direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.Q16, C28.29Q16, Q8⋊2(C7×Q8), (C7×Q8)⋊9Q8, C4⋊C8.7C14, C4⋊Q8.6C14, C4.7(C7×Q16), (C4×Q8).7C14, C4.15(Q8×C14), C2.7(C14×Q16), C2.D8.4C14, (C2×C28).333D4, (Q8×C28).20C2, C28.121(C2×Q8), C14.54(C2×Q16), C42.23(C2×C14), Q8⋊C4.3C14, C22.98(D4×C14), C28.314(C4○D4), (C4×C28).265C22, (C2×C28).933C23, (C2×C56).260C22, C14.96(C22⋊Q8), C14.140(C8⋊C22), (Q8×C14).264C22, (C7×C4⋊C8).20C2, (C2×C8).7(C2×C14), C4.26(C7×C4○D4), (C7×C4⋊Q8).21C2, C4⋊C4.14(C2×C14), (C2×C4).134(C7×D4), C2.15(C7×C8⋊C22), (C7×C2.D8).13C2, C2.15(C7×C22⋊Q8), (C2×C14).654(C2×D4), (C2×Q8).51(C2×C14), (C7×C4⋊C4).236C22, (C7×Q8⋊C4).12C2, (C2×C4).108(C22×C14), SmallGroup(448,885)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.Q16
G = < a,b,c,d | a7=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >
Subgroups: 154 in 96 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C56, C2×C28, C2×C28, C7×Q8, C7×Q8, C4.Q16, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, Q8×C14, Q8×C14, C7×Q8⋊C4, C7×C4⋊C8, C7×C2.D8, Q8×C28, C7×C4⋊Q8, C7×C4.Q16
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, Q16, C2×D4, C2×Q8, C4○D4, C2×C14, C22⋊Q8, C2×Q16, C8⋊C22, C7×D4, C7×Q8, C22×C14, C4.Q16, C7×Q16, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊Q8, C14×Q16, C7×C8⋊C22, C7×C4.Q16
►Regular action on 448 points
Generators in S448
(1 115 387 218 379 210 371)(2 116 388 219 380 211 372)(3 117 389 220 381 212 373)(4 118 390 221 382 213 374)(5 119 391 222 383 214 375)(6 120 392 223 384 215 376)(7 113 385 224 377 216 369)(8 114 386 217 378 209 370)(9 170 331 51 323 43 315)(10 171 332 52 324 44 316)(11 172 333 53 325 45 317)(12 173 334 54 326 46 318)(13 174 335 55 327 47 319)(14 175 336 56 328 48 320)(15 176 329 49 321 41 313)(16 169 330 50 322 42 314)(17 289 206 280 305 25 297)(18 290 207 273 306 26 298)(19 291 208 274 307 27 299)(20 292 201 275 308 28 300)(21 293 202 276 309 29 301)(22 294 203 277 310 30 302)(23 295 204 278 311 31 303)(24 296 205 279 312 32 304)(33 62 243 74 235 66 227)(34 63 244 75 236 67 228)(35 64 245 76 237 68 229)(36 57 246 77 238 69 230)(37 58 247 78 239 70 231)(38 59 248 79 240 71 232)(39 60 241 80 233 72 225)(40 61 242 73 234 65 226)(81 108 269 100 261 92 253)(82 109 270 101 262 93 254)(83 110 271 102 263 94 255)(84 111 272 103 264 95 256)(85 112 265 104 257 96 249)(86 105 266 97 258 89 250)(87 106 267 98 259 90 251)(88 107 268 99 260 91 252)(121 393 417 137 409 129 401)(122 394 418 138 410 130 402)(123 395 419 139 411 131 403)(124 396 420 140 412 132 404)(125 397 421 141 413 133 405)(126 398 422 142 414 134 406)(127 399 423 143 415 135 407)(128 400 424 144 416 136 408)(145 288 441 161 433 153 425)(146 281 442 162 434 154 426)(147 282 443 163 435 155 427)(148 283 444 164 436 156 428)(149 284 445 165 437 157 429)(150 285 446 166 438 158 430)(151 286 447 167 439 159 431)(152 287 448 168 440 160 432)(177 338 362 193 354 185 346)(178 339 363 194 355 186 347)(179 340 364 195 356 187 348)(180 341 365 196 357 188 349)(181 342 366 197 358 189 350)(182 343 367 198 359 190 351)(183 344 368 199 360 191 352)(184 337 361 200 353 192 345)
