direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C14.Q16, C4.61(C2×D28), C4⋊C4.229D14, C4.8(D14⋊C4), (C2×Dic14)⋊9C4, C28.141(C2×D4), (C2×C28).135D4, (C2×C4).140D28, (C2×C14).16Q16, C14.32(C2×Q16), C14⋊1(Q8⋊C4), Dic14⋊19(C2×C4), C28.61(C22×C4), (C2×C14).32SD16, C14.49(C2×SD16), C28.21(C22⋊C4), (C2×C28).322C23, (C22×C4).330D14, (C22×C14).187D4, C23.98(C7⋊D4), C22.8(C7⋊Q16), C22.46(D14⋊C4), C22.11(D4.D7), (C22×C28).137C22, (C22×Dic14).10C2, (C2×Dic14).261C22, C4.50(C2×C4×D7), (C2×C4⋊C4).7D7, C7⋊2(C2×Q8⋊C4), (C14×C4⋊C4).6C2, (C2×C4).76(C4×D7), (C22×C7⋊C8).5C2, C2.2(C2×D4.D7), C2.2(C2×C7⋊Q16), (C2×C28).79(C2×C4), C2.13(C2×D14⋊C4), (C2×C14).442(C2×D4), (C2×C7⋊C8).241C22, C14.40(C2×C22⋊C4), C22.59(C2×C7⋊D4), (C2×C4).125(C7⋊D4), (C7×C4⋊C4).260C22, (C2×C4).422(C22×D7), (C2×C14).59(C22⋊C4), SmallGroup(448,503)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C14.Q16
G = < a,b,c,d | a2=b14=c8=1, d2=b7c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b7c-1 >
Subgroups: 644 in 162 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C28, C2×C14, C2×C14, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C2×C28, C22×C14, C2×Q8⋊C4, C2×C7⋊C8, C2×C7⋊C8, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C22×C28, C14.Q16, C22×C7⋊C8, C14×C4⋊C4, C22×Dic14, C2×C14.Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, SD16, Q16, C22×C4, C2×D4, D14, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C4×D7, D28, C7⋊D4, C22×D7, C2×Q8⋊C4, D14⋊C4, D4.D7, C7⋊Q16, C2×C4×D7, C2×D28, C2×C7⋊D4, C14.Q16, C2×D14⋊C4, C2×D4.D7, C2×C7⋊Q16, C2×C14.Q16
►Regular action on 448 points
Generators in S448
(1 140)(2 127)(3 128)(4 129)(5 130)(6 131)(7 132)(8 133)(9 134)(10 135)(11 136)(12 137)(13 138)(14 139)(15 225)(16 226)(17 227)(18 228)(19 229)(20 230)(21 231)(22 232)(23 233)(24 234)(25 235)(26 236)(27 237)(28 238)(29 348)(30 349)(31 350)(32 337)(33 338)(34 339)(35 340)(36 341)(37 342)(38 343)(39 344)(40 345)(41 346)(42 347)(43 307)(44 308)(45 295)(46 296)(47 297)(48 298)(49 299)(50 300)(51 301)(52 302)(53 303)(54 304)(55 305)(56 306)(57 351)(58 352)(59 353)(60 354)(61 355)(62 356)(63 357)(64 358)(65 359)(66 360)(67 361)(68 362)(69 363)(70 364)(71 409)(72 410)(73 411)(74 412)(75 413)(76 414)(77 415)(78 416)(79 417)(80 418)(81 419)(82 420)(83 407)(84 408)(85 196)(86 183)(87 184)(88 185)(89 186)(90 187)(91 188)(92 189)(93 190)(94 191)(95 192)(96 193)(97 194)(98 195)(99 224)(100 211)(101 212)(102 213)(103 214)(104 215)(105 216)(106 217)(107 218)(108 219)(109 220)(110 221)(111 222)(112 223)(113 403)(114 404)(115 405)(116 406)(117 393)(118 394)(119 395)(120 396)(121 397)(122 398)(123 399)(124 400)(125 401)(126 402)(141 375)(142 376)(143 377)(144 378)(145 365)(146 366)(147 367)(148 368)(149 369)(150 370)(151 371)(152 372)(153 373)(154 374)(155 170)(156 171)(157 172)(158 173)(159 174)(160 175)(161 176)(162 177)(163 178)(164 179)(165 180)(166 181)(167 182)(168 169)(197 275)(198 276)(199 277)(200 278)(201 279)(202 280)(203 267)(204 268)(205 269)(206 270)(207 271)(208 272)(209 273)(210 274)(239 426)(240 427)(241 428)(242 429)(243 430)(244 431)(245 432)(246 433)(247 434)(248 421)(249 422)(250 423)(251 424)(252 425)(253 317)(254 318)(255 319)(256 320)(257 321)(258 322)(259 309)(260 310)(261 311)(262 312)(263 313)(264 314)(265 315)(266 316)(281 437)(282 438)(283 439)(284 440)(285 441)(286 442)(287 443)(288 444)(289 445)(290 446)(291 447)(292 448)(293 435)(294 436)(323 383)(324 384)(325 385)(326 386)(327 387)(328 388)(329 389)(330 390)(331 391)(332 392)(333 379)(334 380)(335 381)(336 382)
