direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×Q16⋊C4, Q16⋊3C28, C8.5(C2×C28), (C7×Q16)⋊9C4, C56.46(C2×C4), C2.16(D4×C28), (C4×Q8).3C14, Q8.2(C2×C28), C4.Q8.2C14, C8⋊C4.1C14, C14.118(C4×D4), (C2×C28).457D4, C42.9(C2×C14), (Q8×C28).16C2, (C2×Q16).6C14, C4.13(C22×C28), Q8⋊C4.6C14, (C14×Q16).13C2, C22.55(D4×C14), C28.260(C4○D4), (C2×C28).908C23, C28.158(C22×C4), (C2×C56).330C22, (C4×C28).250C22, (Q8×C14).257C22, C14.130(C8.C22), C4.5(C7×C4○D4), (C7×C4.Q8).7C2, (C7×C8⋊C4).5C2, C4⋊C4.49(C2×C14), (C2×C8).19(C2×C14), (C2×C4).103(C7×D4), (C7×Q8).20(C2×C4), C2.5(C7×C8.C22), (C2×C14).631(C2×D4), (C2×Q8).42(C2×C14), (C7×C4⋊C4).370C22, (C2×C4).83(C22×C14), (C7×Q8⋊C4).15C2, SmallGroup(448,849)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Q16⋊C4
G = < a,b,c,d | a7=b8=d4=1, c2=b4, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, cd=dc >
Subgroups: 154 in 108 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C14, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C56, C56, C2×C28, C2×C28, C2×C28, C7×Q8, C7×Q8, Q16⋊C4, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×Q16, Q8×C14, C7×C8⋊C4, C7×Q8⋊C4, C7×C4.Q8, Q8×C28, C14×Q16, C7×Q16⋊C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C8.C22, C2×C28, C7×D4, C22×C14, Q16⋊C4, C22×C28, D4×C14, C7×C4○D4, D4×C28, C7×C8.C22, C7×Q16⋊C4
►Regular action on 448 points
Generators in S448
(1 174 225 180 247 188 239)(2 175 226 181 248 189 240)(3 176 227 182 241 190 233)(4 169 228 183 242 191 234)(5 170 229 184 243 192 235)(6 171 230 177 244 185 236)(7 172 231 178 245 186 237)(8 173 232 179 246 187 238)(9 385 417 25 409 17 401)(10 386 418 26 410 18 402)(11 387 419 27 411 19 403)(12 388 420 28 412 20 404)(13 389 421 29 413 21 405)(14 390 422 30 414 22 406)(15 391 423 31 415 23 407)(16 392 424 32 416 24 408)(33 398 441 49 433 41 425)(34 399 442 50 434 42 426)(35 400 443 51 435 43 427)(36 393 444 52 436 44 428)(37 394 445 53 437 45 429)(38 395 446 54 438 46 430)(39 396 447 55 439 47 431)(40 397 448 56 440 48 432)(57 331 355 73 347 65 339)(58 332 356 74 348 66 340)(59 333 357 75 349 67 341)(60 334 358 76 350 68 342)(61 335 359 77 351 69 343)(62 336 360 78 352 70 344)(63 329 353 79 345 71 337)(64 330 354 80 346 72 338)(81 105 379 97 371 89 363)(82 106 380 98 372 90 364)(83 107 381 99 373 91 365)(84 108 382 100 374 92 366)(85 109 383 101 375 93 367)(86 110 384 102 376 94 368)(87 111 377 103 369 95 361)(88 112 378 104 370 96 362)(113 275 298 129 290 121 282)(114 276 299 130 291 122 283)(115 277 300 131 292 123 284)(116 278 301 132 293 124 285)(117 279 302 133 294 125 286)(118 280 303 134 295 126 287)(119 273 304 135 296 127 288)(120 274 297 136 289 128 281)(137 161 322 153 314 145 306)(138 162 323 154 315 146 307)(139 163 324 155 316 147 308)(140 164 325 156 317 148 309)(141 165 326 157 318 149 310)(142 166 327 158 319 150 311)(143 167 328 159 320 151 312)(144 168 321 160 313 152 305)(193 217 268 209 260 201 252)(194 218 269 210 261 202 253)(195 219 270 211 262 203 254)(196 220 271 212 263 204 255)(197 221 272 213 264 205 256)(198 222 265 214 257 206 249)(199 223 266 215 258 207 250)(200 224 267 216 259 208 251)
