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G = C515C8order 408 = 23·3·17

1st semidirect product of C51 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C515C8, C68.2S3, C4.2D51, C6.Dic17, C2.Dic51, C102.3C4, C204.2C2, C12.2D17, C34.2Dic3, C3⋊(C173C8), C173(C3⋊C8), SmallGroup(408,3)

Series: Derived Chief Lower central Upper central

C1C51 — C515C8
C1C17C51C102C204 — C515C8
C51 — C515C8
C1C4

Generators and relations for C515C8
 G = < a,b | a51=b8=1, bab-1=a-1 >

51C8
17C3⋊C8
3C173C8

Smallest permutation representation of C515C8
Regular action on 408 points
Generators in S408
(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)
(1 380 163 285 84 308 111 216)(2 379 164 284 85 307 112 215)(3 378 165 283 86 357 113 214)(4 377 166 282 87 356 114 213)(5 376 167 281 88 355 115 212)(6 375 168 280 89 354 116 211)(7 374 169 279 90 353 117 210)(8 373 170 278 91 352 118 209)(9 372 171 277 92 351 119 208)(10 371 172 276 93 350 120 207)(11 370 173 275 94 349 121 206)(12 369 174 274 95 348 122 205)(13 368 175 273 96 347 123 255)(14 367 176 272 97 346 124 254)(15 366 177 271 98 345 125 253)(16 365 178 270 99 344 126 252)(17 364 179 269 100 343 127 251)(18 363 180 268 101 342 128 250)(19 362 181 267 102 341 129 249)(20 361 182 266 52 340 130 248)(21 360 183 265 53 339 131 247)(22 359 184 264 54 338 132 246)(23 358 185 263 55 337 133 245)(24 408 186 262 56 336 134 244)(25 407 187 261 57 335 135 243)(26 406 188 260 58 334 136 242)(27 405 189 259 59 333 137 241)(28 404 190 258 60 332 138 240)(29 403 191 257 61 331 139 239)(30 402 192 256 62 330 140 238)(31 401 193 306 63 329 141 237)(32 400 194 305 64 328 142 236)(33 399 195 304 65 327 143 235)(34 398 196 303 66 326 144 234)(35 397 197 302 67 325 145 233)(36 396 198 301 68 324 146 232)(37 395 199 300 69 323 147 231)(38 394 200 299 70 322 148 230)(39 393 201 298 71 321 149 229)(40 392 202 297 72 320 150 228)(41 391 203 296 73 319 151 227)(42 390 204 295 74 318 152 226)(43 389 154 294 75 317 153 225)(44 388 155 293 76 316 103 224)(45 387 156 292 77 315 104 223)(46 386 157 291 78 314 105 222)(47 385 158 290 79 313 106 221)(48 384 159 289 80 312 107 220)(49 383 160 288 81 311 108 219)(50 382 161 287 82 310 109 218)(51 381 162 286 83 309 110 217)

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

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

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

108 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D12A12B17A···17H34A···34H51A···51P68A···68P102A···102P204A···204AF
order1234468888121217···1734···3451···5168···68102···102204···204
size11211251515151222···22···22···22···22···22···2

108 irreducible representations

dim1111222222222
type+++-+-+-
imageC1C2C4C8S3Dic3C3⋊C8D17Dic17D51C173C8Dic51C515C8
kernelC515C8C204C102C51C68C34C17C12C6C4C3C2C1
# reps11241128816161632

Matrix representation of C515C8 in GL4(𝔽409) generated by

14125500
15420600
00398186
00129347
,
22619200
118300
00122237
00165287
G:=sub<GL(4,GF(409))| [141,154,0,0,255,206,0,0,0,0,398,129,0,0,186,347],[226,1,0,0,192,183,0,0,0,0,122,165,0,0,237,287] >;

C515C8 in GAP, Magma, Sage, TeX

C_{51}\rtimes_5C_8
% in TeX

G:=Group("C51:5C8");
// GroupNames label

G:=SmallGroup(408,3);
// by ID

G=gap.SmallGroup(408,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,10,26,323,9604]);
// Polycyclic

G:=Group<a,b|a^51=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C515C8 in TeX

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