(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 56 84 311 421 65 92 99)(2 55 71 310 422 64 93 112)(3 54 72 309 423 63 94 111)(4 53 73 322 424 62 95 110)(5 52 74 321 425 61 96 109)(6 51 75 320 426 60 97 108)(7 50 76 319 427 59 98 107)(8 49 77 318 428 58 85 106)(9 48 78 317 429 57 86 105)(10 47 79 316 430 70 87 104)(11 46 80 315 431 69 88 103)(12 45 81 314 432 68 89 102)(13 44 82 313 433 67 90 101)(14 43 83 312 434 66 91 100)(15 341 146 385 285 276 164 406)(16 340 147 384 286 275 165 405)(17 339 148 383 287 274 166 404)(18 338 149 382 288 273 167 403)(19 337 150 381 289 272 168 402)(20 350 151 380 290 271 155 401)(21 349 152 379 291 270 156 400)(22 348 153 392 292 269 157 399)(23 347 154 391 293 268 158 398)(24 346 141 390 294 267 159 397)(25 345 142 389 281 280 160 396)(26 344 143 388 282 279 161 395)(27 343 144 387 283 278 162 394)(28 342 145 386 284 277 163 393)(29 373 332 448 205 172 123 232)(30 372 333 447 206 171 124 231)(31 371 334 446 207 170 125 230)(32 370 335 445 208 169 126 229)(33 369 336 444 209 182 113 228)(34 368 323 443 210 181 114 227)(35 367 324 442 197 180 115 226)(36 366 325 441 198 179 116 225)(37 365 326 440 199 178 117 238)(38 378 327 439 200 177 118 237)(39 377 328 438 201 176 119 236)(40 376 329 437 202 175 120 235)(41 375 330 436 203 174 121 234)(42 374 331 435 204 173 122 233)(127 305 409 260 249 358 190 223)(128 304 410 259 250 357 191 222)(129 303 411 258 251 356 192 221)(130 302 412 257 252 355 193 220)(131 301 413 256 239 354 194 219)(132 300 414 255 240 353 195 218)(133 299 415 254 241 352 196 217)(134 298 416 253 242 351 183 216)(135 297 417 266 243 364 184 215)(136 296 418 265 244 363 185 214)(137 295 419 264 245 362 186 213)(138 308 420 263 246 361 187 212)(139 307 407 262 247 360 188 211)(140 306 408 261 248 359 189 224)
(1 399 428 385)(2 400 429 386)(3 401 430 387)(4 402 431 388)(5 403 432 389)(6 404 433 390)(7 405 434 391)(8 406 421 392)(9 393 422 379)(10 394 423 380)(11 395 424 381)(12 396 425 382)(13 397 426 383)(14 398 427 384)(15 318 292 99)(16 319 293 100)(17 320 294 101)(18 321 281 102)(19 322 282 103)(20 309 283 104)(21 310 284 105)(22 311 285 106)(23 312 286 107)(24 313 287 108)(25 314 288 109)(26 315 289 110)(27 316 290 111)(28 317 291 112)(29 415 198 189)(30 416 199 190)(31 417 200 191)(32 418 201 192)(33 419 202 193)(34 420 203 194)(35 407 204 195)(36 408 205 196)(37 409 206 183)(38 410 207 184)(39 411 208 185)(40 412 209 186)(41 413 210 187)(42 414 197 188)(43 165 59 154)(44 166 60 141)(45 167 61 142)(46 168 62 143)(47 155 63 144)(48 156 64 145)(49 157 65 146)(50 158 66 147)(51 159 67 148)(52 160 68 149)(53 161 69 150)(54 162 70 151)(55 163 57 152)(56 164 58 153)(71 270 86 342)(72 271 87 343)(73 272 88 344)(74 273 89 345)(75 274 90 346)(76 275 91 347)(77 276 92 348)(78 277 93 349)(79 278 94 350)(80 279 95 337)(81 280 96 338)(82 267 97 339)(83 268 98 340)(84 