(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 85 5 81)(2 84 6 88)(3 83 7 87)(4 82 8 86)(9 308 13 312)(10 307 14 311)(11 306 15 310)(12 305 16 309)(17 316 21 320)(18 315 22 319)(19 314 23 318)(20 313 24 317)(25 324 29 328)(26 323 30 327)(27 322 31 326)(28 321 32 325)(33 114 37 118)(34 113 38 117)(35 120 39 116)(36 119 40 115)(41 122 45 126)(42 121 46 125)(43 128 47 124)(44 127 48 123)(49 130 53 134)(50 129 54 133)(51 136 55 132)(52 135 56 131)(57 250 61 254)(58 249 62 253)(59 256 63 252)(60 255 64 251)(65 258 69 262)(66 257 70 261)(67 264 71 260)(68 263 72 259)(73 266 77 270)(74 265 78 269)(75 272 79 268)(76 271 80 267)(89 188 93 192)(90 187 94 191)(91 186 95 190)(92 185 96 189)(97 180 101 184)(98 179 102 183)(99 178 103 182)(100 177 104 181)(105 174 109 170)(106 173 110 169)(107 172 111 176)(108 171 112 175)(137 391 141 387)(138 390 142 386)(139 389 143 385)(140 388 144 392)(145 407 149 403)(146 406 150 402)(147 405 151 401)(148 404 152 408)(153 415 157 411)(154 414 158 410)(155 413 159 409)(156 412 160 416)(161 423 165 419)(162 422 166 418)(163 421 167 417)(164 420 168 424)(193 333 197 329)(194 332 198 336)(195 331 199 335)(196 330 200 334)(201 341 205 337)(202 340 206 344)(203 339 207 343)(204 338 208 342)(209 349 213 345)(210 348 214 352)(211 347 215 351)(212 346 216 350)(217 357 221 353)(218 356 222 360)(219 355 223 359)(220 354 224 358)(225 383 229 379)(226 382 230 378)(227 381 231 377)(228 380 232 384)(233 365 237 361)(234 364 238 368)(235 363 239 367)(236 362 240 366)(241 373 245 369)(242 372 246 376)(243 371 247 375)(244 370 248 374)(273 397 277 393)(274 396 278 400)(275 395 279 399)(276 394 280 398)(281 431 285 427)(282 430 286 426)(283 429 287 425)(284 428 288 432)(289 439 293 435)(290 438 294 434)(291 437 295 433)(292 436 296 440)(297 447 301 443)(298 446 302 442)(299 445 303 441)(300 444 304 448)
(1 199 35 387)(2 196 36 392)(3 193 37 389)(4 198 38 386)(5 195 39 391)(6 200 40 388)(7 197 33 385)(8 194 34 390)(9 237 256 425)(10 234 249 430)(11 239 250 427)(12 236 251 432)(13 233 252 429)(14 238 253 426)(15 235 254 431)(16 240 255 428)(17 245 264 433)(18 242 257 438)(19 247 258 435)(20 244 259 440)(21 241 260 437)(22 246 261 434)(23 243 262 439)(24 248 263 436)(25 231 272 441)(26 228 265 446)(27 225 266 443)(28 230 267 448)(29 227 268 445)(30 232 269 442)(31 229 270 447)(32 226 271 444)(41 401 186 205)(42 406 187 202)(43 403 188 207)(44 408 189 204)(45 405 190 201)(46 402 191 206)(47 407 192 203)(48 404 185 208)(49 409 178 213)(50 414 179 210)(51 411 180 215)(52 416 181 212)(53 413 182 209)(54 410 183 214)(55 415 184 211)(56 412 177 216)(57 285 310 363)(58 282 311 368)(59 287 312 365)(60 284 305 362)(61 281 306 367)(62 286 307 364)(63 283 308 361)(64 288 309 366)(65 293 318 371)(66 290 319 376)(67 295 320 373)(68 292 313 370)(69 289 314 375)(70 294 315 372)(71 291 316 369)(72 296 317 374)(73 301 326 379)(74 298 327 384)(75 303 328 381)(76 300 321 378)(77 297 322 383)(78 302 323 380)(79 299 324 377)(80 304 325 382)(81 331 116 141)(82 336 117 138)(83 333 118 143)(84 330 119 140)(85 335 120 137)(86 332 113 142)(87 329 114 139)(88 334 115 144)(89 339 124 149)(90 344 125 146)(91 341 126 151)(92 338 127 148)(93 343 128 145)(94 340 121 150)(95 337 122 147)(96 342 123 152)(97 347 132 157)(98 352 133 154)(99 349 134 159)(100 346 135 156)(101 351 136 153)(102 348 129 158)(103 345 130 155)(104 350 131 160)(105 355 278 165)(106 360 279 162)(107 357 280 167)(108 354 273 164)(109 359 274 161)(110 356 275 166)(111 353 276 163)(112 358 277 168)(169 222 395 418)(170 219 396 423)(171 224 397 420)(172 221 398 417)(173 218 399 422)(174 223 400 419)(175 220 393 424)(176 217 394 421)
