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G = C14.Q32order 448 = 26·7

3rd non-split extension by C14 of Q32 acting via Q32/Q16=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C14.Q32
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C56⋊1C4 — C14.Q32
 Lower central C7 — C14 — C28 — C56 — C14.Q32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for C14.Q32
G = < a,b,c | a14=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=a7b-1 >

Subgroups: 228 in 58 conjugacy classes, 31 normal (27 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C16, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, Dic7, C28, C28, C2×C14, C2.D8, C2×C16, C2×Q16, C56, C2×Dic7, C2×C28, C2×C28, C7×Q8, C2.Q32, C7⋊C16, C4⋊Dic7, C2×C56, C7×Q16, C7×Q16, Q8×C14, C2×C7⋊C16, C561C4, C14×Q16, C14.Q32
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Dic7, D14, D4⋊C4, SD32, Q32, C2×Dic7, C7⋊D4, C2.Q32, D4⋊D7, D4.D7, C23.D7, C7⋊SD32, C7⋊Q32, D4⋊Dic7, C14.Q32

Smallest permutation representation of C14.Q32
Regular action on 448 points
Generators in S448
```(1 64 67 199 337 285 434 425 183 222 161 105 84 266)(2 267 85 106 162 223 184 426 435 286 338 200 68 49)(3 50 69 201 339 287 436 427 185 224 163 107 86 268)(4 269 87 108 164 209 186 428 437 288 340 202 70 51)(5 52 71 203 341 273 438 429 187 210 165 109 88 270)(6 271 89 110 166 211 188 430 439 274 342 204 72 53)(7 54 73 205 343 275 440 431 189 212 167 111 90 272)(8 257 91 112 168 213 190 432 441 276 344 206 74 55)(9 56 75 207 345 277 442 417 191 214 169 97 92 258)(10 259 93 98 170 215 192 418 443 278 346 208 76 57)(11 58 77 193 347 279 444 419 177 216 171 99 94 260)(12 261 95 100 172 217 178 420 445 280 348 194 78 59)(13 60 79 195 349 281 446 421 179 218 173 101 96 262)(14 263 81 102 174 219 180 422 447 282 350 196 80 61)(15 62 65 197 351 283 448 423 181 220 175 103 82 264)(16 265 83 104 176 221 182 424 433 284 352 198 66 63)(17 41 149 381 290 319 250 322 119 401 131 353 232 391)(18 392 233 354 132 402 120 323 251 320 291 382 150 42)(19 43 151 383 292 305 252 324 121 403 133 355 234 393)(20 394 235 356 134 404 122 325 253 306 293 384 152 44)(21 45 153 369 294 307 254 326 123 405 135 357 236 395)(22 396 237 358 136 406 124 327 255 308 295 370 154 46)(23 47 155 371 296 309 256 328 125 407 137 359 238 397)(24 398 239 360 138 408 126 329 241 310 297 372 156 48)(25 33 157 373 298 311 242 330 127 409 139 361 240 399)(26 400 225 362 140 410 128 331 243 312 299 374 158 34)(27 35 159 375 300 313 244 332 113 411 141 363 226 385)(28 386 227 364 142 412 114 333 245 314 301 376 160 36)(29 37 145 377 302 315 246 334 115 413 143 365 228 387)(30 388 229 366 144 414 116 335 247 316 303 378 146 38)(31 39 147 379 304 317 248 336 117 415 129 367 230 389)(32 390 231 368 130 416 118 321 249 318 289 380 148 40)