269 85 341)(113 245 329 130)(114 246 330 131)(115 247 331 132)(116 248 332 133)(117 249 333 134)(118 250 334 135)(119 251 335 136)(120 252 336 137)(121 239 323 138)(122 240 324 139)(123 241 325 140)(124 242 326 127)(125 243 327 128)(126 244 328 129)(169 356 377 296)(170 357 378 297)(171 358 365 298)(172 359 366 299)(173 360 367 300)(174 361 368 301)(175 362 369 302)(176 363 370 303)(177 364 371 304)(178 351 372 305)(179 352 373 306)(180 353 374 307)(181 354 375 308)(182 355 376 295)(211 226 255 435)(212 227 256 436)(213 228 257 437)(214 229 258 438)(215 230 259 439)(216 231 260 440)(217 232 261 441)(218 233 262 442)(219 234 263 443)(220 235 264 444)(221 236 265 445)(222 237 266 446)(223 238 253 447)(224 225 254 448)
G:=sub<Sym(448)| (1,140)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,133)(9,134)(10,135)(11,136)(12,137)(13,138)(14,139)(15,225)(16,226)(17,227)(18,228)(19,229)(20,230)(21,231)(22,232)(23,233)(24,234)(25,235)(26,236)(27,237)(28,238)(29,348)(30,349)(31,350)(32,337)(33,338)(34,339)(35,340)(36,341)(37,342)(38,343)(39,344)(40,345)(41,346)(42,347)(43,307)(44,308)(45,295)(46,296)(47,297)(48,298)(49,299)(50,300)(51,301)(52,302)(53,303)(54,304)(55,305)(56,306)(57,351)(58,352)(59,353)(60,354)(61,355)(62,356)(63,357)(64,358)(65,359)(66,360)(67,361)(68,362)(69,363)(70,364)(71,409)(72,410)(73,411)(74,412)(75,413)(76,414)(77,415)(78,416)(79,417)(80,418)(81,419)(82,420)(83,407)(84,408)(85,196)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,224)(100,211)(101,212)(102,213)(103,214)(104,215)(105,216)(106,217)(107,218)(108,219)(109,220)(110,221)(111,222)(112,223)(113,403)(114,404)(115,405)(116,406)(117,393)(118,394)(119,395)(120,396)(121,397)(122,398)(123,399)(124,400)(125,401)(126,402)(141,375)(142,376)(143,377)(144,378)(145,365)(146,366)(147,367)(148,368)(149,369)(150,370)(151,371)(152,372)(153,373)(154,374)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(166,181)(167,182)(168,169)(197,275)(198,276)(199,277)(200,278)(201,279)(202,280)(203,267)(204,268)(205,269)(206,270)(207,271)(208,272)(209,273)(210,274)(239,426)(240,427)(241,428)(242,429)(243,430)(244,431)(245,432)(246,433)(247,434)(248,421)(249,422)(250,423)(251,424)(252,425)(253,317)(254,318)(255,319)(256,320)(257,321)(258,322)(259,309)(260,310)(261,311)(262,312)(263,313)(264,314)(265,315)(266,316)(281,437)(282,438)(283,439)(284,440)(285,441)(286,442)(287,443)(288,444)(289,445)(290,446)(291,447)(292,448)(293,435)(294,436)(323,383)(324,384)(325,385)(326,386)(327,387)(328,388)(329,389)(330,390)(331,391)(332,392)(333,379)(334,380)(335,381)(336,382), (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), (1,56,84,311,421,65,92,99)(2,55,71,310,422,64,93,112)(3,54,72,309,423,63,94,111)(4,53,73,322,424,62,95,110)(5,52,74,321,425,61,96,109)(6,51,75,320,426,60,97,108)(7,50,76,319,427,59,98,107)(8,49,77,318,428,58,85,106)(9,48,78,317,429,57,86,105)(10,47,79,316,430,70,87,104)(11,46,80,315,431,69,88,103)(12,45,81,314,432,68,89,102)(13,44,82,313,433,67,90,101)(14,43,83,312,434,66,91,100)(15,341,146,385,285,276,164,406)(16,340,147,384,286,275,165,405)(17,339,148,383,287,274,166,404)(18,338,149,382,288,273,167,403)(19,337,150,381,289,272,168,402)(20,350,151,380,290,271,155,401