G:=sub<Sym(448)| (1,174,225,180,247,188,239)(2,175,226,181,248,189,240)(3,176,227,182,241,190,233)(4,169,228,183,242,191,234)(5,170,229,184,243,192,235)(6,171,230,177,244,185,236)(7,172,231,178,245,186,237)(8,173,232,179,246,187,238)(9,385,417,25,409,17,401)(10,386,418,26,410,18,402)(11,387,419,27,411,19,403)(12,388,420,28,412,20,404)(13,389,421,29,413,21,405)(14,390,422,30,414,22,406)(15,391,423,31,415,23,407)(16,392,424,32,416,24,408)(33,398,441,49,433,41,425)(34,399,442,50,434,42,426)(35,400,443,51,435,43,427)(36,393,444,52,436,44,428)(37,394,445,53,437,45,429)(38,395,446,54,438,46,430)(39,396,447,55,439,47,431)(40,397,448,56,440,48,432)(57,331,355,73,347,65,339)(58,332,356,74,348,66,340)(59,333,357,75,349,67,341)(60,334,358,76,350,68,342)(61,335,359,77,351,69,343)(62,336,360,78,352,70,344)(63,329,353,79,345,71,337)(64,330,354,80,346,72,338)(81,105,379,97,371,89,363)(82,106,380,98,372,90,364)(83,107,381,99,373,91,365)(84,108,382,100,374,92,366)(85,109,383,101,375,93,367)(86,110,384,102,376,94,368)(87,111,377,103,369,95,361)(88,112,378,104,370,96,362)(113,275,298,129,290,121,282)(114,276,299,130,291,122,283)(115,277,300,131,292,123,284)(116,278,301,132,293,124,285)(117,279,302,133,294,125,286)(118,280,303,134,295,126,287)(119,273,304,135,296,127,288)(120,274,297,136,289,128,281)(137,161,322,153,314,145,306)(138,162,323,154,315,146,307)(139,163,324,155,316,147,308)(140,164,325,156,317,148,309)(141,165,326,157,318,149,310)(142,166,327,158,319,150,311)(143,167,328,159,320,151,312)(144,168,321,160,313,152,305)(193,217,268,209,260,201,252)(194,218,269,210,261,202,253)(195,219,270,211,262,203,254)(196,220,271,212,263,204,255)(197,221,272,213,264,205,256)(198,222,265,214,257,206,249)(199,223,266,215,258,207,250)(200,224,267,216,259,208,251), (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), (1,85,5,81)(2,84,6,88)(3,83,7,87)(4,82,8,86)(9,308,13,312)(10,307,14,311)(11,306,15,310)(12,305,16,309)(17,316,21,320)(18,315,22,319)(19,314,23,318)(20,313,24,317)(25,324,29,328)(26,323,30,327)(27,322,31,326)(28,321,32,325)(33,114,37,118)(34,113,38,117)(35,120,39,116)(36,119,40,115)(41,122,45,126)(42,121,46,125)(43,128,47,124)(44,127,48,123)(49,130,53,134)(50,129,54,133)(51,136,55,132)(52,135,56,131)(57,250,61,254)(58,249,62,253)(59,256,63,252)(60,255,64,251)(65,258,69,262)(66,257,70,261)(67,264,71,260)(68,263,72,259)(73,266,77,270)(74,265,78,269)(75,272,79,268)(76,271,80,267)(89,188,93,192)(90,187,94,191)(91,186,95,190)(92,185,96,189)(97,180,101,184)(98,179,102,183)(99,178,103,182)(100,177,104,181)(105,174,109,170)(106,173,110,169)(107,172,111,176)(108,171,112,175)(137,391,141,387)(138,390,142,386)(139,389,143,385)(140,388,144,392)(145,407,149,403)(146,406,150,402)(147,405,151,401)(148,404,152,408)(153,415,157