(1 289 229 151)(2 152 230 290)(3 291 231 145)(4 146 232 292)(5 293 225 147)(6 148 226 294)(7 295 227 149)(8 150 228 296)(9 83 419 367)(10 368 420 84)(11 85 421 361)(12 362 422 86)(13 87 423 363)(14 364 424 88)(15 81 417 365)(16 366 418 82)(17 68 431 371)(18 372 432 69)(19 70 425 373)(20 374 426 71)(21 72 427 375)(22 376 428 65)(23 66 429 369)(24 370 430 67)(25 76 439 379)(26 380 440 77)(27 78 433 381)(28 382 434 79)(29 80 435 383)(30 384 436 73)(31 74 437 377)(32 378 438 75)(33 284 113 204)(34 205 114 285)(35 286 115 206)(36 207 116 287)(37 288 117 208)(38 201 118 281)(39 282 119 202)(40 203 120 283)(41 92 121 180)(42 181 122 93)(43 94 123 182)(44 183 124 95)(45 96 125 184)(46 177 126 89)(47 90 127 178)(48 179 128 91)(49 100 129 188)(50 189 130 101)(51 102 131 190)(52 191 132 103)(53 104 133 192)(54 185 134 97)(55 98 135 186)(56 187 136 99)(57 273 388 448)(58 441 389 274)(59 275 390 442)(60 443 391 276)(61 277 392 444)(62 445 385 278)(63 279 386 446)(64 447 387 280)(105 173 193 142)(106 143 194 174)(107 175 195 144)(108 137 196 176)(109 169 197 138)(110 139 198 170)(111 171 199 140)(112 141 200 172)(153 212 299 239)(154 240 300 213)(155 214 301 233)(156 234 302 215)(157 216 303 235)(158 236 304 209)(159 210 297 237)(160 238 298 211)(161 220 307 247)(162 248 308 221)(163 222 309 241)(164 242 310 223)(165 224 311 243)(166 244 312 217)(167 218 305 245)(168 246 306 219)(249 397 337 317)(250 318 338 398)(251 399 339 319)(252 320 340 400)(253 393 341 313)(254 314 342 394)(255 395 343 315)(256 316 344 396)(257 405 345 325)(258 326 346 406)(259 407 347 327)(260 328 348 408)(261 401 349 321)(262 322 350 402)(263 403 351 323)(264 324 352 404)(265 413 353 333)(266 334 354 414)(267 415 355 335)(268 336 356 416)(269 409 357 329)(270 330 358 410)(271 411 359 331)(272 332 360 412)
(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)
(1 343 5 339)(2 254 6 250)(3 341 7 337)(4 252 8 256)(9 202 13 206)(10 281 14 285)(11 208 15 204)(12 287 16 283)(17 43 21 47)(18 122 22 126)(19 41 23 45)(20 128 24 124)(25 51 29 55)(26 130 30 134)(27 49 31 53)(28 136 32 132)(33 85 37 81)(34 368 38 364)(35 83 39 87)(36 366 40 362)(42 428 46 432)(44 426 48 430)(50 436 54 440)(52 434 56 438)(57 197 61 193)(58 108 62 112)(59 195 63 199)(60 106 64 110)(65 177 69 181)(66 96 70 92)(67 183 71 179)(68 94 72 90)(73 185 77 189)(74 104 78 100)(75 191 79 187)(76 102 80 98)(82 120 86 116)(84 118 88 114)(89 372 93 376)(91 370 95 374)(97 380 101 384)(99 378 103 382)(105 388 109 392)(107 386 111 390)(113 361 117 365)(115 367 119 363)(121 429 125 425)(123 427 127 431)(129 437 133 433)(131 435 135 439)(137 445 141 441)(138 277 142 273)(139 443 143 447)(140 275 144 279)(145 393 149 397)(146 320 150 316)(147 399 151 395)(148 318 152 314)(153 401 157 405)(154 328 158 324)(155 407 159 403)(156 326 160 322)(161 409 165 413)(162 336 166 332)(163 415 167 411)(164 334 168 330)(169 444 173 448)(170 276 174 280)(171 442 175 446)(172 274 176 278)(178 371 182 375)(180 369 184 373)(186 379 190 383)(188 377 192 381)(194 387 198 391)(196 385 200 389)(201 424 205 