direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4⋊Q16, C28⋊6Q16, C56.71D4, C4⋊1(C7×Q16), C8.8(C7×D4), (C4×C8).7C14, C4.3(D4×C14), C4⋊Q8.9C14, (C4×C56).25C2, (C2×C28).423D4, C28.310(C2×D4), (C2×Q16).3C14, C14.57(C2×Q16), C2.10(C14×Q16), C42.81(C2×C14), (C14×Q16).10C2, C14.44(C4⋊1D4), (C2×C28).950C23, (C2×C56).424C22, (C4×C28).365C22, C22.115(D4×C14), (Q8×C14).177C22, (C2×C4).79(C7×D4), C2.7(C7×C4⋊1D4), (C7×C4⋊Q8).24C2, (C2×C8).80(C2×C14), (C2×C14).671(C2×D4), (C2×Q8).21(C2×C14), (C2×C4).125(C22×C14), SmallGroup(448,902)
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, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 194 in 122 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C28, C28, C2×C14, C4×C8, C4⋊Q8, C2×Q16, C56, C2×C28, C2×C28, C2×C28, C7×Q8, C4⋊Q16, C4×C28, C7×C4⋊C4, C2×C56, C7×Q16, Q8×C14, C4×C56, C7×C4⋊Q8, C14×Q16, C7×C4⋊Q16
Quotients: C1, C2, C22, C7, D4, C23, C14, Q16, C2×D4, C2×C14, C4⋊1D4, C2×Q16, C7×D4, C22×C14, C4⋊Q16, C7×Q16, D4×C14, C7×C4⋊1D4, C14×Q16, C7×C4⋊Q16
►Regular action on 448 points
Generators in S448
(1 145 33 137 25 129 17)(2 146 34 138 26 130 18)(3 147 35 139 27 131 19)(4 148 36 140 28 132 20)(5 149 37 141 29 133 21)(6 150 38 142 30 134 22)(7 151 39 143 31 135 23)(8 152 40 144 32 136 24)(9 122 58 162 50 154 42)(10 123 59 163 51 155 43)(11 124 60 164 52 156 44)(12 125 61 165 53 157 45)(13 126 62 166 54 158 46)(14 127 63 167 55 159 47)(15 128 64 168 56 160 48)(16 121 57 161 49 153 41)(65 169 193 81 185 73 177)(66 170 194 82 186 74 178)(67 171 195 83 187 75 179)(68 172 196 84 188 76 180)(69 173 197 85 189 77 181)(70 174 198 86 190 78 182)(71 175 199 87 191 79 183)(72 176 200 88 192 80 184)(89 229 217 105 209 97 201)(90 230 218 106 210 98 202)(91 231 219 107 211 99 203)(92 232 220 108 212 100 204)(93 225 221 109 213 101 205)(94 226 222 110 214 102 206)(95 227 223 111 215 103 207)(96 228 224 112 216 104 208)(113 444 336 436 328 428 320)(114 445 329 437 321 429 313)(115 446 330 438 322 430 314)(116 447 331 439 323 431 315)(117 448 332 440 324 432 316)(118 441 333 433 325 425 317)(119 442 334 434 326 426 318)(120 443 335 435 327 427 319)(233 257 365 249 357 241 349)(234 258 366 250 358 242 350)(235 259 367 251 359 243 351)(236 260 368 252 360 244 352)(237 261 361 253 353 245 345)(238 262 362 254 354 246 346)(239 263 363 255 355 247 347)(240 264 364 256 356 248 348)(265 373 342 281 389 273 381)(266 374 343 282 390 274 382)(267 375 344 283 391 275 383)(268 376 337 284 392 276 384)(269 369 338 285 385 277 377)(270 370 339 286 386 278 378)(271 371 340 287 387 279 379)(272 372 341 288 388 280 380)(289 397 421 305 413 297 405)(290 398 422 306 414 298 406)(291 399 423 307 415 299 407)(292 400 424 308 416 300 408)(293 393 417 309 409 301 401)(294 394 418 310 410 302 402)(295 395 419 311 411 303 403)(296 396 420 312 412 304 404)