,411)(154,414,158,410)(155,413,159,409)(156,412,160,416)(161,423,165,419)(162,422,166,418)(163,421,167,417)(164,420,168,424)(193,333,197,329)(194,332,198,336)(195,331,199,335)(196,330,200,334)(201,341,205,337)(202,340,206,344)(203,339,207,343)(204,338,208,342)(209,349,213,345)(210,348,214,352)(211,347,215,351)(212,346,216,350)(217,357,221,353)(218,356,222,360)(219,355,223,359)(220,354,224,358)(225,383,229,379)(226,382,230,378)(227,381,231,377)(228,380,232,384)(233,365,237,361)(234,364,238,368)(235,363,239,367)(236,362,240,366)(241,373,245,369)(242,372,246,376)(243,371,247,375)(244,370,248,374)(273,397,277,393)(274,396,278,400)(275,395,279,399)(276,394,280,398)(281,431,285,427)(282,430,286,426)(283,429,287,425)(284,428,288,432)(289,439,293,435)(290,438,294,434)(291,437,295,433)(292,436,296,440)(297,447,301,443)(298,446,302,442)(299,445,303,441)(300,444,304,448), (1,199,35,387)(2,196,36,392)(3,193,37,389)(4,198,38,386)(5,195,39,391)(6,200,40,388)(7,197,33,385)(8,194,34,390)(9,237,256,425)(10,234,249,430)(11,239,250,427)(12,236,251,432)(13,233,252,429)(14,238,253,426)(15,235,254,431)(16,240,255,428)(17,245,264,433)(18,242,257,438)(19,247,258,435)(20,244,259,440)(21,241,260,437)(22,246,261,434)(23,243,262,439)(24,248,263,436)(25,231,272,441)(26,228,265,446)(27,225,266,443)(28,230,267,448)(29,227,268,445)(30,232,269,442)(31,229,270,447)(32,226,271,444)(41,401,186,205)(42,406,187,202)(43,403,188,207)(44,408,189,204)(45,405,190,201)(46,402,191,206)(47,407,192,203)(48,404,185,208)(49,409,178,213)(50,414,179,210)(51,411,180,215)(52,416,181,212)(53,413,182,209)(54,410,183,214)(55,415,184,211)(56,412,177,216)(57,285,310,363)(58,282,311,368)(59,287,312,365)(60,284,305,362)(61,281,306,367)(62,286,307,364)(63,283,308,361)(64,288,309,366)(65,293,318,371)(66,290,319,376)(67,295,320,373)(68,292,313,370)(69,289,314,375)(70,294,315,372)(71,291,316,369)(72,296,317,374)(73,301,326,379)(74,298,327,384)(75,303,328,381)(76,300,321,378)(77,297,322,383)(78,302,323,380)(79,299,324,377)(80,304,325,382)(81,331,116,141)(82,336,117,138)(83,333,118,143)(84,330,119,140)(85,335,120,137)(86,332,113,142)(87,329,114,139)(88,334,115,144)(89,339,124,149)(90,344,125,146)(91,341,126,151)(92,338,127,148)(93,343,128,145)(94,340,121,150)(95,337,122,147)(96,342,123,152)(97,347,132,157)(98,352,133,154)(99,349,134,159)(100,346,135,156)(101,351,136,153)(102,348,129,158)(103,345,130,155)(104,350,131,160)(105,355,278,165)(106,360,279,162)(107,357,280,167)(108,354,273,164)(109,359,274,161)(110,356,275,166)(111,353,276,163)(112,358,277,168)(169,222,395,418)(170,219,396,423)(171,224,397,420)(172,221,398,417)(173,218,399,422)(174,223,400,419)(175,220,393,424)(176,217,394,421)>;
G:=Group( (1,174,225,180,247,188,239)(2,175,226,181,248,189,240)(3,176,227,182,241,190,233)(4,169,228,183,242,191,234)(5,170,229,184,243,192,235)(6,171,230,177,244,185,236)(7,172,231,178,245,186,237)(8,173,232,179,246,187,238)(9,385,417,25,409,17,401)(10,386,418,26,410,18,402)(11,387,419,27,411,19,403)(12,388,420,28,412,20,404)(13,389,421,29,413,21,405)(14,390,422,30,414,22,406)(15,391,423,31,415,23,407)(16,392,424,32,416,24,408)(33,398,441,49,433,41,425)(34,399,442,50,434,42,426)(35,400,443