420)(203 422 207 418)(209 264 213 260)(210 351 214 347)(211 262 215 258)(212 349 216 345)(217 272 221 268)(218 359 222 355)(219 270 223 266)(220 357 224 353)(225 251 229 255)(226 338 230 342)(227 249 231 253)(228 344 232 340)(233 259 237 263)(234 346 238 350)(235 257 239 261)(236 352 240 348)(241 267 245 271)(242 354 246 358)(243 265 247 269)(244 360 248 356)(282 423 286 419)(284 421 288 417)(289 315 293 319)(290 394 294 398)(291 313 295 317)(292 400 296 396)(297 323 301 327)(298 402 302 406)(299 321 303 325)(300 408 304 404)(305 331 309 335)(306 410 310 414)(307 329 311 333)(308 416 312 412)
G:=sub<Sym(448)| (1,115,387,218,379,210,371)(2,116,388,219,380,211,372)(3,117,389,220,381,212,373)(4,118,390,221,382,213,374)(5,119,391,222,383,214,375)(6,120,392,223,384,215,376)(7,113,385,224,377,216,369)(8,114,386,217,378,209,370)(9,170,331,51,323,43,315)(10,171,332,52,324,44,316)(11,172,333,53,325,45,317)(12,173,334,54,326,46,318)(13,174,335,55,327,47,319)(14,175,336,56,328,48,320)(15,176,329,49,321,41,313)(16,169,330,50,322,42,314)(17,289,206,280,305,25,297)(18,290,207,273,306,26,298)(19,291,208,274,307,27,299)(20,292,201,275,308,28,300)(21,293,202,276,309,29,301)(22,294,203,277,310,30,302)(23,295,204,278,311,31,303)(24,296,205,279,312,32,304)(33,62,243,74,235,66,227)(34,63,244,75,236,67,228)(35,64,245,76,237,68,229)(36,57,246,77,238,69,230)(37,58,247,78,239,70,231)(38,59,248,79,240,71,232)(39,60,241,80,233,72,225)(40,61,242,73,234,65,226)(81,108,269,100,261,92,253)(82,109,270,101,262,93,254)(83,110,271,102,263,94,255)(84,111,272,103,264,95,256)(85,112,265,104,257,96,249)(86,105,266,97,258,89,250)(87,106,267,98,259,90,251)(88,107,268,99,260,91,252)(121,393,417,137,409,129,401)(122,394,418,138,410,130,402)(123,395,419,139,411,131,403)(124,396,420,140,412,132,404)(125,397,421,141,413,133,405)(126,398,422,142,414,134,406)(127,399,423,143,415,135,407)(128,400,424,144,416,136,408)(145,288,441,161,433,153,425)(146,281,442,162,434,154,426)(147,282,443,163,435,155,427)(148,283,444,164,436,156,428)(149,284,445,165,437,157,429)(150,285,446,166,438,158,430)(151,286,447,167,439,159,431)(152,287,448,168,440,160,432)(177,338,362,193,354,185,346)(178,339,363,194,355,186,347)(179,340,364,195,356,187,348)(180,341,365,196,357,188,349)(181,342,366,197,358,189,350)(182,343,367,198,359,190,351)(183,344,368,199,360,191,352)(184,337,361,200,353,192,345), (1,289,229,151)(2,152,230,290)(3,291,231,145)(4,146,232,292)(5,293,225,147)(6,148,226,294)(7,295,227,149)(8,150,228,296)(9,83,419,367)(10,368,420,84)(11,85,421,361)(12,362,422,86)(13,87,423,363)(14,364,424,88)(15,81,417,365)(16,366,418,82)(17,68,431,371)(18,372,432,69)(19,70,425,373)(20,374,426,71)(21,72,427,375)(22,376,428,65)(23,66,429,369)(24,370,430,67)(25,76,439,379)(26,380,440,77)(27,78,433,381)(28,382,434,79)(29,80,435,383)(30,384,436,73)(31,74,437,377)(32,378,438,75)(33,284,113,204)(34,205,114,285)(35,286,115,206)(36,207,116,287)(37,288,117,208)(38,201,118,281)(39,282,119,202)(40,203,120,283)(41,92,121,180)(42,181,122,93)(43,94,123,182)(44,183,124,95)(45,96,125,184)(46,177,126,89)(47,90,127,178)(48,179,