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G = C52⋊C8order 416 = 25·13

1st semidirect product of C52 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C521C8, Dic13.4Q8, C26.1M4(2), Dic13.12D4, C4⋊(C13⋊C8), C131(C4⋊C8), C26.8(C2×C8), (C2×C52).5C4, C26.5(C4⋊C4), C2.1(C52⋊C4), (C2×Dic13).9C4, C2.1(C52.C4), (C4×Dic13).12C2, (C2×Dic13).49C22, C2.4(C2×C13⋊C8), (C2×C13⋊C8).1C2, (C2×C4).6(C13⋊C4), (C2×C26).4(C2×C4), C22.9(C2×C13⋊C4), SmallGroup(416,76)

Series: Derived Chief Lower central Upper central

C1C26 — C52⋊C8
C1C13C26Dic13C2×Dic13C2×C13⋊C8 — C52⋊C8
C13C26 — C52⋊C8
C1C22C2×C4

Generators and relations for C52⋊C8
 G = < a,b | a52=b8=1, bab-1=a31 >

13C4
13C4
26C4
13C2×C4
13C2×C4
26C8
26C8
2Dic13
13C42
13C2×C8
13C2×C8
2C13⋊C8
2C13⋊C8
13C4⋊C8

Smallest permutation representation of C52⋊C8
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)
(1 317 269 94 406 155 188 236)(2 364 294 73 407 150 161 215)(3 359 267 104 408 145 186 246)(4 354 292 83 409 140 159 225)(5 349 265 62 410 135 184 256)(6 344 290 93 411 130 157 235)(7 339 263 72 412 125 182 214)(8 334 288 103 413 120 207 245)(9 329 261 82 414 115 180 224)(10 324 286 61 415 110 205 255)(11 319 311 92 416 105 178 234)(12 314 284 71 365 152 203 213)(13 361 309 102 366 147 176 244)(14 356 282 81 367 142 201 223)(15 351 307 60 368 137 174 254)(16 346 280 91 369 132 199 233)(17 341 305 70 370 127 172 212)(18 336 278 101 371 122 197 243)(19 331 303 80 372 117 170 222)(20 326 276 59 373 112 195 253)(21 321 301 90 374 107 168 232)(22 316 274 69 375 154 193 211)(23 363 299 100 376 149 166 242)(24 358 272 79 377 144 191 221)(25 353 297 58 378 139 164 252)(26 348 270 89 379 134 189 231)(27 343 295 68 380 129 162 210)(28 338 268 99 381 124 187 241)(29 333 293 78 382 119 160 220)(30 328 266 57 383 114 185 251)(31 323 291 88 384 109 158 230)(32 318 264 67 385 156 183 209)(33 313 289 98 386 151 208 240)(34 360 262 77 387 146 181 219)(35 355 287 56 388 141 206 250)(36 350 312 87 389 136 179 229)(37 345 285 66 390 131 204 260)(38 340 310 97 391 126 177 239)(39 335 283 76 392 121 202 218)(40 330 308 55 393 116 175 249)(41 325 281 86 394 111 200 228)(42 320 306 65 395 106 173 259)(43 315 279 96 396 153 198 238)(44 362 304 75 397 148 171 217)(45 357 277 54 398 143 196 248)(46 352 302 85 399 138 169 227)(47 347 275 64 400 133 194 258)(48 342 300 95 401 128 167 237)(49 337 273 74 402 123 192 216)(50 332 298 53 403 118 165 247)(51 327 271 84 404 113 190 226)(52 322 296 63 405 108 163 257)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A···8H13A13B13C26A···26I52A···52L
order1222444444448···813131326···2652···52
size11112213131313262626···264444···44···4

44 irreducible representations

dim11111122244444
type++++-+-+
imageC1C2C2C4C4C8D4Q8M4(2)C13⋊C4C13⋊C8C2×C13⋊C4C52.C4C52⋊C4
kernelC52⋊C8C4×Dic13C2×C13⋊C8C2×Dic13C2×C52C52Dic13Dic13C26C2×C4C4C22C2C2
# reps11222811236366

Matrix representation of C52⋊C8 in GL6(𝔽313)

1061990000
2992070000
00283217069
0012062291
0027529413178
0004330418
,
2162510000
283970000
00746314160
002460197101
007170235150
00292582344

G:=sub<GL(6,GF(313))| [106,299,0,0,0,0,199,207,0,0,0,0,0,0,283,1,275,0,0,0,21,20,294,43,0,0,70,62,131,304,0,0,69,291,78,18],[216,283,0,0,0,0,251,97,0,0,0,0,0,0,74,246,71,29,0,0,63,0,70,258,0,0,141,197,235,234,0,0,60,101,150,4] >;

C52⋊C8 in GAP, Magma, Sage, TeX

C_{52}\rtimes C_8
% in TeX

G:=Group("C52:C8");
// GroupNames label

G:=SmallGroup(416,76);
// by ID

G=gap.SmallGroup(416,76);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,86,9221,3473]);
// Polycyclic

G:=Group<a,b|a^52=b^8=1,b*a*b^-1=a^31>;
// generators/relations

Export

Subgroup lattice of C52⋊C8 in TeX

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