,51,435,43,427)(36,393,444,52,436,44,428)(37,394,445,53,437,45,429)(38,395,446,54,438,46,430)(39,396,447,55,439,47,431)(40,397,448,56,440,48,432)(57,331,355,73,347,65,339)(58,332,356,74,348,66,340)(59,333,357,75,349,67,341)(60,334,358,76,350,68,342)(61,335,359,77,351,69,343)(62,336,360,78,352,70,344)(63,329,353,79,345,71,337)(64,330,354,80,346,72,338)(81,105,379,97,371,89,363)(82,106,380,98,372,90,364)(83,107,381,99,373,91,365)(84,108,382,100,374,92,366)(85,109,383,101,375,93,367)(86,110,384,102,376,94,368)(87,111,377,103,369,95,361)(88,112,378,104,370,96,362)(113,275,298,129,290,121,282)(114,276,299,130,291,122,283)(115,277,300,131,292,123,284)(116,278,301,132,293,124,285)(117,279,302,133,294,125,286)(118,280,303,134,295,126,287)(119,273,304,135,296,127,288)(120,274,297,136,289,128,281)(137,161,322,153,314,145,306)(138,162,323,154,315,146,307)(139,163,324,155,316,147,308)(140,164,325,156,317,148,309)(141,165,326,157,318,149,310)(142,166,327,158,319,150,311)(143,167,328,159,320,151,312)(144,168,321,160,313,152,305)(193,217,268,209,260,201,252)(194,218,269,210,261,202,253)(195,219,270,211,262,203,254)(196,220,271,212,263,204,255)(197,221,272,213,264,205,256)(198,222,265,214,257,206,249)(199,223,266,215,258,207,250)(200,224,267,216,259,208,251), (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), (1,85,5,81)(2,84,6,88)(3,83,7,87)(4,82,8,86)(9,308,13,312)(10,307,14,311)(11,306,15,310)(12,305,16,309)(17,316,21,320)(18,315,22,319)(19,314,23,318)(20,313,24,317)(25,324,29,328)(26,323,30,327)(27,322,31,326)(28,321,32,325)(33,114,37,118)(34,113,38,117)(35,120,39,116)(36,119,40,115)(41,122,45,126)(42,121,46,125)(43,128,47,124)(44,127,48,123)(49,130,53,134)(50,129,54,133)(51,136,55,132)(52,135,56,131)(57,250,61,254)(58,249,62,253)(59,256,63,252)(60,255,64,251)(65,258,69,262)(66,257,70,261)(67,264,71,260)(68,263,72,259)(73,266,77,270)(74,265,78,269)(75,272,79,268)(76,271,80,267)(89,188,93,192)(90,187,94,191)(91,186,95,190)(92,185,96,189)(97,180,101,184)(98,179,102,183)(99,178,103,182)(100,177,104,181)(105,174,109,170)(106,173,110,169)(107,172,111,176)(108,171,112,175)(137,391,141,387)(138,390,142,386)(139,389,143,385)(140,388,144,392)(145,407,149,403)(146,406,150,402)(147,405,151,401)(148,404,152,408)(153,415,157,411)(154,414,158,410)(155,413,159,409)(156,412,160,416)(161,423,165,419)(162,422,166,418)(163,421,167,417)(164,420,168,424)(193,333,197,329)(194,332,198,336)(195,331,199,335)(196,330,200,334)(201,341,205,337)(202,340,206,344)(203,339,207,343)(204,338,208,342)(209,349,213,345)(210,348,214,352)(211,347,215,351)(212,346,216,350)(217,357,221,353)(218,356,222,360)(219,355,223,359)(220,354,224,358)(225,383,229,379)(226,382,230,378)(227,381,231,377)(228,380,232,384)(233,365,237,361)(234,364,238,368)(235,363,239,367)(236,362,240,366)(241,373,245,369)(242,372,246,376)(243,371,247,375)(244,370,248,374)(273,397,277,393)(274,396,278,400)(275,395,279,399)(276,394,280,398)(281,431,285,427)(282,430,286,426)(283,429,287,425)(284,428,288,432)(289,439,293,435)(290,438,294,434)(291,437,295,433)(292,436,296,440)(297,447,301,443)(298,446,302,442)(299,445,303,441)(300,444,304,448), (1,199,35,387)(2,196,36,392)(3,193,37,389)(4,198,