)(21,349,152,379,291,270,156,400)(22,348,153,392,292,269,157,399)(23,347,154,391,293,268,158,398)(24,346,141,390,294,267,159,397)(25,345,142,389,281,280,160,396)(26,344,143,388,282,279,161,395)(27,343,144,387,283,278,162,394)(28,342,145,386,284,277,163,393)(29,373,332,448,205,172,123,232)(30,372,333,447,206,171,124,231)(31,371,334,446,207,170,125,230)(32,370,335,445,208,169,126,229)(33,369,336,444,209,182,113,228)(34,368,323,443,210,181,114,227)(35,367,324,442,197,180,115,226)(36,366,325,441,198,179,116,225)(37,365,326,440,199,178,117,238)(38,378,327,439,200,177,118,237)(39,377,328,438,201,176,119,236)(40,376,329,437,202,175,120,235)(41,375,330,436,203,174,121,234)(42,374,331,435,204,173,122,233)(127,305,409,260,249,358,190,223)(128,304,410,259,250,357,191,222)(129,303,411,258,251,356,192,221)(130,302,412,257,252,355,193,220)(131,301,413,256,239,354,194,219)(132,300,414,255,240,353,195,218)(133,299,415,254,241,352,196,217)(134,298,416,253,242,351,183,216)(135,297,417,266,243,364,184,215)(136,296,418,265,244,363,185,214)(137,295,419,264,245,362,186,213)(138,308,420,263,246,361,187,212)(139,307,407,262,247,360,188,211)(140,306,408,261,248,359,189,224), (1,399,428,385)(2,400,429,386)(3,401,430,387)(4,402,431,388)(5,403,432,389)(6,404,433,390)(7,405,434,391)(8,406,421,392)(9,393,422,379)(10,394,423,380)(11,395,424,381)(12,396,425,382)(13,397,426,383)(14,398,427,384)(15,318,292,99)(16,319,293,100)(17,320,294,101)(18,321,281,102)(19,322,282,103)(20,309,283,104)(21,310,284,105)(22,311,285,106)(23,312,286,107)(24,313,287,108)(25,314,288,109)(26,315,289,110)(27,316,290,111)(28,317,291,112)(29,415,198,189)(30,416,199,190)(31,417,200,191)(32,418,201,192)(33,419,202,193)(34,420,203,194)(35,407,204,195)(36,408,205,196)(37,409,206,183)(38,410,207,184)(39,411,208,185)(40,412,209,186)(41,413,210,187)(42,414,197,188)(43,165,59,154)(44,166,60,141)(45,167,61,142)(46,168,62,143)(47,155,63,144)(48,156,64,145)(49,157,65,146)(50,158,66,147)(51,159,67,148)(52,160,68,149)(53,161,69,150)(54,162,70,151)(55,163,57,152)(56,164,58,153)(71,270,86,342)(72,271,87,343)(73,272,88,344)(74,273,89,345)(75,274,90,346)(76,275,91,347)(77,276,92,348)(78,277,93,349)(79,278,94,350)(80,279,95,337)(81,280,96,338)(82,267,97,339)(83,268,98,340)(84,269,85,341)(113,245,329,130)(114,246,330,131)(115,247,331,132)(116,248,332,133)(117,249,333,134)(118,250,334,135)(119,251,335,136)(120,252,336,137)(121,239,323,138)(122,240,324,139)(123,241,325,140)(124,242,326,127)(125,243,327,128)(126,244,328,129)(169,356,377,296)(170,357,378,297)(171,358,365,298)(172,359,366,299)(173,360,367,300)(174,361,368,301)(175,362,369,302)(176,363,370,303)(177,364,371,304)(178,351,372,305)(179,352,373,306)(180,353,374,307)(181,354,375,308)(182,355,376,295)(211,226,255,435)(212,227,256,436)(213,228,257,437)(214,229,258,438)(215,230,259,439)(216,231,260,440)(217,232,261,441)(218,233,262,442)(219,234,263,443)(220,235,264,444)(221,236,265,445)(222,237,266,446)(223,238,253,447)(224,225,254,448)>;
