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## G = C2×C13⋊Q16order 416 = 25·13

### Direct product of C2 and C13⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C13⋊Q16
 Chief series C1 — C13 — C26 — C52 — Dic26 — C2×Dic26 — C2×C13⋊Q16
 Lower central C13 — C26 — C52 — C2×C13⋊Q16
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×C13⋊Q16
G = < a,b,c,d | a2=b13=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 304 in 60 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C2×C4, Q8, Q8, C13, C2×C8, Q16, C2×Q8, C2×Q8, C26, C26, C2×Q16, Dic13, C52, C52, C2×C26, C132C8, Dic26, Dic26, C2×Dic13, C2×C52, C2×C52, Q8×C13, Q8×C13, C2×C132C8, C13⋊Q16, C2×Dic26, Q8×C26, C2×C13⋊Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, D13, C2×Q16, D26, C13⋊D4, C22×D13, C13⋊Q16, C2×C13⋊D4, C2×C13⋊Q16

Smallest permutation representation of C2×C13⋊Q16
Regular action on 416 points
Generators in S416
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 165)(62 166)(63 167)(64 168)(65 169)(66 170)(67 171)(68 172)(69 173)(70 174)(71 175)(72 176)(73 177)(74 178)(75 179)(76 180)(77 181)(78 182)(79 183)(80 184)(81 185)(82 186)(83 187)(84 188)(85 189)(86 190)(87 191)(88 192)(89 193)(90 194)(91 195)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(209 321)(210 322)(211 323)(212 324)(213 325)(214 313)(215 314)(216 315)(217 316)(218 317)(219 318)(220 319)(221 320)(222 328)(223 329)(224 330)(225 331)(226 332)(227 333)(228 334)(229 335)(230 336)(231 337)(232 338)(233 326)(234 327)(235 343)(236 344)(237 345)(238 346)(239 347)(240 348)(241 349)(242 350)(243 351)(244 339)(245 340)(246 341)(247 342)(248 353)(249 354)(250 355)(251 356)(252 357)(253 358)(254 359)(255 360)(256 361)(257 362)(258 363)(259 364)(260 352)(261 368)(262 369)(263 370)(264 371)(265 372)(266 373)(267 374)(268 375)(269 376)(270 377)(271 365)(272 366)(273 367)(274 384)(275 385)(276 386)(277 387)(278 388)(279 389)(280 390)(281 378)(282 379)(283 380)(284 381)(285 382)(286 383)(287 397)(288 398)(289 399)(290 400)(291 401)(292 402)(293 403)(294 391)(295 392)(296 393)(297 394)(298 395)(299 396)(300 410)(301 411)(302 412)(303 413)(304 414)(305 415)(306 416)(307 404)(308 405)(309 406)(310 407)(311 408)(312 409)
(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)
(1 289 27 279 14 309 40 263)(2 288 28 278 15 308 41 262)(3 287 29 277 16 307 42 261)(4 299 30 276 17 306 43 273)(5 298 31 275 18 305 44 272)(6 297 32 274 19 304 45 271)(7 296 33 286 20 303 46 270)(8 295 34 285 21 302 47 269)(9 294 35 284 22 301 48 268)(10 293 36 283 23 300 49 267)(11 292 37 282 24 312 50 266)(12 291 38 281 25 311 51 265)(13 290 39 280 26 310 52 264)(53 240 92 211 66 256 79 223)(54 239 93 210 67 255 80 222)(55 238 94 209 68 254 81 234)(56 237 95 221 69 253 82 233)(57 236 96 220 70 252 83 232)(58 235 97 219 71 251 84 231)(59 247 98 218 72 250 85 230)(60 246 99 217 73 249 86 229)(61 245 100 216 74 248 87 228)(62 244 101 215 75 260 88 227)(63 243 102 214 76 259 89 226)(64 242 103 213 77 258 90 225)(65 241 104 212 78 257 91 224)(105 399 131 389 118 406 144 370)(106 398 132 388 119 405 145 369)(107 397 133 387 120 404 146 368)(108 396 134 386 121 416 147 367)(109 395 135 385 122 415 148 366)(110 394 136 384 123 414 149 365)(111 393 137 383 124 413 150 377)(112 392 138 382 125 412 151 376)(113 391 139 381 126 411 152 375)(114 403 140 380 127 410 153 374)(115 402 141 379 128 409 154 373)(116 401 142 378 129 408 155 372)(117 400 143 390 130 407 156 371)(157 348 196 323 170 361 183 329)(158 347 197 322 171 360 184 328)(159 346 198 321 172 359 185 327)(160 345 199 320 173 358 186 326)(161 344 200 319 174 357 187 338)(162 343 201 318 175 356 188 337)(163 342 202 317 176 355 189 336)(164 341 203 316 177 354 190 335)(165 340 204 315 178 353 191 334)(166 339 205 314 179 352 192 333)(167 351 206 313 180 364 193 332)(168 350 207 325 181 363 194 331)(169 349 208 324 182 362 195 330)
(1 170 14 157)(2 171 15 158)(3 172 16 159)(4 173 17 160)(5 174 18 161)(6 175 19 162)(7 176 20 163)(8 177 21 164)(9 178 22 165)(10 179 23 166)(11 180 24 167)(12 181 25 168)(13 182 26 169)(27 196 40 183)(28 197 41 184)(29 198 42 185)(30 199 43 186)(31 200 44 187)(32 201 45 188)(33 202 46 189)(34 203 47 190)(35 204 48 191)(36 205 49 192)(37 206 50 193)(38 207 51 194)(39 208 52 195)(53 105 66 118)(54 106 67 119)(55 107 68 120)(56 108 69 121)(57 109 70 122)(58 110 71 123)(59 111 72 124)(60 112 73 125)(61 113 74 126)(62 114 75 127)(63 115 76 128)(64 116 77 129)(65 117 78 130)(79 131 92 144)(80 132 93 145)(81 133 94 146)(82 134 95 147)(83 135 96 148)(84 136 97 149)(85 137 98 150)(86 138 99 151)(87 139 100 152)(88 140 101 153)(89 141 102 154)(90 142 103 155)(91 143 104 156)(209 404 234 397)(210 405 222 398)(211 406 223 399)(212 407 224 400)(213 408 225 401)(214 409 226 402)(215 410 227 403)(216 411 228 391)(217 412 229 392)(218 413 230 393)(219 414 231 394)(220 415 232 395)(221 416 233 396)(235 365 251 384)(236 366 252 385)(237 367 253 386)(238 368 254 387)(239 369 255 388)(240 370 256 389)(241 371 257 390)(242 372 258 378)(243 373 259 379)(244 374 260 380)(245 375 248 381)(246 376 249 382)(247 377 250 383)(261 359 277 346)(262 360 278 347)(263 361 279 348)(264 362 280 349)(265 363 281 350)(266 364 282 351)(267 352 283 339)(268 353 284 340)(269 354 285 341)(270 355 286 342)(271 356 274 343)(272 357 275 344)(273 358 276 345)(287 321 307 327)(288 322 308 328)(289 323 309 329)(290 324 310 330)(291 325 311 331)(292 313 312 332)(293 314 300 333)(294 315 301 334)(295 316 302 335)(296 317 303 336)(297 318 304 337)(298 319 305 338)(299 320 306 326)

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 4 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 4 4 52 52 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 D4 D4 Q16 D13 D26 D26 C13⋊D4 C13⋊D4 C13⋊Q16 kernel C2×C13⋊Q16 C2×C13⋊2C8 C13⋊Q16 C2×Dic26 Q8×C26 C52 C2×C26 C26 C2×Q8 C2×C4 Q8 C4 C22 C2 # reps 1 1 4 1 1 1 1 4 6 6 12 12 12 12

Matrix representation of C2×C13⋊Q16 in GL4(𝔽313) generated by

 312 0 0 0 0 312 0 0 0 0 1 0 0 0 0 1
,
 85 312 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 70 84 0 0 87 243 0 0 0 0 120 203 0 0 37 0
,
 312 0 0 0 0 312 0 0 0 0 120 105 0 0 152 193
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[85,1,0,0,312,0,0,0,0,0,1,0,0,0,0,1],[70,87,0,0,84,243,0,0,0,0,120,37,0,0,203,0],[312,0,0,0,0,312,0,0,0,0,120,152,0,0,105,193] >;

C2×C13⋊Q16 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes Q_{16}
% in TeX

G:=Group("C2xC13:Q16");
// GroupNames label

G:=SmallGroup(416,164);
// by ID

G=gap.SmallGroup(416,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,218,86,579,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^13=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