38,386)(5,195,39,391)(6,200,40,388)(7,197,33,385)(8,194,34,390)(9,237,256,425)(10,234,249,430)(11,239,250,427)(12,236,251,432)(13,233,252,429)(14,238,253,426)(15,235,254,431)(16,240,255,428)(17,245,264,433)(18,242,257,438)(19,247,258,435)(20,244,259,440)(21,241,260,437)(22,246,261,434)(23,243,262,439)(24,248,263,436)(25,231,272,441)(26,228,265,446)(27,225,266,443)(28,230,267,448)(29,227,268,445)(30,232,269,442)(31,229,270,447)(32,226,271,444)(41,401,186,205)(42,406,187,202)(43,403,188,207)(44,408,189,204)(45,405,190,201)(46,402,191,206)(47,407,192,203)(48,404,185,208)(49,409,178,213)(50,414,179,210)(51,411,180,215)(52,416,181,212)(53,413,182,209)(54,410,183,214)(55,415,184,211)(56,412,177,216)(57,285,310,363)(58,282,311,368)(59,287,312,365)(60,284,305,362)(61,281,306,367)(62,286,307,364)(63,283,308,361)(64,288,309,366)(65,293,318,371)(66,290,319,376)(67,295,320,373)(68,292,313,370)(69,289,314,375)(70,294,315,372)(71,291,316,369)(72,296,317,374)(73,301,326,379)(74,298,327,384)(75,303,328,381)(76,300,321,378)(77,297,322,383)(78,302,323,380)(79,299,324,377)(80,304,325,382)(81,331,116,141)(82,336,117,138)(83,333,118,143)(84,330,119,140)(85,335,120,137)(86,332,113,142)(87,329,114,139)(88,334,115,144)(89,339,124,149)(90,344,125,146)(91,341,126,151)(92,338,127,148)(93,343,128,145)(94,340,121,150)(95,337,122,147)(96,342,123,152)(97,347,132,157)(98,352,133,154)(99,349,134,159)(100,346,135,156)(101,351,136,153)(102,348,129,158)(103,345,130,155)(104,350,131,160)(105,355,278,165)(106,360,279,162)(107,357,280,167)(108,354,273,164)(109,359,274,161)(110,356,275,166)(111,353,276,163)(112,358,277,168)(169,222,395,418)(170,219,396,423)(171,224,397,420)(172,221,398,417)(173,218,399,422)(174,223,400,419)(175,220,393,424)(176,217,394,421) );
G=PermutationGroup([[(1,174,225,180,247,188,239),(2,175,226,181,248,189,240),(3,176,227,182,241,190,233),(4,169,228,183,242,191,234),(5,170,229,184,243,192,235),(6,171,230,177,244,185,236),(7,172,231,178,245,186,237),(8,173,232,179,246,187,238),(9,385,417,25,409,17,401),(10,386,418,26,410,18,402),(11,387,419,27,411,19,403),(12,388,420,28,412,20,404),(13,389,421,29,413,21,405),(14,390,422,30,414,22,406),(15,391,423,31,415,23,407),(16,392,424,32,416,24,408),(33,398,441,49,433,41,425),(34,399,442,50,434,42,426),(35,400,443,51,435,43,427),(36,393,444,52,436,44,428),(37,394,445,53,437,45,429),(38,395,446,54,438,46,430),(39,396,447,55,439,47,431),(40,397,448,56,440,48,432),(57,331,355,73,347,65,339),(58,332,356,74,348,66,340),(59,333,357,75,349,67,341),(60,334,358,76,350,68,342),(61,335,359,77,351,69,343),(62,336,360,78,352,70,344),(63,329,353,79,345,71,337),(64,330,354,80,346,72,338),(81,105,379,97,371,89,363),(82,106,380,98,372,90,364),(83,107,381,99,373,91,365),(84,108,382,100,374,92,366),(85,109,383,101,375,93,367),(86,110,384,102,376,94,368),(87,111,377,103,369,95,361),(88,112,378,104,370,96,362),(113,275,298,129,290,121,282),(114,276,299,130,291,122,283),(115,277,300,131,292,123,284),(116,278,301,132,293,124,285),(117,279,302,133,294,125,286),(118,280,303,134,295,126,287),(119,273,304,135,296,127,288),(120,274,297,136,289,128,281),(137,161,322,153,314,145,306),(138,162,323,154,315,146,307),(139