128,91)(49,100,129,188)(50,189,130,101)(51,102,131,190)(52,191,132,103)(53,104,133,192)(54,185,134,97)(55,98,135,186)(56,187,136,99)(57,273,388,448)(58,441,389,274)(59,275,390,442)(60,443,391,276)(61,277,392,444)(62,445,385,278)(63,279,386,446)(64,447,387,280)(105,173,193,142)(106,143,194,174)(107,175,195,144)(108,137,196,176)(109,169,197,138)(110,139,198,170)(111,171,199,140)(112,141,200,172)(153,212,299,239)(154,240,300,213)(155,214,301,233)(156,234,302,215)(157,216,303,235)(158,236,304,209)(159,210,297,237)(160,238,298,211)(161,220,307,247)(162,248,308,221)(163,222,309,241)(164,242,310,223)(165,224,311,243)(166,244,312,217)(167,218,305,245)(168,246,306,219)(249,397,337,317)(250,318,338,398)(251,399,339,319)(252,320,340,400)(253,393,341,313)(254,314,342,394)(255,395,343,315)(256,316,344,396)(257,405,345,325)(258,326,346,406)(259,407,347,327)(260,328,348,408)(261,401,349,321)(262,322,350,402)(263,403,351,323)(264,324,352,404)(265,413,353,333)(266,334,354,414)(267,415,355,335)(268,336,356,416)(269,409,357,329)(270,330,358,410)(271,411,359,331)(272,332,360,412), (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), (1,343,5,339)(2,254,6,250)(3,341,7,337)(4,252,8,256)(9,202,13,206)(10,281,14,285)(11,208,15,204)(12,287,16,283)(17,43,21,47)(18,122,22,126)(19,41,23,45)(20,128,24,124)(25,51,29,55)(26,130,30,134)(27,49,31,53)(28,136,32,132)(33,85,37,81)(34,368,38,364)(35,83,39,87)(36,366,40,362)(42,428,46,432)(44,426,48,430)(50,436,54,440)(52,434,56,438)(57,197,61,193)(58,108,62,112)(59,195,63,199)(60,106,64,110)(65,177,69,181)(66,96,70,92)(67,183,71,179)(68,94,72,90)(73,185,77,189)(74,104,78,100)(75,191,79,187)(76,102,80,98)(82,120,86,116)(84,118,88,114)(89,372,93,376)(91,370,95,374)(97,380,101,384)(99,378,103,382)(105,388,109,392)(107,386,111,390)(113,361,117,365)(115,367,119,363)(121,429,125,425)(123,427,127,431)(129,437,133,433)(131,435,135,439)(137,445,141,441)(138,277,142,273)(139,443,143,447)(140,275,144,279)(145,393,149,397)(146,320,150,316)(147,399,151,395)(148,318,152,314)(153,401,157,405)(154,328,158,324)(155,407,159,403)(156,326,160,322)(161,409,165,413)(162,336,166,332)(163,415,167,411)(164,334,168,330)(169,444,173,448)(170,276,174,280)(171,442,175,446)(172,274,176,278)(178,371,182,375)(180,369,184,373)(186,379,190,383)(188,377,192,381)(194,387,198,391)(196,385,200,389)(201,424,205,420)(203,422,207,418)(209,264,213,260)(210,351,214,347)(211,262,215,258)(212,349,216,345)(217,272,221,268)(218,359,222,355)(219,270,223,266)(220,357,224,353)(225,251,229,255)(226,338,230,342)(227,249,231,253)(228,344,232,340)(233,259,237,263)(234,346,238,350)(235,257,239,261)(236,352,240,348)(241,267,245,271)(242,354,246,358)(243,265,247,269)(244,360,248,356)(282,423,286,419)(284,421,288,417)(289,315,293,319)(290,394,294,398)(291,313,295,317)(292,400,296,396)(297,323,301,327)(298,402,302,406)(299,321,303,325)(300,408,304,404)(305,331,309,335)(306,410,310,414)(307,329,311,333)(308,416,312,412)>;