G:=Group( (1,140)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,133)(9,134)(10,135)(11,136)(12,137)(13,138)(14,139)(15,225)(16,226)(17,227)(18,228)(19,229)(20,230)(21,231)(22,232)(23,233)(24,234)(25,235)(26,236)(27,237)(28,238)(29,348)(30,349)(31,350)(32,337)(33,338)(34,339)(35,340)(36,341)(37,342)(38,343)(39,344)(40,345)(41,346)(42,347)(43,307)(44,308)(45,295)(46,296)(47,297)(48,298)(49,299)(50,300)(51,301)(52,302)(53,303)(54,304)(55,305)(56,306)(57,351)(58,352)(59,353)(60,354)(61,355)(62,356)(63,357)(64,358)(65,359)(66,360)(67,361)(68,362)(69,363)(70,364)(71,409)(72,410)(73,411)(74,412)(75,413)(76,414)(77,415)(78,416)(79,417)(80,418)(81,419)(82,420)(83,407)(84,408)(85,196)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,224)(100,211)(101,212)(102,213)(103,214)(104,215)(105,216)(106,217)(107,218)(108,219)(109,220)(110,221)(111,222)(112,223)(113,403)(114,404)(115,405)(116,406)(117,393)(118,394)(119,395)(120,396)(121,397)(122,398)(123,399)(124,400)(125,401)(126,402)(141,375)(142,376)(143,377)(144,378)(145,365)(146,366)(147,367)(148,368)(149,369)(150,370)(151,371)(152,372)(153,373)(154,374)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(166,181)(167,182)(168,169)(197,275)(198,276)(199,277)(200,278)(201,279)(202,280)(203,267)(204,268)(205,269)(206,270)(207,271)(208,272)(209,273)(210,274)(239,426)(240,427)(241,428)(242,429)(243,430)(244,431)(245,432)(246,433)(247,434)(248,421)(249,422)(250,423)(251,424)(252,425)(253,317)(254,318)(255,319)(256,320)(257,321)(258,322)(259,309)(260,310)(261,311)(262,312)(263,313)(264,314)(265,315)(266,316)(281,437)(282,438)(283,439)(284,440)(285,441)(286,442)(287,443)(288,444)(289,445)(290,446)(291,447)(292,448)(293,435)(294,436)(323,383)(324,384)(325,385)(326,386)(327,387)(328,388)(329,389)(330,390)(331,391)(332,392)(333,379)(334,380)(335,381)(336,382), (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), (1,56,84,311,421,65,92,99)(2,55,71,310,422,64,93,112)(3,54,72,309,423,63,94,111)(4,53,73,322,424,62,95,110)(5,52,74,321,425,61,96,109)(6,51,75,320,426,60,97,108)(7,50,76,319,427,59,98,107)(8,49,77,318,428,58,85,106)(9,48,78,317,429,57,86,105)(10,47,79,316,430,70,87,104)(11,46,80,315,431,69,88,103)(12,45,81,314,432,68,89,102)(13,44,82,313,433,67,90,101)(14,43,83,312,434,66,91,100)(15,341,146,385,285,276,164,406)(16,340,147,384,286,275,165,405)(17,339,148,383,287,274,166,404)(18,338,149,382,288,273,167,403)(19,337,150,381,289,272,168,402)(20,350,151,380,290,271,155,401)(21,349,152,379,291,270,156,400)(22,348,153,392,292,269,157,399)(23,347,154,391,293,268,158,398)(24,346,141,390,294,267,159,397)(25,345,142,389,281,280,160,396)(26,344,143,388,282,279,161,395)(27,343,144,387,283,278,162,394)(28,342,145,386,284,277,163,393)(29,373,332,448,205,172,123,232)(30,372,333,447,206,171,124,231)(31,371,334,446,207,170,125,230)(32,370,335,445,208,169,126,229)(33,369,336,444,209,182,113,228)(34,368,323,443,210,181,114,227)(35,367,324,442,197,180,115,226)(36,366,325,441,198,179,116,225)(37,365,326,440,199,178,117,238)(38,378,327,439,200,177,118,237)(39,377,328,438,201,176,119,236)(40,376,329,437,202,175,120,235)(41,375,330,436,203,174,121,234)(42,374,331,435,204,173,122,233)(127,305,409,260,249,358,190,223)(128,304,410,259,250,357,191,222)(129,303,411,258,251,356,192,221)(130,302,412,257,252,355,193,220)(131,301,413,256,239,354,194,219)(132,300,414,255,240,353,195,218)(133,299,415,254,241,352,196,217)(134,298,416,253,242