,163,324,155,316,147,308),(140,164,325,156,317,148,309),(141,165,326,157,318,149,310),(142,166,327,158,319,150,311),(143,167,328,159,320,151,312),(144,168,321,160,313,152,305),(193,217,268,209,260,201,252),(194,218,269,210,261,202,253),(195,219,270,211,262,203,254),(196,220,271,212,263,204,255),(197,221,272,213,264,205,256),(198,222,265,214,257,206,249),(199,223,266,215,258,207,250),(200,224,267,216,259,208,251)], [(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)], [(1,85,5,81),(2,84,6,88),(3,83,7,87),(4,82,8,86),(9,308,13,312),(10,307,14,311),(11,306,15,310),(12,305,16,309),(17,316,21,320),(18,315,22,319),(19,314,23,318),(20,313,24,317),(25,324,29,328),(26,323,30,327),(27,322,31,326),(28,321,32,325),(33,114,37,118),(34,113,38,117),(35,120,39,116),(36,119,40,115),(41,122,45,126),(42,121,46,125),(43,128,47,124),(44,127,48,123),(49,130,53,134),(50,129,54,133),(51,136,55,132),(52,135,56,131),(57,250,61,254),(58,249,62,253),(59,256,63,252),(60,255,64,251),(65,258,69,262),(66,257,70,261),(67,264,71,260),(68,263,72,259),(73,266,77,270),(74,265,78,269),(75,272,79,268),(76,271,80,267),(89,188,93,192),(90,187,94,191),(91,186,95,190),(92,185,96,189),(97,180,101,184),(98,179,102,183),(99,178,103,182),(100,177,104,181),(105,174,109,170),(106,173,110,169),(107,172,111,176),(108,171,112,175),(137,391,141,387),(138,390,142,386),(139,389,143,385),(140,388,144,392),(145,407,149,403),(146,406,150,402),(147,405,151,401),(148,404,152,408),(153,415,157,411),(154,414,158,410),(155,413,159,409),(156,412,160,416),(161,423,165,419),(162,422,166,418),(163,421,167,417),(164,420,168,424),(193,333,197,329),(194,332,198,336),(195,331,199,335),(196,330,200,334),(201,341,205,337),(202,340,206,344),(203,339,207,343),(204,338,208,342),(209,349,213,345),(210,348,214,352),(211,347,215,351),(212,346,216,350),(217,357,221,353),(218,356,222,360),(219,355,223,359),(220,354,224,358),(225,383,229,379),(226,382,230,378),(227,381,231,377),(228,380,232,384),(233,365,237,361),(234,364,238,368),(235,363,239,367),(236,362,240,366),(241,373,245,369),(242,372,246,376),(243,371,247,375),(244,370,248,374),(273,397,277,393),(274,396,278,400),(275,395,279,399),(276,394,280,398),(281,431,285,427),(282,430,286,426),(283,429,287,425),(284,428,288,432),(289,439,293,435),(290,438,294,434),(291,437,295,433),(292,436,296,440),(297,447,301,443),(298,446,302,442),(299,445,303,441),(300,444,304,448)], [(1,199,35,387),(2,196,36,392),(3,193,37,389),(4,198,38,386),(5,195,39,391),(6,200,40,388),(7,197,33,385),(8,194,34,390),(9,237,256,425),(10,234,249,430),(11,239,250,427),(12,236,251,432),(13,233,252,429),(14,238,253,426),(15,235,254,431),(16,240,255,428),(17,245,264,433),(18,242,257,438),(19,247,258,435),(20,244,259,440),(21,241,260,437),(22,246,261,434),(23,243,262,439),(24,248,263,436),(25,231,272,441),(26,228,265,446),(27,225,266,443),(28,230,267,448),(29,227,268,445),(30,232,269,442),(31,229,270,447),(32,226,271,444),(41,401,186,205),(42,406,187,202),(43,403,188,207),(44,408,189,204),(45,405,190,201),(46,402,191,206),(47,407,192,203),(48,404,185,208),(49,409,178,213),(50,414,179,210),(51,411,180,215),(52,416,181,212),(53,413,182,209),(54,410,183,214),(55,415,184,211