G:=Group( (1,115,387,218,379,210,371)(2,116,388,219,380,211,372)(3,117,389,220,381,212,373)(4,118,390,221,382,213,374)(5,119,391,222,383,214,375)(6,120,392,223,384,215,376)(7,113,385,224,377,216,369)(8,114,386,217,378,209,370)(9,170,331,51,323,43,315)(10,171,332,52,324,44,316)(11,172,333,53,325,45,317)(12,173,334,54,326,46,318)(13,174,335,55,327,47,319)(14,175,336,56,328,48,320)(15,176,329,49,321,41,313)(16,169,330,50,322,42,314)(17,289,206,280,305,25,297)(18,290,207,273,306,26,298)(19,291,208,274,307,27,299)(20,292,201,275,308,28,300)(21,293,202,276,309,29,301)(22,294,203,277,310,30,302)(23,295,204,278,311,31,303)(24,296,205,279,312,32,304)(33,62,243,74,235,66,227)(34,63,244,75,236,67,228)(35,64,245,76,237,68,229)(36,57,246,77,238,69,230)(37,58,247,78,239,70,231)(38,59,248,79,240,71,232)(39,60,241,80,233,72,225)(40,61,242,73,234,65,226)(81,108,269,100,261,92,253)(82,109,270,101,262,93,254)(83,110,271,102,263,94,255)(84,111,272,103,264,95,256)(85,112,265,104,257,96,249)(86,105,266,97,258,89,250)(87,106,267,98,259,90,251)(88,107,268,99,260,91,252)(121,393,417,137,409,129,401)(122,394,418,138,410,130,402)(123,395,419,139,411,131,403)(124,396,420,140,412,132,404)(125,397,421,141,413,133,405)(126,398,422,142,414,134,406)(127,399,423,143,415,135,407)(128,400,424,144,416,136,408)(145,288,441,161,433,153,425)(146,281,442,162,434,154,426)(147,282,443,163,435,155,427)(148,283,444,164,436,156,428)(149,284,445,165,437,157,429)(150,285,446,166,438,158,430)(151,286,447,167,439,159,431)(152,287,448,168,440,160,432)(177,338,362,193,354,185,346)(178,339,363,194,355,186,347)(179,340,364,195,356,187,348)(180,341,365,196,357,188,349)(181,342,366,197,358,189,350)(182,343,367,198,359,190,351)(183,344,368,199,360,191,352)(184,337,361,200,353,192,345), (1,289,229,151)(2,152,230,290)(3,291,231,145)(4,146,232,292)(5,293,225,147)(6,148,226,294)(7,295,227,149)(8,150,228,296)(9,83,419,367)(10,368,420,84)(11,85,421,361)(12,362,422,86)(13,87,423,363)(14,364,424,88)(15,81,417,365)(16,366,418,82)(17,68,431,371)(18,372,432,69)(19,70,425,373)(20,374,426,71)(21,72,427,375)(22,376,428,65)(23,66,429,369)(24,370,430,67)(25,76,439,379)(26,380,440,77)(27,78,433,381)(28,382,434,79)(29,80,435,383)(30,384,436,73)(31,74,437,377)(32,378,438,75)(33,284,113,204)(34,205,114,285)(35,286,115,206)(36,207,116,287)(37,288,117,208)(38,201,118,281)(39,282,119,202)(40,203,120,283)(41,92,121,180)(42,181,122,93)(43,94,123,182)(44,183,124,95)(45,96,125,184)(46,177,126,89)(47,90,127,178)(48,179,128,91)(49,100,129,188)(50,189,130,101)(51,102,131,190)(52,191,132,103)(53,104,133,192)(54,185,134,97)(55,98,135,186)(56,187,136,99)(57,273,388,448)(58,441,389,274)(59,275,390,442)(60,443,391,276)(61,277,392,444)(62,445,385,278)(63,279,386,446)(64,447,387,280)(105,173,193,142)(106,143,194,174)(107,175,195,144)(108,137,196,176)(109,169,197,138)(110,139,198,170)(111,171,199,140)(112,141,200,172)(153,212,299,239)(154,240,300,213)(155,214,301,233)(156,234,302,215)(157,216,303,235)(158,236,304,209)(159,210,297,237)(160,238,298,211)(161,220,307,247)(162,248,308,221)(163,222,309,241)(164,242,310,223)(165,224,311,243)(166,244,312,217)(167,218,305,245)(168,246,306,219)(249,397,337,317)(250,318,338,398)(251,399,339,319)(252,320,340,400)(253,393,341,313