,351,183,216)(135,297,417,266,243,364,184,215)(136,296,418,265,244,363,185,214)(137,295,419,264,245,362,186,213)(138,308,420,263,246,361,187,212)(139,307,407,262,247,360,188,211)(140,306,408,261,248,359,189,224), (1,399,428,385)(2,400,429,386)(3,401,430,387)(4,402,431,388)(5,403,432,389)(6,404,433,390)(7,405,434,391)(8,406,421,392)(9,393,422,379)(10,394,423,380)(11,395,424,381)(12,396,425,382)(13,397,426,383)(14,398,427,384)(15,318,292,99)(16,319,293,100)(17,320,294,101)(18,321,281,102)(19,322,282,103)(20,309,283,104)(21,310,284,105)(22,311,285,106)(23,312,286,107)(24,313,287,108)(25,314,288,109)(26,315,289,110)(27,316,290,111)(28,317,291,112)(29,415,198,189)(30,416,199,190)(31,417,200,191)(32,418,201,192)(33,419,202,193)(34,420,203,194)(35,407,204,195)(36,408,205,196)(37,409,206,183)(38,410,207,184)(39,411,208,185)(40,412,209,186)(41,413,210,187)(42,414,197,188)(43,165,59,154)(44,166,60,141)(45,167,61,142)(46,168,62,143)(47,155,63,144)(48,156,64,145)(49,157,65,146)(50,158,66,147)(51,159,67,148)(52,160,68,149)(53,161,69,150)(54,162,70,151)(55,163,57,152)(56,164,58,153)(71,270,86,342)(72,271,87,343)(73,272,88,344)(74,273,89,345)(75,274,90,346)(76,275,91,347)(77,276,92,348)(78,277,93,349)(79,278,94,350)(80,279,95,337)(81,280,96,338)(82,267,97,339)(83,268,98,340)(84,269,85,341)(113,245,329,130)(114,246,330,131)(115,247,331,132)(116,248,332,133)(117,249,333,134)(118,250,334,135)(119,251,335,136)(120,252,336,137)(121,239,323,138)(122,240,324,139)(123,241,325,140)(124,242,326,127)(125,243,327,128)(126,244,328,129)(169,356,377,296)(170,357,378,297)(171,358,365,298)(172,359,366,299)(173,360,367,300)(174,361,368,301)(175,362,369,302)(176,363,370,303)(177,364,371,304)(178,351,372,305)(179,352,373,306)(180,353,374,307)(181,354,375,308)(182,355,376,295)(211,226,255,435)(212,227,256,436)(213,228,257,437)(214,229,258,438)(215,230,259,439)(216,231,260,440)(217,232,261,441)(218,233,262,442)(219,234,263,443)(220,235,264,444)(221,236,265,445)(222,237,266,446)(223,238,253,447)(224,225,254,448) );
G=PermutationGroup([[(1,140),(2,127),(3,128),(4,129),(5,130),(6,131),(7,132),(8,133),(9,134),(10,135),(11,136),(12,137),(13,138),(14,139),(15,225),(16,226),(17,227),(18,228),(19,229),(20,230),(21,231),(22,232),(23,233),(24,234),(25,235),(26,236),(27,237),(28,238),(29,348),(30,349),(31,350),(32,337),(33,338),(34,339),(35,340),(36,341),(37,342),(38,343),(39,344),(40,345),(41,346),(42,347),(43,307),(44,308),(45,295),(46,296),(47,297),(48,298),(49,299),(50,300),(51,301),(52,302),(53,303),(54,304),(55,305),(56,306),(57,351),(58,352),(59,353),(60,354),(61,355),(62,356),(63,357),(64,358),(65,359),(66,360),(67,361),(68,362),(69,363),(70,364),(71,409),(72,410),(73,411),(74,412),(75,413),(76,414),(77,415),(78,416),(79,417),(80,418),(81,419),(82,420),(83,407),(84,408),(85,196),(86,183),(87,184),(88,185),(89,186),(90,187),(91,188),(92,189),(93,190),(94,191),(95,192),(96,193),(97,194),(98,195),(99,224),(100,211),(101,212),(102,213),(103,214),(104,215),(105,216),(106,217),(107,218),(108,219),(109,220),(110,221),(111,222),(112,223),(113,403),(114,404),(115,405),(116,406),(117,393),(118,394),(119,395),(120,396),(121,397),(122,398),(123,399),(124,400),(125,401),(126,402),(141