(1 93 175 47)(2 94 176 48)(3 95 169 41)(4 96 170 42)(5 89 171 43)(6 90 172 44)(7 91 173 45)(8 92 174 46)(9 148 228 194)(10 149 229 195)(11 150 230 196)(12 151 231 197)(13 152 232 198)(14 145 225 199)(15 146 226 200)(16 147 227 193)(17 205 71 159)(18 206 72 160)(19 207 65 153)(20 208 66 154)(21 201 67 155)(22 202 68 156)(23 203 69 157)(24 204 70 158)(25 213 79 167)(26 214 80 168)(27 215 73 161)(28 216 74 162)(29 209 75 163)(30 210 76 164)(31 211 77 165)(32 212 78 166)(33 221 87 127)(34 222 88 128)(35 223 81 121)(36 224 82 122)(37 217 83 123)(38 218 84 124)(39 219 85 125)(40 220 86 126)(49 131 103 177)(50 132 104 178)(51 133 97 179)(52 134 98 180)(53 135 99 181)(54 136 100 182)(55 129 101 183)(56 130 102 184)(57 139 111 185)(58 140 112 186)(59 141 105 187)(60 142 106 188)(61 143 107 189)(62 144 108 190)(63 137 109 191)(64 138 110 192)(113 264 343 418)(114 257 344 419)(115 258 337 420)(116 259 338 421)(117 260 339 422)(118 261 340 423)(119 262 341 424)(120 263 342 417)(233 375 395 313)(234 376 396 314)(235 369 397 315)(236 370 398 316)(237 371 399 317)(238 372 400 318)(239 373 393 319)(240 374 394 320)(241 383 403 321)(242 384 404 322)(243 377 405 323)(244 378 406 324)(245 379 407 325)(246 380 408 326)(247 381 401 327)(248 382 402 328)(249 391 411 329)(250 392 412 330)(251 385 413 331)(252 386 414 332)(253 387 415 333)(254 388 416 334)(255 389 409 335)(256 390 410 336)(265 293 427 347)(266 294 428 348)(267 295 429 349)(268 296 430 350)(269 289 431 351)(270 290 432 352)(271 291 425 345)(272 292 426 346)(273 301 435 355)(274 302 436 356)(275 303 437 357)(276 304 438 358)(277 297 439 359)(278 298 440 360)(279 299 433 353)(280 300 434 354)(281 309 443 363)(282 310 444 364)(283 311 445 365)(284 312 446 366)(285 305 447 367)(286 306 448 368)(287 307 441 361)(288 308 442 362)
(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)
(1 235 5 239)(2 234 6 238)(3 233 7 237)(4 240 8 236)(9 343 13 339)(10 342 14 338)(11 341 15 337)(12 340 16 344)(17 351 21 347)(18 350 22 346)(19 349 23 345)(20 348 24 352)(25 359 29 355)(26 358 30 354)(27 357 31 353)(28 356 32 360)(33 367 37 363)(34 366 38 362)(35 365 39 361)(36 364 40 368)(41 375 45 371)(42 374 46 370)(43 373 47 369)(44 372 48 376)(49 383 53 379)(50 382 54 378)(51 381 55 377)(52 380 56 384)(57 391 61 387)(58 390 62 386)(59 389 63 385)(60 388 64 392)(65 295 69 291)(66 294 70 290)(67 293 71 289)(68 292 72 296)(73 303 77 299)(74 302 78 298)(75 301 79 297)(76 300 80 304)(81 311 85 307)(82 310 86 306)(83 309 87 305)(84 308 88 312)(89 319 93 315)(90 318 94 314)(91 317 95 313)(92 316 96 320)(97 327 101 323)(98 326 102 322)(99 325 103 321)(100 324 104 328)(105 335 109 331)(106 334 110 330)(107 333 111 329)(108 332 112 336)(113 232 117 228)(114 231 118 227)(115 230 119 226)(116 229 120 225)(121 283 125 287)(122 282 126 286)(123 281 127 285)(124 288 128 284)(129 243 133 247)(130 242 134 246)(131 241 135 245)(132 248 136 244)(137 251 141 255)(138 250 142 254)(139 249 143 253)(140 256 144 252)(145 259 149 263)(146 258 150 262)(147 257 151 261)(148 264 152 260)(153 267 157 271)(154 266 158 270)(155 265 159 269)(156 272 160 268)(161 275 165 279)(162 274 166 278)(163 273 167 277)(164 280 168 276)(169 395 173 399)(170 394 174 398)(171 393 175 397)(172 400 176 396)(177 403 181 407)(178 402 182 406)(179 401 183 405)(180 408 184 404)(185 411 189 415)(186 410 190 414)(187 409 191 413)(188 416 192 412)(193 419 197 423)(194 418 198 422)(195 417 199 421)(196 424 200 420)(201 427 205 431)(202 426 206 430)(203 425 207 429)(204 432 208 428)(209 435 213 439)(210 434 214 438)(211 433 215 437)(212 440 216 436)(217 443 221 447)(218 442 222 446)(219 441 223 445)(220 448 224 444)