)(254,314,342,394)(255,395,343,315)(256,316,344,396)(257,405,345,325)(258,326,346,406)(259,407,347,327)(260,328,348,408)(261,401,349,321)(262,322,350,402)(263,403,351,323)(264,324,352,404)(265,413,353,333)(266,334,354,414)(267,415,355,335)(268,336,356,416)(269,409,357,329)(270,330,358,410)(271,411,359,331)(272,332,360,412), (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), (1,343,5,339)(2,254,6,250)(3,341,7,337)(4,252,8,256)(9,202,13,206)(10,281,14,285)(11,208,15,204)(12,287,16,283)(17,43,21,47)(18,122,22,126)(19,41,23,45)(20,128,24,124)(25,51,29,55)(26,130,30,134)(27,49,31,53)(28,136,32,132)(33,85,37,81)(34,368,38,364)(35,83,39,87)(36,366,40,362)(42,428,46,432)(44,426,48,430)(50,436,54,440)(52,434,56,438)(57,197,61,193)(58,108,62,112)(59,195,63,199)(60,106,64,110)(65,177,69,181)(66,96,70,92)(67,183,71,179)(68,94,72,90)(73,185,77,189)(74,104,78,100)(75,191,79,187)(76,102,80,98)(82,120,86,116)(84,118,88,114)(89,372,93,376)(91,370,95,374)(97,380,101,384)(99,378,103,382)(105,388,109,392)(107,386,111,390)(113,361,117,365)(115,367,119,363)(121,429,125,425)(123,427,127,431)(129,437,133,433)(131,435,135,439)(137,445,141,441)(138,277,142,273)(139,443,143,447)(140,275,144,279)(145,393,149,397)(146,320,150,316)(147,399,151,395)(148,318,152,314)(153,401,157,405)(154,328,158,324)(155,407,159,403)(156,326,160,322)(161,409,165,413)(162,336,166,332)(163,415,167,411)(164,334,168,330)(169,444,173,448)(170,276,174,280)(171,442,175,446)(172,274,176,278)(178,371,182,375)(180,369,184,373)(186,379,190,383)(188,377,192,381)(194,387,198,391)(196,385,200,389)(201,424,205,420)(203,422,207,418)(209,264,213,260)(210,351,214,347)(211,262,215,258)(212,349,216,345)(217,272,221,268)(218,359,222,355)(219,270,223,266)(220,357,224,353)(225,251,229,255)(226,338,230,342)(227,249,231,253)(228,344,232,340)(233,259,237,263)(234,346,238,350)(235,257,239,261)(236,352,240,348)(241,267,245,271)(242,354,246,358)(243,265,247,269)(244,360,248,356)(282,423,286,419)(284,421,288,417)(289,315,293,319)(290,394,294,398)(291,313,295,317)(292,400,296,396)(297,323,301,327)(298,402,302,406)(299,321,303,325)(300,408,304,404)(305,331,309,335)(306,410,310,414)(307,329,311,333)(308,416,312,412) );
G=PermutationGroup([[(1,115,387,218,379,210,371),(2,116,388,219,380,211,372),(3,117,389,220,381,212,373),(4,118,390,221,382,213,374),(5,119,391,222,383,214,375),(6,120,392,223,384,215,376),(7,113,385,224,377,216,369),(8,114,386,217,378,209,370),(9,170,331,51,323,43,315),(10,171,332,52,324,44,316),(11,172,333,53,325,45,317),(12,173,334,54,326,46,318),(13,174,335,55,327,47,319),(14,175,336,56,328,48,320),(15,176,329,49,321,41,313),(16,169,330,50,322,42,314),(17,289,206,280,305,25,297),(18,290,207,273,306,26,298),(19,291,208,274,307,27,299),(20,292,201,275,308,28,300),(21,293,202,276,309,29,301),(22,294,203,277,310,30,302),(23,295,204,278,311,31,303),(24,296,205,279,312,32,304),(33,62,243,74,235,66,227),(34,63,244,75,236,67,228),(35,64,245,76,237,68,229),(36,57,246,77,238,69,230),(37,58,247,78,239,70,231),(38,59,248,79,240,71,232),(39,60,241,80,233,72,225),(40,61,242,73,234,65,226),(81,108