,375),(142,376),(143,377),(144,378),(145,365),(146,366),(147,367),(148,368),(149,369),(150,370),(151,371),(152,372),(153,373),(154,374),(155,170),(156,171),(157,172),(158,173),(159,174),(160,175),(161,176),(162,177),(163,178),(164,179),(165,180),(166,181),(167,182),(168,169),(197,275),(198,276),(199,277),(200,278),(201,279),(202,280),(203,267),(204,268),(205,269),(206,270),(207,271),(208,272),(209,273),(210,274),(239,426),(240,427),(241,428),(242,429),(243,430),(244,431),(245,432),(246,433),(247,434),(248,421),(249,422),(250,423),(251,424),(252,425),(253,317),(254,318),(255,319),(256,320),(257,321),(258,322),(259,309),(260,310),(261,311),(262,312),(263,313),(264,314),(265,315),(266,316),(281,437),(282,438),(283,439),(284,440),(285,441),(286,442),(287,443),(288,444),(289,445),(290,446),(291,447),(292,448),(293,435),(294,436),(323,383),(324,384),(325,385),(326,386),(327,387),(328,388),(329,389),(330,390),(331,391),(332,392),(333,379),(334,380),(335,381),(336,382)], [(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)], [(1,56,84,311,421,65,92,99),(2,55,71,310,422,64,93,112),(3,54,72,309,423,63,94,111),(4,53,73,322,424,62,95,110),(5,52,74,321,425,61,96,109),(6,51,75,320,426,60,97,108),(7,50,76,319,427,59,98,107),(8,49,77,318,428,58,85,106),(9,48,78,317,429,57,86,105),(10,47,79,316,430,70,87,104),(11,46,80,315,431,69,88,103),(12,45,81,314,432,68,89,102),(13,44,82,313,433,67,90,101),(14,43,83,312,434,66,91,100),(15,341,146,385,285,276,164,406),(16,340,147,384,286,275,165,405),(17,339,148,383,287,274,166,404),(18,338,149,382,288,273,167,403),(19,337,150,381,289,272,168,402),(20,350,151,380,290,271,155,401),(21,349,152,379,291,270,156,400),(22,348,153,392,292,269,157,399),(23,347,154,391,293,268,158,398),(24,346,141,390,294,267,159,397),(25,345,142,389,281,280,160,396),(26,344,143,388,282,279,161,395),(27,343,144,387,283,278,162,394),(28,342,145,386,284,277,163,393),(29,373,332,448,205,172,123,232),(30,372,333,447,206,171,124,231),(31,371,334,446,207,170,125,230),(32,370,335,445,208,169,126,229),(33,369,336,444,209,182,113,228),(34,368,323,443,210,181,114,227),(35,367,324,442,197,180,115,226),(36,366,325,441,198,179,116,225),(37,365,326,440,199,178,117,238),(38,378,327,439,200,177,118,237),(39,377,328,438,201,176,119,236),(40,376,329,437,202,175,120,235),(41,375,330,436,203,174,121,234),(42,374,331,435,204,173,122,233),(127,305,409,260,249,358,190,223),(128,304,410,259,250,357,191,222),(129,303,411,258,251,356,192,221),(130,302,412,257,252,355,193,220),(131,301,413,256,239,354,194,219),(132,300,414,255,240,353,195,218),(133,299,415,254,241,352,196,217),(134,298,416,253,242,351,183,216),(135,297,417,266,243,364,184,215),(136,296,418,265,244,363,185,214),(137,295,419,264,245,362,186,213),(138,308,420,263,246,361,187,212),(139,307,407,262,247,360,188,211),(140,306,408,261,248,359,189,224)], [(1,399,428,385),(2,400,429,386),(3,401,430,387),(4,402,431,388),(5,403,432,389),(6,404,433,390),(7,405,434,391),(8,406,421,392),(9,393,422,379),(10,394,423,380),(11,395,424,381),(12,396,425,382),(13,397,426,383),(14,398,427,384),(15,318,292,99),(16,319,293,100),(17,320,294,101),(18,321,281,102),(19,322,282,103),(20,309,283,104),(21,310,284,105),(22,311