G:=sub<Sym(448)| (1,145,33,137,25,129,17)(2,146,34,138,26,130,18)(3,147,35,139,27,131,19)(4,148,36,140,28,132,20)(5,149,37,141,29,133,21)(6,150,38,142,30,134,22)(7,151,39,143,31,135,23)(8,152,40,144,32,136,24)(9,122,58,162,50,154,42)(10,123,59,163,51,155,43)(11,124,60,164,52,156,44)(12,125,61,165,53,157,45)(13,126,62,166,54,158,46)(14,127,63,167,55,159,47)(15,128,64,168,56,160,48)(16,121,57,161,49,153,41)(65,169,193,81,185,73,177)(66,170,194,82,186,74,178)(67,171,195,83,187,75,179)(68,172,196,84,188,76,180)(69,173,197,85,189,77,181)(70,174,198,86,190,78,182)(71,175,199,87,191,79,183)(72,176,200,88,192,80,184)(89,229,217,105,209,97,201)(90,230,218,106,210,98,202)(91,231,219,107,211,99,203)(92,232,220,108,212,100,204)(93,225,221,109,213,101,205)(94,226,222,110,214,102,206)(95,227,223,111,215,103,207)(96,228,224,112,216,104,208)(113,444,336,436,328,428,320)(114,445,329,437,321,429,313)(115,446,330,438,322,430,314)(116,447,331,439,323,431,315)(117,448,332,440,324,432,316)(118,441,333,433,325,425,317)(119,442,334,434,326,426,318)(120,443,335,435,327,427,319)(233,257,365,249,357,241,349)(234,258,366,250,358,242,350)(235,259,367,251,359,243,351)(236,260,368,252,360,244,352)(237,261,361,253,353,245,345)(238,262,362,254,354,246,346)(239,263,363,255,355,247,347)(240,264,364,256,356,248,348)(265,373,342,281,389,273,381)(266,374,343,282,390,274,382)(267,375,344,283,391,275,383)(268,376,337,284,392,276,384)(269,369,338,285,385,277,377)(270,370,339,286,386,278,378)(271,371,340,287,387,279,379)(272,372,341,288,388,280,380)(289,397,421,305,413,297,405)(290,398,422,306,414,298,406)(291,399,423,307,415,299,407)(292,400,424,308,416,300,408)(293,393,417,309,409,301,401)(294,394,418,310,410,302,402)(295,395,419,311,411,303,403)(296,396,420,312,412,304,404), (1,93,175,47)(2,94,176,48)(3,95,169,41)(4,96,170,42)(5,89,171,43)(6,90,172,44)(7,91,173,45)(8,92,174,46)(9,148,228,194)(10,149,229,195)(11,150,230,196)(12,151,231,197)(13,152,232,198)(14,145,225,199)(15,146,226,200)(16,147,227,193)(17,205,71,159)(18,206,72,160)(19,207,65,153)(20,208,66,154)(21,201,67,155)(22,202,68,156)(23,203,69,157)(24,204,70,158)(25,213,79,167)(26,214,80,168)(27,215,73,161)(28,216,74,162)(29,209,75,163)(30,210,76,164)(31,211,77,165)(32,212,78,166)(33,221,87,127)(34,222,88,128)(35,223,81,121)(36,224,82,122)(37,217,83,123)(38,218,84,124)(39,219,85,125)(40,220,86,126)(49,131,103,177)(50,132,104,178)(51,133,97,179)(52,134,98,180)(53,135,99,181)(54,136,100,182)(55,129,101,183)(56,130,102,184)(57,139,111,185)(58,140,112,186)(59,141,105,187)(60,142,106,188)(61,143,107,189)(62,144,108,190)(63,137,109,191)(64,138,110