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## G = C7×C4⋊2Q16order 448 = 26·7

### Direct product of C7 and C4⋊2Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×C4⋊2Q16
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — Q8×C14 — C7×C4⋊Q8 — C7×C4⋊2Q16
 Lower central C1 — C2 — C2×C4 — C7×C4⋊2Q16
 Upper central C1 — C2×C14 — C4×C28 — C7×C4⋊2Q16

Generators and relations for C7×C42Q16
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=c-1 >

Subgroups: 170 in 108 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, C2×C8, Q16, C2×Q8, C2×Q8, C28, C28, C28, C2×C14, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C56, C2×C28, C2×C28, C7×Q8, C7×Q8, C42Q16, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×Q16, Q8×C14, Q8×C14, C7×Q8⋊C4, C7×C4⋊C8, Q8×C28, C7×C4⋊Q8, C14×Q16, C7×C42Q16
Quotients: C1, C2, C22, C7, D4, C23, C14, Q16, C2×D4, C4○D4, C2×C14, C4⋊D4, C2×Q16, C8.C22, C7×D4, C22×C14, C42Q16, C7×Q16, D4×C14, C7×C4○D4, C7×C4⋊D4, C14×Q16, C7×C8.C22, C7×C42Q16

Smallest permutation representation of C7×C42Q16
Regular action on 448 points
Generators in S448
(1 43 35 241 80 233 72)(2 44 36 242 73 234 65)(3 45 37 243 74 235 66)(4 46 38 244 75 236 67)(5 47 39 245 76 237 68)(6 48 40 246 77 238 69)(7 41 33 247 78 239 70)(8 42 34 248 79 240 71)(9 273 64 174 335 55 327)(10 274 57 175 336 56 328)(11 275 58 176 329 49 321)(12 276 59 169 330 50 322)(13 277 60 170 331 51 323)(14 278 61 171 332 52 324)(15 279 62 172 333 53 325)(16 280 63 173 334 54 326)(17 289 229 95 305 25 297)(18 290 230 96 306 26 298)(19 291 231 89 307 27 299)(20 292 232 90 308 28 300)(21 293 225 91 309 29 301)(22 294 226 92 310 30 302)(23 295 227 93 311 31 303)(24 296 228 94 312 32 304)(81 108 269 100 261 256 317)(82 109 270 101 262 249 318)(83 110 271 102 263 250 319)(84 111 272 103 264 251 320)(85 112 265 104 257 252 313)(86 105 266 97 258 253 314)(87 106 267 98 259 254 315)(88 107 268 99 260 255 316)(113 385 224 377 216 369 208)(114 386 217 378 209 370 201)(115 387 218 379 210 371 202)(116 388 219 380 211 372 203)(117 389 220 381 212 373 204)(118 390 221 382 213 374 205)(119 391 222 383 214 375 206)(120 392 223 384 215 376 207)(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)(145 284 441 161 433 153 425)(146 285 442 162 434 154 426)(147 286 443 163 435 155 427)(148 287 444 164 436 156 428)(149 288 445 165 437 157 429)(150 281 446 166 438 158 430)(151 282 447 167 439 159 431)(152 283 448 168 440 160 432)(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)
(1 151 202 293)(2 294 203 152)(3 145 204 295)(4 296 205 146)(5 147 206 289)(6 290 207 148)(7 149 208 291)(8 292 201 150)(9 254 423 363)(10 364 424 255)(11 256 417 365)(12 366 418 249)(13 250 419 367)(14 368 420 251)(15 252 421 361)(16 362 422 253)(17 68 427 375)(18 376 428 69)(19 70 429 369)(20 370 430 71)(21 72 431 371)(22 372 432 65)(23 66 425 373)(24 374 426 67)(25 76 435 383)(26 384 436 77)(27 78 437 377)(28 378 438 79)(29 80 439 379)(30 380 440 73)(31 74 433 381)(32 382 434 75)(33 445 385 89)(34 90 386 446)(35 447 387 91)(36 92 388 448)(37 441 389 93)(38 94 390 442)(39 443 391 95)(40 96 392 444)(41 288 113 231)(42 232 114 281)(43 282 115 225)(44 226 116 283)(45 284 117 227)(46 228 118 285)(47 286 119 229)(48 230 120 287)(49 100 121 180)(50 181 122 101)(51 102 123 182)(52 183 124 103)(53 104 125 184)(54 177 126 97)(55 98 127 178)(56 179 128 99)(57 356 416 88)(58 81 409 357)(59 358 410 82)(60 83 411 359)(61 360 412 84)(62 85 413 353)(63 354 414 86)(64 87 415 355)(105 173 185 134)(106 135 186 174)(107 175 187 136)(108 129 188 176)(109 169 189 130)(110 131 190 170)(111 171 191 132)(112 133 192 172)(137 196 275 317)(138 318 276 197)(139 198 277 319)(140 320 278 199)(141 200 279 313)(142 314 280 193)(143 194 273 315)(144 316 274 195)(153 212 303 235)(154 236 304 213)(155 214 297 237)(156 238 298 215)(157 216 299 239)(158 240 300 209)(159 210 301 233)(160 234 302 211)(161 220 311 243)(162 244 312 221)(163 222 305 245)(164 246 306 223)(165 224 307 247)(166 248 308 217)(167 218 309 241)(168 242 310 219)(257 397 337 325)(258 326 338 398)(259 399 339 327)(260 328 340 400)(261 393 341 321)(262 322 342 394)(263 395 343 323)(264 324 344 396)(265 405 345 333)(266 334 346 406)(267 407 347 335)(268 336 348 408)(269 401 349 329)(270 330 350 402)(271 403 351 331)(272 332 352 404)
(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)
(1 263 5 259)(2 262 6 258)(3 261 7 257)(4 260 8 264)(9 225 13 229)(10 232 14 228)(11 231 15 227)(12 230 16 226)(17 55 21 51)(18 54 22 50)(19 53 23 49)(20 52 24 56)(25 174 29 170)(26 173 30 169)(27 172 31 176)(28 171 32 175)(33 313 37 317)(34 320 38 316)(35 319 39 315)(36 318 40 314)(41 252 45 256)(42 251 46 255)(43 250 47 254)(44 249 48 253)(57 308 61 312)(58 307 62 311)(59 306 63 310)(60 305 64 309)(65 101 69 97)(66 100 70 104)(67 99 71 103)(68 98 72 102)(73 109 77 105)(74 108 78 112)(75 107 79 111)(76 106 80 110)(81 247 85 243)(82 246 86 242)(83 245 87 241)(84 244 88 248)(89 279 93 275)(90 278 94 274)(91 277 95 273)(92 276 96 280)(113 361 117 365)(114 368 118 364)(115 367 119 363)(116 366 120 362)(121 429 125 425)(122 428 126 432)(123 427 127 431)(124 426 128 430)(129 437 133 433)(130 436 134 440)(131 435 135 439)(132 434 136 438)(137 445 141 441)(138 444 142 448)(139 443 143 447)(140 442 144 446)(145 393 149 397)(146 400 150 396)(147 399 151 395)(148 398 152 394)(153 401 157 405)(154 408 158 404)(155 407 159 403)(156 406 160 402)(161 409 165 413)(162 416 166 412)(163 415 167 411)(164 414 168 410)(177 372 181 376)(178 371 182 375)(179 370 183 374)(180 369 184 373)(185 380 189 384)(186 379 190 383)(187 378 191 382)(188 377 192 381)(193 388 197 392)(194 387 198 391)(195 386 199 390)(196 385 200 389)(201 344 205 340)(202 343 206 339)(203 342 207 338)(204 341 208 337)(209 352 213 348)(210 351 214 347)(211 350 215 346)(212 349 216 345)(217 360 221 356)(218 359 222 355)(219 358 223 354)(220 357 224 353)(233 271 237 267)(234 270 238 266)(235 269 239 265)(236 268 240 272)(281 420 285 424)(282 419 286 423)(283 418 287 422)(284 417 288 421)(289 327 293 323)(290 326 294 322)(291 325 295 321)(292 324 296 328)(297 335 301 331)(298 334 302 330)(299 333 303 329)(300 332 304 336)