),(56,412,177,216),(57,285,310,363),(58,282,311,368),(59,287,312,365),(60,284,305,362),(61,281,306,367),(62,286,307,364),(63,283,308,361),(64,288,309,366),(65,293,318,371),(66,290,319,376),(67,295,320,373),(68,292,313,370),(69,289,314,375),(70,294,315,372),(71,291,316,369),(72,296,317,374),(73,301,326,379),(74,298,327,384),(75,303,328,381),(76,300,321,378),(77,297,322,383),(78,302,323,380),(79,299,324,377),(80,304,325,382),(81,331,116,141),(82,336,117,138),(83,333,118,143),(84,330,119,140),(85,335,120,137),(86,332,113,142),(87,329,114,139),(88,334,115,144),(89,339,124,149),(90,344,125,146),(91,341,126,151),(92,338,127,148),(93,343,128,145),(94,340,121,150),(95,337,122,147),(96,342,123,152),(97,347,132,157),(98,352,133,154),(99,349,134,159),(100,346,135,156),(101,351,136,153),(102,348,129,158),(103,345,130,155),(104,350,131,160),(105,355,278,165),(106,360,279,162),(107,357,280,167),(108,354,273,164),(109,359,274,161),(110,356,275,166),(111,353,276,163),(112,358,277,168),(169,222,395,418),(170,219,396,423),(171,224,397,420),(172,221,398,417),(173,218,399,422),(174,223,400,419),(175,220,393,424),(176,217,394,421)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | ··· | 4N | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 28A | ··· | 28AJ | 28AK | ··· | 28CF | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | C4○D4 | C7×D4 | C7×C4○D4 | C8.C22 | C7×C8.C22 |
| kernel | C7×Q16⋊C4 | C7×C8⋊C4 | C7×Q8⋊C4 | C7×C4.Q8 | Q8×C28 | C14×Q16 | C7×Q16 | Q16⋊C4 | C8⋊C4 | Q8⋊C4 | C4.Q8 | C4×Q8 | C2×Q16 | Q16 | C2×C28 | C28 | C2×C4 | C4 | C14 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 6 | 6 | 12 | 6 | 12 | 6 | 48 | 2 | 2 | 12 | 12 | 2 | 12 |
Matrix representation of C7×Q16⋊C4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 109 | 0 | 0 | 0 |
| 0 | 0 | 0 | 109 | 0 | 0 |
| 0 | 0 | 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 0 | 0 | 109 |
| 90 | 9 | 0 | 0 | 0 | 0 |
| 29 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 95 | 111 | 75 |
| 0 | 0 | 28 | 83 | 93 | 95 |
| 0 | 0 | 101 | 10 | 20 | 85 |
| 0 | 0 | 111 | 105 | 8 | 2 |
| 30 | 2 | 0 | 0 | 0 | 0 |
| 59 | 83 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 7 | 80 | 99 |
| 0 | 0 | 6 | 14 | 47 | 33 |
| 0 | 0 | 80 | 53 | 86 | 46 |
| 0 | 0 | 53 | 20 | 93 | 60 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 111 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 1 | 0 | 112 | 0 |
| 0 | 0 | 1 | 112 | 112 | 0 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,109,0,0,0,0,0,0,109,0,0,0,0,0,0,109,0,0,0,0,0,0,109],[90,29,0,0,0,0,9,23,0,0,0,0,0,0,8,28,101,111,0,0,95,83,10,105,0,0,111,93,20,8,0,0,75,95,85,2],[30,59,0,0,0,0,2,83,0,0,0,0,0,0,66,6,80,53,0,0,7,14,53,20,0,0,80,47,86,93,0,0,99,33,46,60],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,112,0,0,111,112,112,112,0,0,0,1,0,0] >;
C7×Q16⋊C4 in GAP, Magma, Sage, TeX
C_7\times Q_{16}\rtimes C_4 % in TeX
G:=Group("C7xQ16:C4"); // GroupNames label
G:=SmallGroup(448,849);
// by ID
G=gap.SmallGroup(448,849);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576,4790,604,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=d^4=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations