 7F 8A ··· 8H 14A ··· 14AP 28A ··· 28X 28Y ··· 28BT 56A ··· 56AV order 1 2 ··· 2 4 4 4 4 4 ··· 4 7 ··· 7 8 ··· 8 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 D4 Q8 D4 SD16 C7×D4 C7×Q8 C7×D4 C7×SD16 kernel C14×C4.Q8 C7×C4.Q8 C14×C4⋊C4 C22×C56 C2×C56 C2×C4.Q8 C4.Q8 C2×C4⋊C4 C22×C8 C2×C8 C2×C28 C2×C28 C22×C14 C2×C14 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 8 6 24 12 6 48 1 2 1 8 6 12 6 48

Matrix representation of C14×C4.Q8 in GL5(𝔽113)

 112 0 0 0 0 0 49 0 0 0 0 0 49 0 0 0 0 0 30 0 0 0 0 0 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 112 0
,
 112 0 0 0 0 0 0 112 0 0 0 1 0 0 0 0 0 0 13 100 0 0 0 13 13
,
 112 0 0 0 0 0 101 104 0 0 0 104 12 0 0 0 0 0 31 31 0 0 0 31 82

G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,49,0,0,0,0,0,49,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,1,0],[112,0,0,0,0,0,0,1,0,0,0,112,0,0,0,0,0,0,13,13,0,0,0,100,13],[112,0,0,0,0,0,101,104,0,0,0,104,12,0,0,0,0,0,31,31,0,0,0,31,82] >;

C14×C4.Q8 in GAP, Magma, Sage, TeX

C_{14}\times C_4.Q_8
% in TeX

G:=Group("C14xC4.Q8");
// GroupNames label

G:=SmallGroup(448,833);
// by ID

G=gap.SmallGroup(448,833);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,9804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^4=1,c^4=b^2,d^2=b^-1*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^3>;
// generators/relations

׿
×
𝔽