(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 29 9 21)(2 333 10 325)(3 27 11 19)(4 331 12 323)(5 25 13 17)(6 329 14 321)(7 23 15 31)(8 327 16 335)(18 428 26 420)(20 426 28 418)(22 424 30 432)(24 422 32 430)(33 60 41 52)(34 178 42 186)(35 58 43 50)(36 192 44 184)(37 56 45 64)(38 190 46 182)(39 54 47 62)(40 188 48 180)(49 114 57 122)(51 128 59 120)(53 126 61 118)(55 124 63 116)(65 147 73 155)(66 414 74 406)(67 145 75 153)(68 412 76 404)(69 159 77 151)(70 410 78 402)(71 157 79 149)(72 408 80 416)(81 318 89 310)(82 230 90 238)(83 316 91 308)(84 228 92 236)(85 314 93 306)(86 226 94 234)(87 312 95 320)(88 240 96 232)(97 357 105 365)(98 293 106 301)(99 355 107 363)(100 291 108 299)(101 353 109 361)(102 289 110 297)(103 367 111 359)(104 303 112 295)(113 177 121 185)(115 191 123 183)(117 189 125 181)(119 187 127 179)(129 167 137 175)(130 204 138 196)(131 165 139 173)(132 202 140 194)(133 163 141 171)(134 200 142 208)(135 161 143 169)(136 198 144 206)(146 213 154 221)(148 211 156 219)(150 209 158 217)(152 223 160 215)(162 376 170 384)(164 374 172 382)(166 372 174 380)(168 370 176 378)(193 383 201 375)(195 381 203 373)(197 379 205 371)(199 377 207 369)(210 409 218 401)(212 407 220 415)(214 405 222 413)(216 403 224 411)(225 280 233 288)(227 278 235 286)(229 276 237 284)(231 274 239 282)(241 263 249 271)(242 446 250 438)(243 261 251 269)(244 444 252 436)(245 259 253 267)(246 442 254 434)(247 257 255 265)(248 440 256 448)(258 395 266 387)(260 393 268 385)(262 391 270 399)(264 389 272 397)(273 311 281 319)(275 309 283 317)(277 307 285 315)(279 305 287 313)(290 341 298 349)(292 339 300 347)(294 337 302 345)(296 351 304 343)(322 429 330 421)(324 427 332 419)(326 425 334 417)(328 423 336 431)(338 364 346 356)(340 362 348 354)(342 360 350 368)(344 358 352 366)(386 443 394 435)(388 441 396 433)(390 439 398 447)(392 437 400 445)```

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 16A ··· 16H 28A ··· 28F 28G ··· 28R 56A ··· 56L order 1 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 8 8 56 56 2 2 2 2 2 2 2 2 ··· 2 14 ··· 14 4 ··· 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - - + + - image C1 C2 C2 C2 C4 D4 D4 D7 SD16 D8 D14 Dic7 SD32 Q32 C7⋊D4 C7⋊D4 D4.D7 D4⋊D7 C7⋊SD32 C7⋊Q32 kernel C14.Q32 C2×C7⋊C16 C56⋊1C4 C14×Q16 C7×Q16 C56 C2×C28 C2×Q16 C28 C2×C14 C2×C8 Q16 C14 C14 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 3 2 2 3 6 4 4 6 6 3 3 6 6

Matrix representation of C14.Q32 in GL4(𝔽113) generated by

 88 80 0 0 59 1 0 0 0 0 112 0 0 0 0 112
,
 73 82 0 0 48 40 0 0 0 0 104 25 0 0 44 16
,
 112 0 0 0 0 112 0 0 0 0 20 9 0 0 81 93
`G:=sub<GL(4,GF(113))| [88,59,0,0,80,1,0,0,0,0,112,0,0,0,0,112],[73,48,0,0,82,40,0,0,0,0,104,44,0,0,25,16],[112,0,0,0,0,112,0,0,0,0,20,81,0,0,9,93] >;`

C14.Q32 in GAP, Magma, Sage, TeX

`C_{14}.Q_{32}`
`% in TeX`

`G:=Group("C14.Q32");`
`// GroupNames label`

`G:=SmallGroup(448,121);`
`// by ID`

`G=gap.SmallGroup(448,121);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,675,346,192,1684,851,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^14=b^16=1,c^2=b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b^-1>;`
`// generators/relations`

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