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## G = C13×C4⋊Q8order 416 = 25·13

### Direct product of C13 and C4⋊Q8

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

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 84 in 68 conjugacy classes, 52 normal (12 characteristic)
C1, C2, C2 [×2], C4 [×6], C4 [×4], C22, C2×C4, C2×C4 [×6], Q8 [×4], C13, C42, C4⋊C4 [×4], C2×Q8 [×2], C26, C26 [×2], C4⋊Q8, C52 [×6], C52 [×4], C2×C26, C2×C52, C2×C52 [×6], Q8×C13 [×4], C4×C52, C13×C4⋊C4 [×4], Q8×C26 [×2], C13×C4⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, C13, C2×D4, C2×Q8 [×2], C26 [×7], C4⋊Q8, C2×C26 [×7], D4×C13 [×2], Q8×C13 [×4], C22×C26, D4×C26, Q8×C26 [×2], C13×C4⋊Q8

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

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 13A ··· 13L 26A ··· 26AJ 52A ··· 52BT 52BU ··· 52DP order 1 2 2 2 4 ··· 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C13 C26 C26 C26 D4 Q8 D4×C13 Q8×C13 kernel C13×C4⋊Q8 C4×C52 C13×C4⋊C4 Q8×C26 C4⋊Q8 C42 C4⋊C4 C2×Q8 C52 C52 C4 C4 # reps 1 1 4 2 12 12 48 24 2 4 24 48

Matrix representation of C13×C4⋊Q8 in GL4(𝔽53) generated by

 42 0 0 0 0 42 0 0 0 0 16 0 0 0 0 16
,
 30 0 0 0 0 23 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 23 51 0 0 0 30
,
 0 1 0 0 1 0 0 0 0 0 10 15 0 0 18 43
G:=sub<GL(4,GF(53))| [42,0,0,0,0,42,0,0,0,0,16,0,0,0,0,16],[30,0,0,0,0,23,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,23,0,0,0,51,30],[0,1,0,0,1,0,0,0,0,0,10,18,0,0,15,43] >;

C13×C4⋊Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(416,189);
// by ID

G=gap.SmallGroup(416,189);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,1273,631,3818,950]);
// Polycyclic

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

׿
×
𝔽