,269,100,261,92,253),(82,109,270,101,262,93,254),(83,110,271,102,263,94,255),(84,111,272,103,264,95,256),(85,112,265,104,257,96,249),(86,105,266,97,258,89,250),(87,106,267,98,259,90,251),(88,107,268,99,260,91,252),(121,393,417,137,409,129,401),(122,394,418,138,410,130,402),(123,395,419,139,411,131,403),(124,396,420,140,412,132,404),(125,397,421,141,413,133,405),(126,398,422,142,414,134,406),(127,399,423,143,415,135,407),(128,400,424,144,416,136,408),(145,288,441,161,433,153,425),(146,281,442,162,434,154,426),(147,282,443,163,435,155,427),(148,283,444,164,436,156,428),(149,284,445,165,437,157,429),(150,285,446,166,438,158,430),(151,286,447,167,439,159,431),(152,287,448,168,440,160,432),(177,338,362,193,354,185,346),(178,339,363,194,355,186,347),(179,340,364,195,356,187,348),(180,341,365,196,357,188,349),(181,342,366,197,358,189,350),(182,343,367,198,359,190,351),(183,344,368,199,360,191,352),(184,337,361,200,353,192,345)], [(1,289,229,151),(2,152,230,290),(3,291,231,145),(4,146,232,292),(5,293,225,147),(6,148,226,294),(7,295,227,149),(8,150,228,296),(9,83,419,367),(10,368,420,84),(11,85,421,361),(12,362,422,86),(13,87,423,363),(14,364,424,88),(15,81,417,365),(16,366,418,82),(17,68,431,371),(18,372,432,69),(19,70,425,373),(20,374,426,71),(21,72,427,375),(22,376,428,65),(23,66,429,369),(24,370,430,67),(25,76,439,379),(26,380,440,77),(27,78,433,381),(28,382,434,79),(29,80,435,383),(30,384,436,73),(31,74,437,377),(32,378,438,75),(33,284,113,204),(34,205,114,285),(35,286,115,206),(36,207,116,287),(37,288,117,208),(38,201,118,281),(39,282,119,202),(40,203,120,283),(41,92,121,180),(42,181,122,93),(43,94,123,182),(44,183,124,95),(45,96,125,184),(46,177,126,89),(47,90,127,178),(48,179,128,91),(49,100,129,188),(50,189,130,101),(51,102,131,190),(52,191,132,103),(53,104,133,192),(54,185,134,97),(55,98,135,186),(56,187,136,99),(57,273,388,448),(58,441,389,274),(59,275,390,442),(60,443,391,276),(61,277,392,444),(62,445,385,278),(63,279,386,446),(64,447,387,280),(105,173,193,142),(106,143,194,174),(107,175,195,144),(108,137,196,176),(109,169,197,138),(110,139,198,170),(111,171,199,140),(112,141,200,172),(153,212,299,239),(154,240,300,213),(155,214,301,233),(156,234,302,215),(157,216,303,235),(158,236,304,209),(159,210,297,237),(160,238,298,211),(161,220,307,247),(162,248,308,221),(163,222,309,241),(164,242,310,223),(165,224,311,243),(166,244,312,217),(167,218,305,245),(168,246,306,219),(249,397,337,317),(250,318,338,398),(251,399,339,319),(252,320,340,400),(253,393,341,313),(254,314,342,394),(255,395,343,315),(256,316,344,396),(257,405,345,325),(258,326,346,406),(259,407,347,327),(260,328,348,408),(261,401,349,321),(262,322,350,402),(263,403,351,323),(264,324,352,404),(265,413,353,333),(266,334,354,414),(267,415,355,335),(268,336,356,416),(269,409,357,329),(270,330,358,410),(271,411,359,331),(272,332,360,412)], [(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)], [(1,343,5,339),(2,254,6,250),(3,341,7,337),(4,252,8,256),(9,202,13,206),(10,281,14,285),(11,208,15,204),(12,287,16,283),(17,43,21,47),(18,122,22,126),(19,41,23,45),(20,128,24,124