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## G = C13×C4.Q8order 416 = 25·13

### Direct product of C13 and C4.Q8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C13×C4.Q8
G = < a,b,c,d | a13=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Smallest permutation representation of C13×C4.Q8
Regular action on 416 points
Generators in S416
(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)
(1 252 87 338)(2 253 88 326)(3 254 89 327)(4 255 90 328)(5 256 91 329)(6 257 79 330)(7 258 80 331)(8 259 81 332)(9 260 82 333)(10 248 83 334)(11 249 84 335)(12 250 85 336)(13 251 86 337)(14 113 354 70)(15 114 355 71)(16 115 356 72)(17 116 357 73)(18 117 358 74)(19 105 359 75)(20 106 360 76)(21 107 361 77)(22 108 362 78)(23 109 363 66)(24 110 364 67)(25 111 352 68)(26 112 353 69)(27 370 178 51)(28 371 179 52)(29 372 180 40)(30 373 181 41)(31 374 182 42)(32 375 170 43)(33 376 171 44)(34 377 172 45)(35 365 173 46)(36 366 174 47)(37 367 175 48)(38 368 176 49)(39 369 177 50)(53 312 289 345)(54 300 290 346)(55 301 291 347)(56 302 292 348)(57 303 293 349)(58 304 294 350)(59 305 295 351)(60 306 296 339)(61 307 297 340)(62 308 298 341)(63 309 299 342)(64 310 287 343)(65 311 288 344)(92 143 398 226)(93 131 399 227)(94 132 400 228)(95 133 401 229)(96 134 402 230)(97 135 403 231)(98 136 391 232)(99 137 392 233)(100 138 393 234)(101 139 394 222)(102 140 395 223)(103 141 396 224)(104 142 397 225)(118 238 156 184)(119 239 144 185)(120 240 145 186)(121 241 146 187)(122 242 147 188)(123 243 148 189)(124 244 149 190)(125 245 150 191)(126 246 151 192)(127 247 152 193)(128 235 153 194)(129 236 154 195)(130 237 155 183)(157 383 279 320)(158 384 280 321)(159 385 281 322)(160 386 282 323)(161 387 283 324)(162 388 284 325)(163 389 285 313)(164 390 286 314)(165 378 274 315)(166 379 275 316)(167 380 276 317)(168 381 277 318)(169 382 278 319)(196 217 271 408)(197 218 272 409)(198 219 273 410)(199 220 261 411)(200 221 262 412)(201 209 263 413)(202 210 264 414)(203 211 265 415)(204 212 266 416)(205 213 267 404)(206 214 268 405)(207 215 269 406)(208 216 270 407)
(1 211 342 352 87 415 309 25)(2 212 343 353 88 416 310 26)(3 213 344 354 89 404 311 14)(4 214 345 355 90 405 312 15)(5 215 346 356 91 406 300 16)(6 216 347 357 79 407 301 17)(7 217 348 358 80 408 302 18)(8 218 349 359 81 409 303 19)(9 219 350 360 82 410 304 20)(10 220 351 361 83 411 305 21)(11 221 339 362 84 412 306 22)(12 209 340 363 85 413 307 23)(13 210 341 364 86 414 308 24)(27 283 183 397 178 161 237 104)(28 284 184 398 179 162 238 92)(29 285 185 399 180 163 239 93)(30 286 186 400 181 164 240 94)(31 274 187 401 182 165 241 95)(32 275 188 402 170 166 242 96)(33 276 189 403 171 167 243 97)(34 277 190 391 172 168 244 98)(35 278 191 392 173 169 245 99)(36 279 192 393 174 157 246 100)(37 280 193 394 175 158 247 101)(38 281 194 395 176 159 235 102)(39 282 195 396 177 160 236 103)(40 389 144 131 372 313 119 227)(41 390 145 132 373 314 120 228)(42 378 146 133 374 315 121 229)(43 379 147 134 375 316 122 230)(44 380 148 135 376 317 123 231)(45 381 149 136 377 318 124 232)(46 382 150 137 365 319 125 233)(47 383 151 138 366 320 126 234)(48 384 152 139 367 321 127 222)(49 385 153 140 368 322 128 223)(50 386 154 141 369 323 129 224)(51 387 155 142 370 324 130 225)(52 388 156 143 371 325 118 226)(53 71 328 206 289 114 255 268)(54 72 329 207 290 115 256 269)(55 73 330 208 291 116 257 270)(56 74 331 196 292 117 258 271)(57 75 332 197 293 105 259 272)(58 76 333 198 294 106 260 273)(59 77 334 199 295 107 248 261)(60 78 335 200 296 108 249 262)(61 66 336 201 297 109 250 263)(62 67 337 202 298 110 251 264)(63 68 338 203 299 111 252 265)(64 69 326 204 287 112 253 266)(65 70 327 205 288 113 254 267)
(1 321 299 394)(2 322 287 395)(3 323 288 396)(4 324 289 397)(5 325 290 398)(6 313 291 399)(7 314 292 400)(8 315 293 401)(9 316 294 402)(10 317 295 403)(11 318 296 391)(12 319 297 392)(13 320 298 393)(14 154 205 39)(15 155 206 27)(16 156 207 28)(17 144 208 29)(18 145 196 30)(19 146 197 31)(20 147 198 32)(21 148 199 33)(22 149 200 34)(23 150 201 35)(24 151 202 36)(25 152 203 37)(26 153 204 38)(40 116 239 216)(41 117 240 217)(42 105 241 218)(43 106 242 219)(44 107 243 220)(45 108 244 221)(46 109 245 209)(47 110 246 210)(48 111 247 211)(49 112 235 212)(50 113 236 213)(51 114 237 214)(52 115 238 215)(53 104 90 387)(54 92 91 388)(55 93 79 389)(56 94 80 390)(57 95 81 378)(58 96 82 379)(59 97 83 380)(60 98 84 381)(61 99 85 382)(62 100 86 383)(63 101 87 384)(64 102 88 385)(65 103 89 386)(66 191 413 365)(67 192 414 366)(68 193 415 367)(69 194 416 368)(70 195 404 369)(71 183 405 370)(72 184 406 371)(73 185 407 372)(74 186 408 373)(75 187 409 374)(76 188 410 375)(77 189 411 376)(78 190 412 377)(118 269 179 356)(119 270 180 357)(120 271 181 358)(121 272 182 359)(122 273 170 360)(123 261 171 361)(124 262 172 362)(125 263 173 363)(126 264 174 364)(127 265 175 352)(128 266 176 353)(129 267 177 354)(130 268 178 355)(131 257 285 347)(132 258 286 348)(133 259 274 349)(134 260 275 350)(135 248 276 351)(136 249 277 339)(137 250 278 340)(138 251 279 341)(139 252 280 342)(140 253 281 343)(141 254 282 344)(142 255 283 345)(143 256 284 346)(157 308 234 337)(158 309 222 338)(159 310 223 326)(160 311 224 327)(161 312 225 328)(162 300 226 329)(163 301 227 330)(164 302 228 331)(165 303 229 332)(166 304 230 333)(167 305 231 334)(168 306 232 335)(169 307 233 336)

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 13A ··· 13L 26A ··· 26AJ 52A ··· 52X 52Y ··· 52BT 104A ··· 104AV order 1 2 2 2 4 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - + image C1 C2 C2 C4 C13 C26 C26 C52 Q8 D4 SD16 Q8×C13 D4×C13 C13×SD16 kernel C13×C4.Q8 C13×C4⋊C4 C2×C104 C104 C4.Q8 C4⋊C4 C2×C8 C8 C52 C2×C26 C26 C4 C22 C2 # reps 1 2 1 4 12 24 12 48 1 1 4 12 12 48

Matrix representation of C13×C4.Q8 in GL4(𝔽313) generated by

 150 0 0 0 0 150 0 0 0 0 113 0 0 0 0 113
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 312 0
,
 1 311 0 0 1 312 0 0 0 0 248 65 0 0 248 248
,
 37 103 0 0 245 276 0 0 0 0 221 232 0 0 232 92
G:=sub<GL(4,GF(313))| [150,0,0,0,0,150,0,0,0,0,113,0,0,0,0,113],[1,0,0,0,0,1,0,0,0,0,0,312,0,0,1,0],[1,1,0,0,311,312,0,0,0,0,248,248,0,0,65,248],[37,245,0,0,103,276,0,0,0,0,221,232,0,0,232,92] >;

C13×C4.Q8 in GAP, Magma, Sage, TeX

C_{13}\times C_4.Q_8
% in TeX

G:=Group("C13xC4.Q8");
// GroupNames label

G:=SmallGroup(416,56);
// by ID

G=gap.SmallGroup(416,56);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,319,6243,117]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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