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

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

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

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 D4 Q16 C4○D4 C7×D4 C7×D4 C7×Q16 C7×C4○D4 C8.C22 C7×C8.C22 kernel C7×C4⋊2Q16 C7×Q8⋊C4 C7×C4⋊C8 Q8×C28 C7×C4⋊Q8 C14×Q16 C4⋊2Q16 Q8⋊C4 C4⋊C8 C4×Q8 C4⋊Q8 C2×Q16 C2×C28 C7×Q8 C28 C28 C2×C4 Q8 C4 C4 C14 C2 # reps 1 2 1 1 1 2 6 12 6 6 6 12 2 2 4 2 12 12 24 12 1 6

Matrix representation of C7×C42Q16 in GL4(𝔽113) generated by

 49 0 0 0 0 49 0 0 0 0 109 0 0 0 0 109
,
 0 1 0 0 112 0 0 0 0 0 1 0 0 0 0 1
,
 62 15 0 0 15 51 0 0 0 0 0 51 0 0 31 51
,
 1 0 0 0 0 1 0 0 0 0 4 31 0 0 76 109
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,109,0,0,0,0,109],[0,112,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[62,15,0,0,15,51,0,0,0,0,0,31,0,0,51,51],[1,0,0,0,0,1,0,0,0,0,4,76,0,0,31,109] >;

C7×C42Q16 in GAP, Magma, Sage, TeX

C_7\times C_4\rtimes_2Q_{16}
% in TeX

G:=Group("C7xC4:2Q16");
// GroupNames label

G:=SmallGroup(448,870);
// by ID

G=gap.SmallGroup(448,870);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,400,2438,1192,14117,3547,124]);
// 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=c^-1>;
// generators/relations

׿
×
𝔽