,192)(113,264,343,418)(114,257,344,419)(115,258,337,420)(116,259,338,421)(117,260,339,422)(118,261,340,423)(119,262,341,424)(120,263,342,417)(233,375,395,313)(234,376,396,314)(235,369,397,315)(236,370,398,316)(237,371,399,317)(238,372,400,318)(239,373,393,319)(240,374,394,320)(241,383,403,321)(242,384,404,322)(243,377,405,323)(244,378,406,324)(245,379,407,325)(246,380,408,326)(247,381,401,327)(248,382,402,328)(249,391,411,329)(250,392,412,330)(251,385,413,331)(252,386,414,332)(253,387,415,333)(254,388,416,334)(255,389,409,335)(256,390,410,336)(265,293,427,347)(266,294,428,348)(267,295,429,349)(268,296,430,350)(269,289,431,351)(270,290,432,352)(271,291,425,345)(272,292,426,346)(273,301,435,355)(274,302,436,356)(275,303,437,357)(276,304,438,358)(277,297,439,359)(278,298,440,360)(279,299,433,353)(280,300,434,354)(281,309,443,363)(282,310,444,364)(283,311,445,365)(284,312,446,366)(285,305,447,367)(286,306,448,368)(287,307,441,361)(288,308,442,362), (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), (1,235,5,239)(2,234,6,238)(3,233,7,237)(4,240,8,236)(9,343,13,339)(10,342,14,338)(11,341,15,337)(12,340,16,344)(17,351,21,347)(18,350,22,346)(19,349,23,345)(20,348,24,352)(25,359,29,355)(26,358,30,354)(27,357,31,353)(28,356,32,360)(33,367,37,363)(34,366,38,362)(35,365,39,361)(36,364,40,368)(41,375,45,371)(42,374,46,370)(43,373,47,369)(44,372,48,376)(49,383,53,379)(50,382,54,378)(51,381,55,377)(52,380,56,384)(57,391,61,387)(58,390,62,386)(59,389,63,385)(60,388,64,392)(65,295,69,291)(66,294,70,290)(67,293,71,289)(68,292,72,296)(73,303,77,299)(74,302,78,298)(75,301,79,297)(76,300,80,304)(81,311,85,307)(82,310,86,306)(83,309,87,305)(84,308,88,312)(89,319,93,315)(90,318,94,314)(91,317,95,313)(92,316,96,320)(97,327,101,323)(98,326,102,322)(99,325,103,321)(100,324,104,328)(105,335,109,331)(106,334,110,330)(107,333,111,329)(108,332,112,336)(113,232,117,228)(114,231,118,227)(115,230,119,226)(116,229,120,225)(121,283,125,287)(122,282,126,286)(123,281,127,285)(124,288,128,284)(129,243,133,247)(130,242,134,246)(131,241,135,245)(132,248,136,244)(137,251,141,255)(138,250,142,254)(139,249,143,253)(140,256,144,252)(145,259,149,263)(146,258,150,262)(147,257,151,261)(148,264,152,260)(153,267,157,271)(154,266,158,270)(155,265,159,269)(156,272,160,268)(161,275,165,279)(162,274,166,278)(163,273,167,277)(164,280,168,276)(169,395,173,399)(170,394,174,398)(171,393,175,397)(172,400,176,396)(177,403,181,407)(178,402,182,406)(179,401,183,405)(180,408,184,404)(185,411,189,415)(186,410,190,414)(187,409,191,413)(188,416,192,412)(193,419,197,423)(194,418,198,422)(195,417,199,421)(196,424,200,420)(201,427,205,431)(202,426,206,430)(203,425,207,429)(204,432,208,428)(209,435,213,439)(210,434,214,438)(211,433,215,437)(212,440,216,436)(217,443,221,447)(218,442,222,446)(219,441,223,445)(220,448,224,444)>;
G:=Group( (1,145,33,137,25,129,17)(2,146,34,138,26,130,18)(3,147,35,139,27,131,19)(4,148,36,140,28,132,20)(5,149,37,141,29,133,21)(6,150,38,142,30,134,22)(7,151,39,143,31,135,23)(8,152,40,144,32,136,24)(9,122,58,162,50,154,42)(10,123,59,163,51,155,43)(11,124,60,164,52,156,44)(12,125,61,165,53,157,45)(13,126,62,166,54,158,46)(14,127,63,167,55,159,47)(15,128,64,168,56,160,48)(16,121,57,161,49,153,41)(65,169,193,81,185,73,177)(66,170,194,82,186,74,178)(67,171,195,83,187,75,179)(68,172,196,84,188,76,180)(69,173,197,85,189,77,181)(70,174,198,86,190,78,182)(71,175,199,87,191,79,183)(72,176,200,88,192,80,184)(89,229,217,105,209,97,201)(90,230,218,106,210,98,202)(91,231,219,107,211,99,203)(92,232,220,108,212,100,204)(93,225,221,109,213,101,205)(94,226,222,110,214,102,206)(95,227,223,111,215,103,207)(96,228,224,112,216,104,208)(113,444,336,436,328,428,320)(114,445,329,437,321,429,313)(115,446,330,438,322,430,314)(116,447,331,439,323,431,315)(117,448,332,440,324,432,316)(118,441,333,433,325,425,317)(119,442,334,434,326,426,318)(120,443,335,435,327,427,319)(233,257,365,249,357,241,349)(234,258,366,250,358,242,350)(235,259,367,251,359,243,351)(236,260,368,252,360,244,352)(237,261,361,253,353,245,345)(238,262,362,254,354,246,346)(239,263,363,255,355,247,347)(240,264,364,256,356,248,348)(265,373,342,281,389,273,381)(266,374,343,282,390,274,382)(267,375,344,283,391,275,383)(268,376,337,284,392,276,384)(269,369,338,285,385,277,377)(270,370,339,286,386,278,378)(271,371,340,287,387,279,379)(272,372,341,288,388,280,380)(289,397,421,305,413,297,405)(290,398,422,306,414,298,406)(291,399,423,307,415,299,407)(292,400,424,308,416,300,408)(293,393,417,309,409,301,401)(294,394,418,310,410,302,402)(295,395,419,311,411,303,403)(296,396,420,312,412,304,404), (1,93,175,47)(2,94,176,48)(3,95,169,41)(4,96,170,42)(5,89,171,43)(6,90,172,44)(7,91,173,45)(8,92,174,46)(9,148,228,194)(10,149,229,195)(11,150,230,196)(12,151,231,197)(13,152,232,198)(14,145,225,199)(15,146,226,200)(16,147,227,193)(17,205,71,159)(18,206,72,160)(19,207,65,153)(20,208,66,154)(21,201,67,155)(22,202,68,156)(23,203,69,157)(24,204,70,158)(25,213,79,167)(26,214,80,168)(27,215,73,161)(28,216,74,162)(29,209,75,163)(30,210,76,164)(31,211,77,165)(32,212,78,166)(33,221,87,127)(34,222,88,128)(35,223,81,121)(36,224,82,122)(37,217,83,123)(38,218,84,124)(39,219,85,125)(40,220,86,126)(49,131,103,177)(50,132,104,178)(51,133,97,179)(52,134,98,180)(53,135,99,181)(54,136,100,182)(55,129,101,183)(56,130,102,184)(57,139,111,185)(58,140,112,186)(59,141,105,187)(60,142,106,188)(61,143,107,189)(62,144,108,190)(63,137,109,191)(64,138,110,192)(113,264,343,418)(114,257,344,419)(115,258,337,420)(116,259,338,421)(117,260,339,422)(118,261,340,423)(119,262,341,424)(120,263,342,417)(233,375,395,313)(234,376,396,314)(235,369,397,315)(236,370,398,316)(237,371,399,317)(238,372,400,318)(239,373,393,319)(240,374,394,320)(241,383,403,321)(242,384,404,322)(243,377,405,323)(244,378,406,324)(245,379,407,325)(246,380,408,326)(247,381,401,327)(248,382,402,328)(249,391,411,329)(250,392,412,330)(251,385,413,331)(252,386,414,332)(253,387,415,333)(254,388,416,334)(255,389,409,335)(256,390,410,336)(265,293,427,347)(266,294,428,348)(267,295,429,349)(268,296,430,350)(269,289,431,351)(270,290,432,352)(271,291,425,345)(272,292,426,346)(273,301,435,355)(274,302,436,356)(275,303,437,357)(276,304,438,358)(277,297,439,359)(278,298,440,360)(279,299,433,353)(280,300,434,354)(281,309,443,363)(282,310,444,364)(283,311,445,365)(284,312,446,366)(285,305,447,367)(286,306,448,368)(287,307,441,361)(288,308,442,362), (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), (1,235,5,239)(2,234,6,238)(3,233,7,237)(4,240,8,236)(9,343,13,339)(10,342,14,338)(11,341,15,337)(12,340,16,344)(17,351,21,347)(18,350,22,346)(19,349,23,345)(20,348,24,352)(25,359,29,355)(26,358,30,354)(27,357,31,353)(28,356,32,360)(33,367,37,363)(34,366,38,362)(35,365,39,361)(36,364,40,368)(41,375,45,371)(42,374,46,370)(43,373,47,369)(44,372,48,376)(49,383,53,379)(50,382,54,378)(51,381,55,377)(52,380,56,384)(57,391,61,387)(58,390,62,386)(59,389,63,385)(60,388,64,392)(65,295,69,291)(66,294,70,290)(67,293,71,289)(68,292,72,296)(73,303,77,299)(74,302,78,298)(75,301,79,297)(76,300,80,304)(81,311,85,307)(82,310,86,306)(83,309,87,305)(84,308,88,312)(89,319,93,315)(90,318,94,314)(91,317,95,313)(92,316,96,320)(97,327,101,323)(98,326,102,322)(99,325,103,321)(100,324,104,328)(105,335,109,331)(106,334,110,330)(107,333,111,329)(108,332,112,336)(113,232,117,228)(114,231,118,227)(115,230,119,226)(116,229,120,225)(121,283,125,287)(122,282,126,286)(123,281,127,285)(124,288,128,284)(129,243,133,247)(130,242,134,246)(131,241,135,245)(132,248,136,244)(137,251,141,255)(138,250,142,254)(139,249,143,253)(140,256,144,252)(145,259,149,263)(146,258,150,262)(147,257,151,261)(148,264,152,260)(153,267,157,271)(154,266,158,270)(155,265,159,269)(156,272,160,268)(161,275,165,279)(162,274,166,278)(163,273,167,277)(164,280,168,276)(169,395,173,399)(170,394,174,398)(171,393,175,397)(172,400,176,396)(177,403,181,407)(178,402,182,406)(179,401,183,405)(180,408,184,404)(185,411,189,415)(186,410,190,414)(187,409,191,413)(188,416,192,412)(193,419,197,423)(194,418,198,422)(195,417,199,421)(196,424,200,420)(201,427,205,431)(202,426,206,430)(203,425,207,429)(204,432,208,428)(209,435,213,439)(210,434,214,438)(211,433,215,437)(212,440,216,436)(217,443,221,447)(218,442,222,446)(219,441,223,445)(220,448,224,444) );
G=PermutationGroup([[(1,145,33,137,25,129,17),(2,146,34,138,26,130,18),(3,147,35,139,27,131,19),(4,148,36,140,28,132,20),(5,149,37,141,29,133,21),(6,150,38,142,30,134,22),(7,151,39,143,31,135,23),(8,152,40,144,32,136,24),(9,122,58,162,50,154,42),(10,123,59,163,51,155,43),(11,124,60,164,52,156,44),(12,125,61,165,53,157,45),(13,126,62,166,54,158,46),(14,127,63,167,55,159,47),(15,128,64,168,56,160,48),(16,121,57,161,49,153,41),(65,169,193,81,185,73,177),(66,170,194,82,186,74,178),(67,171,195,83,187,75,179),(68,172,196,84,188,76,180),(69,173,197,85,189,77,181),(70,174,198,86,190,78,182),(71,175,199,87,191,79,183),(72,176,200,88,192,80,184),(89,229,217,105,209,97,201),(90,230,218,106,210,98,202),(91,231,219,107,211,99,203),(92,232,220,108,212,100,204),(93,225,221,109,213,101,205),(94,226,222,110,214,102,206),(95,227,223,111,215,103,207),(96,228,224,112,216,104,208),(113,444,336,436,328,428,320),(114,445,329,437,321,429,313),(115,446,330,438,322,430,314),(116,447,331,439,323,431,315),(117,448,332,440,324,432,316),(118,441,333,433,325,425,317),(119,442,334,434,326,426,318),(120,443,335,435,327,427,319),(233,257,365,249,357,241,349),(234,258,366,250,358,242,350),(235,259,367,251,359,243,351),(236,260,368,252,360,244,352),(237,261,361,253,353,245,345),(238,262,362,254,354,246,346),(239,263,363,255,355,247,347),(240,264,364,256,356,248,348),(265,373,342,281,389,273,381),(266,374,343,282,390,274,382),(267,375,344,283,391,275,383),(268,376,337,284,392,276,384),(269,369,338,285,385,277,377),(270,370,339,286,386,278,378),(271,371,340,287,387,279,379),(272,372,341,288,388,280,380),(289,397,421,305,413,297,405),(290,398,422,306,414,298,406),(291,399,423,307,415,299,407),(292,400,424,308,416,300,408),(293,393,417,309,409,301,401),(294,394,418,310,410,302,402),(295,395,419,311,411,303,403),(296,396,420,312,412,304,404)], [(1,93,175,47),(2,94,176,48),(3,95,169,41),(4,96,170,42),(5,89,171,43),(6,90,172,44),(7,91,173,45),(8,92,174,46),(9,148,228,194),(10,149,229,195),(11,150,230,196),(12,151,231,197),(13,152,232,198),(14,145,225,199),(15,146,226,200),(16,147,227,193),(17,205,71,159),(18,206,72,160),(19,207,65,153),(20,208,66,154),(21,201,67,155),(22,202,68,156),(23,203,69,157),(24,204,70,158),(25,213,79,167),(26,214,80,168),(27,215,73,161),(28,216,74,162),(29,209,75,163),(30,210,76,164),(31,211,77,165),(32,212,78,166),(33,221,87,127),(34,222,88,128),(35,223,81,121),(36,224,82,122),(37,217,83,123),(38,218,84,124),(39,219,85,125),(40,220,86,126),(49,131,103,177),(50,132,104,178),(51,133,97,179),(52,134,98,180),(53,135,99,181),(54,136,100,182),(55,129,101,183),(56,130,102,184),(57,139,111,185),(58,140,112,186),(59,141,105,187),(60,142,106,188),(61,143,107,189),(62,144,108,190),(63,137,109,191),(64,138,110,192),(113,264,343,418),(114,257,344,419),(115,258,337,420),(116,259,338,421),(117,260,339,422),(118,261,340,423),(119,262,341,424),(120,263,342,417),(233,375,395,313),(234,376,396,314),(235,369,397,315),(236,370,398,316),(237,371,399,317),(238,372,400,318),(239,373,393,319),(240,374,394,320),(241,383,403,321),(242,384,404,322),(243,377,405,323),(244,378,406,324),(245,379,407,325),(246,380,408,326),(247,381,401,327),(248,382,402,328),(249,391,411,329),(250,392,412,330),(251,385,413,331),(252,386,414,332),(253,387,415,333),(254,388,416,334),(255,389,409,335),(256,390,410,336),(265,293,427,347),(266,294,428,348),(267,295,429,349),(268,296,430,350),(269,289,431,351),(270,290,432,352),(271,291,425,345),(272,292,426,346),(273,301,435,355),(274,302,436,356),(275,303,437,357),(276,304,438,358),(277,297,439,359),(278,298,440,360),(279,299,433,353),(280,300,434,354),(281,309,443,363),(282,310,444,364),(283,311,445,365),(284,312,446,366),(285,305,447,367),(286,306,448,368),(287,307,441,361),(288,308,442,362)], [(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)], [(1,235,5,239),(2,234,6,238),(3,233,7,237),(4,240,8,236),