,285,106),(23,312,286,107),(24,313,287,108),(25,314,288,109),(26,315,289,110),(27,316,290,111),(28,317,291,112),(29,415,198,189),(30,416,199,190),(31,417,200,191),(32,418,201,192),(33,419,202,193),(34,420,203,194),(35,407,204,195),(36,408,205,196),(37,409,206,183),(38,410,207,184),(39,411,208,185),(40,412,209,186),(41,413,210,187),(42,414,197,188),(43,165,59,154),(44,166,60,141),(45,167,61,142),(46,168,62,143),(47,155,63,144),(48,156,64,145),(49,157,65,146),(50,158,66,147),(51,159,67,148),(52,160,68,149),(53,161,69,150),(54,162,70,151),(55,163,57,152),(56,164,58,153),(71,270,86,342),(72,271,87,343),(73,272,88,344),(74,273,89,345),(75,274,90,346),(76,275,91,347),(77,276,92,348),(78,277,93,349),(79,278,94,350),(80,279,95,337),(81,280,96,338),(82,267,97,339),(83,268,98,340),(84,269,85,341),(113,245,329,130),(114,246,330,131),(115,247,331,132),(116,248,332,133),(117,249,333,134),(118,250,334,135),(119,251,335,136),(120,252,336,137),(121,239,323,138),(122,240,324,139),(123,241,325,140),(124,242,326,127),(125,243,327,128),(126,244,328,129),(169,356,377,296),(170,357,378,297),(171,358,365,298),(172,359,366,299),(173,360,367,300),(174,361,368,301),(175,362,369,302),(176,363,370,303),(177,364,371,304),(178,351,372,305),(179,352,373,306),(180,353,374,307),(181,354,375,308),(182,355,376,295),(211,226,255,435),(212,227,256,436),(213,228,257,437),(214,229,258,438),(215,230,259,439),(216,231,260,440),(217,232,261,441),(218,233,262,442),(219,234,263,443),(220,235,264,444),(221,236,265,445),(222,237,266,446),(223,238,253,447),(224,225,254,448)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | SD16 | Q16 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | C7⋊D4 | D4.D7 | C7⋊Q16 |
| kernel | C2×C14.Q16 | C14.Q16 | C22×C7⋊C8 | C14×C4⋊C4 | C22×Dic14 | C2×Dic14 | C2×C28 | C22×C14 | C2×C4⋊C4 | C2×C14 | C2×C14 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 4 | 4 | 6 | 3 | 12 | 12 | 6 | 6 | 6 | 6 |
Matrix representation of C2×C14.Q16 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 89 | 33 | 0 | 0 | 0 | 0 |
| 89 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 25 | 0 | 0 |
| 0 | 0 | 88 | 88 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 102 | 40 | 0 | 0 | 0 | 0 |
| 93 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 82 | 0 | 0 |
| 0 | 0 | 12 | 91 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 12 | 0 | 0 |
| 0 | 0 | 101 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 101 |
| 0 | 0 | 0 | 0 | 101 | 99 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[89,89,0,0,0,0,33,0,0,0,0,0,0,0,34,88,0,0,0,0,25,88,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[102,93,0,0,0,0,40,11,0,0,0,0,0,0,22,12,0,0,0,0,82,91,0,0,0,0,0,0,13,13,0,0,0,0,100,13],[15,0,0,0,0,0,0,15,0,0,0,0,0,0,91,101,0,0,0,0,12,22,0,0,0,0,0,0,14,101,0,0,0,0,101,99] >;
C2×C14.Q16 in GAP, Magma, Sage, TeX
C_2\times C_{14}.Q_{16} % in TeX
G:=Group("C2xC14.Q16"); // GroupNames label
G:=SmallGroup(448,503);
// by ID
G=gap.SmallGroup(448,503);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,422,58,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^8=1,d^2=b^7*c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^7*c^-1>;
// generators/relations