),(25,51,29,55),(26,130,30,134),(27,49,31,53),(28,136,32,132),(33,85,37,81),(34,368,38,364),(35,83,39,87),(36,366,40,362),(42,428,46,432),(44,426,48,430),(50,436,54,440),(52,434,56,438),(57,197,61,193),(58,108,62,112),(59,195,63,199),(60,106,64,110),(65,177,69,181),(66,96,70,92),(67,183,71,179),(68,94,72,90),(73,185,77,189),(74,104,78,100),(75,191,79,187),(76,102,80,98),(82,120,86,116),(84,118,88,114),(89,372,93,376),(91,370,95,374),(97,380,101,384),(99,378,103,382),(105,388,109,392),(107,386,111,390),(113,361,117,365),(115,367,119,363),(121,429,125,425),(123,427,127,431),(129,437,133,433),(131,435,135,439),(137,445,141,441),(138,277,142,273),(139,443,143,447),(140,275,144,279),(145,393,149,397),(146,320,150,316),(147,399,151,395),(148,318,152,314),(153,401,157,405),(154,328,158,324),(155,407,159,403),(156,326,160,322),(161,409,165,413),(162,336,166,332),(163,415,167,411),(164,334,168,330),(169,444,173,448),(170,276,174,280),(171,442,175,446),(172,274,176,278),(178,371,182,375),(180,369,184,373),(186,379,190,383),(188,377,192,381),(194,387,198,391),(196,385,200,389),(201,424,205,420),(203,422,207,418),(209,264,213,260),(210,351,214,347),(211,262,215,258),(212,349,216,345),(217,272,221,268),(218,359,222,355),(219,270,223,266),(220,357,224,353),(225,251,229,255),(226,338,230,342),(227,249,231,253),(228,344,232,340),(233,259,237,263),(234,346,238,350),(235,257,239,261),(236,352,240,348),(241,267,245,271),(242,354,246,358),(243,265,247,269),(244,360,248,356),(282,423,286,419),(284,421,288,417),(289,315,293,319),(290,394,294,398),(291,313,295,317),(292,400,296,396),(297,323,301,327),(298,402,302,406),(299,321,303,325),(300,408,304,404),(305,331,309,335),(306,410,310,414),(307,329,311,333),(308,416,312,412)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28BB | 28BC | ··· | 28BN | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | Q8 | Q16 | C4○D4 | C7×D4 | C7×Q8 | C7×Q16 | C7×C4○D4 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×C4.Q16 | C7×Q8⋊C4 | C7×C4⋊C8 | C7×C2.D8 | Q8×C28 | C7×C4⋊Q8 | C4.Q16 | Q8⋊C4 | C4⋊C8 | C2.D8 | C4×Q8 | C4⋊Q8 | C2×C28 | C7×Q8 | C28 | C28 | C2×C4 | Q8 | C4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 6 | 12 | 6 | 12 | 6 | 6 | 2 | 2 | 4 | 2 | 12 | 12 | 24 | 12 | 1 | 6 |
Matrix representation of C7×C4.Q16 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 27 | 15 |
| 82 | 31 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 0 | 0 | 66 | 111 |
| 0 | 0 | 88 | 47 |
| 9 | 101 | 0 | 0 |
| 101 | 104 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,49,0,0,0,0,49],[1,0,0,0,0,1,0,0,0,0,98,27,0,0,0,15],[82,82,0,0,31,82,0,0,0,0,66,88,0,0,111,47],[9,101,0,0,101,104,0,0,0,0,112,0,0,0,0,112] >;
C7×C4.Q16 in GAP, Magma, Sage, TeX
C_7\times C_4.Q_{16} % in TeX
G:=Group("C7xC4.Q16"); // GroupNames label
G:=SmallGroup(448,885);
// by ID
G=gap.SmallGroup(448,885);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1960,813,400,2438,1192,14117,3547,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^8=1,d^2=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^2*c^-1>;
// generators/relations