(9,343,13,339),(10,342,14,338),(11,341,15,337),(12,340,16,344),(17,351,21,347),(18,350,22,346),(19,349,23,345),(20,348,24,352),(25,359,29,355),(26,358,30,354),(27,357,31,353),(28,356,32,360),(33,367,37,363),(34,366,38,362),(35,365,39,361),(36,364,40,368),(41,375,45,371),(42,374,46,370),(43,373,47,369),(44,372,48,376),(49,383,53,379),(50,382,54,378),(51,381,55,377),(52,380,56,384),(57,391,61,387),(58,390,62,386),(59,389,63,385),(60,388,64,392),(65,295,69,291),(66,294,70,290),(67,293,71,289),(68,292,72,296),(73,303,77,299),(74,302,78,298),(75,301,79,297),(76,300,80,304),(81,311,85,307),(82,310,86,306),(83,309,87,305),(84,308,88,312),(89,319,93,315),(90,318,94,314),(91,317,95,313),(92,316,96,320),(97,327,101,323),(98,326,102,322),(99,325,103,321),(100,324,104,328),(105,335,109,331),(106,334,110,330),(107,333,111,329),(108,332,112,336),(113,232,117,228),(114,231,118,227),(115,230,119,226),(116,229,120,225),(121,283,125,287),(122,282,126,286),(123,281,127,285),(124,288,128,284),(129,243,133,247),(130,242,134,246),(131,241,135,245),(132,248,136,244),(137,251,141,255),(138,250,142,254),(139,249,143,253),(140,256,144,252),(145,259,149,263),(146,258,150,262),(147,257,151,261),(148,264,152,260),(153,267,157,271),(154,266,158,270),(155,265,159,269),(156,272,160,268),(161,275,165,279),(162,274,166,278),(163,273,167,277),(164,280,168,276),(169,395,173,399),(170,394,174,398),(171,393,175,397),(172,400,176,396),(177,403,181,407),(178,402,182,406),(179,401,183,405),(180,408,184,404),(185,411,189,415),(186,410,190,414),(187,409,191,413),(188,416,192,412),(193,419,197,423),(194,418,198,422),(195,417,199,421),(196,424,200,420),(201,427,205,431),(202,426,206,430),(203,425,207,429),(204,432,208,428),(209,435,213,439),(210,434,214,438),(211,433,215,437),(212,440,216,436),(217,443,221,447),(218,442,222,446),(219,441,223,445),(220,448,224,444)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 28A | ··· | 28AJ | 28AK | ··· | 28BH | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D4 | Q16 | C7×D4 | C7×D4 | C7×Q16 |
| kernel | C7×C4⋊Q16 | C4×C56 | C7×C4⋊Q8 | C14×Q16 | C4⋊Q16 | C4×C8 | C4⋊Q8 | C2×Q16 | C56 | C2×C28 | C28 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 6 | 6 | 12 | 24 | 4 | 2 | 8 | 24 | 12 | 48 |
Matrix representation of C7×C4⋊Q16 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 |
| 31 | 82 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 31 | 82 |
| 0 | 0 | 31 | 31 |
| 18 | 50 | 0 | 0 |
| 50 | 95 | 0 | 0 |
| 0 | 0 | 100 | 100 |
| 0 | 0 | 100 | 13 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,112,0],[31,31,0,0,82,31,0,0,0,0,31,31,0,0,82,31],[18,50,0,0,50,95,0,0,0,0,100,100,0,0,100,13] >;
C7×C4⋊Q16 in GAP, Magma, Sage, TeX
C_7\times C_4\rtimes Q_{16} % in TeX
G:=Group("C7xC4:Q16"); // GroupNames label
G:=SmallGroup(448,902);
// by ID
G=gap.SmallGroup(448,902);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1968,2438,604,9804,172]);
// 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,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations