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G = C523C8order 416 = 25·13

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

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C523C8, C52.7Q8, C52.32D4, C4.16D52, C4.7Dic26, C42.2D13, C26.10M4(2), C4⋊(C132C8), C133(C4⋊C8), (C4×C52).4C2, C26.8(C4⋊C4), C26.16(C2×C8), (C2×C52).18C4, (C2×C4).89D26, (C2×C4).3Dic13, C2.1(C523C4), C2.2(C52.4C4), (C2×C52).103C22, C22.8(C2×Dic13), C2.3(C2×C132C8), (C2×C132C8).8C2, (C2×C26).46(C2×C4), SmallGroup(416,11)

Series: Derived Chief Lower central Upper central

C1C26 — C523C8
C1C13C26C52C2×C52C2×C132C8 — C523C8
C13C26 — C523C8
C1C2×C4C42

Generators and relations for C523C8
 G = < a,b | a52=b8=1, bab-1=a-1 >

2C4
26C8
26C8
2C52
13C2×C8
13C2×C8
2C132C8
2C132C8
13C4⋊C8

Smallest permutation representation of C523C8
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)
(1 269 67 350 223 119 367 182)(2 268 68 349 224 118 368 181)(3 267 69 348 225 117 369 180)(4 266 70 347 226 116 370 179)(5 265 71 346 227 115 371 178)(6 264 72 345 228 114 372 177)(7 263 73 344 229 113 373 176)(8 262 74 343 230 112 374 175)(9 261 75 342 231 111 375 174)(10 312 76 341 232 110 376 173)(11 311 77 340 233 109 377 172)(12 310 78 339 234 108 378 171)(13 309 79 338 235 107 379 170)(14 308 80 337 236 106 380 169)(15 307 81 336 237 105 381 168)(16 306 82 335 238 156 382 167)(17 305 83 334 239 155 383 166)(18 304 84 333 240 154 384 165)(19 303 85 332 241 153 385 164)(20 302 86 331 242 152 386 163)(21 301 87 330 243 151 387 162)(22 300 88 329 244 150 388 161)(23 299 89 328 245 149 389 160)(24 298 90 327 246 148 390 159)(25 297 91 326 247 147 391 158)(26 296 92 325 248 146 392 157)(27 295 93 324 249 145 393 208)(28 294 94 323 250 144 394 207)(29 293 95 322 251 143 395 206)(30 292 96 321 252 142 396 205)(31 291 97 320 253 141 397 204)(32 290 98 319 254 140 398 203)(33 289 99 318 255 139 399 202)(34 288 100 317 256 138 400 201)(35 287 101 316 257 137 401 200)(36 286 102 315 258 136 402 199)(37 285 103 314 259 135 403 198)(38 284 104 313 260 134 404 197)(39 283 53 364 209 133 405 196)(40 282 54 363 210 132 406 195)(41 281 55 362 211 131 407 194)(42 280 56 361 212 130 408 193)(43 279 57 360 213 129 409 192)(44 278 58 359 214 128 410 191)(45 277 59 358 215 127 411 190)(46 276 60 357 216 126 412 189)(47 275 61 356 217 125 413 188)(48 274 62 355 218 124 414 187)(49 273 63 354 219 123 415 186)(50 272 64 353 220 122 416 185)(51 271 65 352 221 121 365 184)(52 270 66 351 222 120 366 183)

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

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

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

116 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A···8H13A···13F26A···26R52A···52BT
order1222444444448···813···1326···2652···52
size11111111222226···262···22···22···2

116 irreducible representations

dim111112222222222
type++++-+-+-+
imageC1C2C2C4C8D4Q8M4(2)D13Dic13D26C132C8Dic26D52C52.4C4
kernelC523C8C2×C132C8C4×C52C2×C52C52C52C52C26C42C2×C4C2×C4C4C4C4C2
# reps12148112612624121224

Matrix representation of C523C8 in GL3(𝔽313) generated by

100
0259123
0190199
,
18800
0153274
0207160
G:=sub<GL(3,GF(313))| [1,0,0,0,259,190,0,123,199],[188,0,0,0,153,207,0,274,160] >;

C523C8 in GAP, Magma, Sage, TeX

C_{52}\rtimes_3C_8
% in TeX

G:=Group("C52:3C8");
// GroupNames label

G:=SmallGroup(416,11);
// by ID

G=gap.SmallGroup(416,11);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,86,13829]);
// Polycyclic

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

Export

Subgroup lattice of C523C8 in TeX

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