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G = Dic115order 460 = 22·5·23

Dicyclic group

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

Aliases: Dic115, C46.D5, C23⋊Dic5, C1153C4, C2.D115, C10.D23, C52Dic23, C230.1C2, SmallGroup(460,3)

Series: Derived Chief Lower central Upper central

C1C115 — Dic115
C1C23C115C230 — Dic115
C115 — Dic115
C1C2

Generators and relations for Dic115
 G = < a,b | a230=1, b2=a115, bab-1=a-1 >

115C4
23Dic5
5Dic23

Smallest permutation representation of Dic115
Regular action on 460 points
Generators in S460
(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)(231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460)
(1 432 116 317)(2 431 117 316)(3 430 118 315)(4 429 119 314)(5 428 120 313)(6 427 121 312)(7 426 122 311)(8 425 123 310)(9 424 124 309)(10 423 125 308)(11 422 126 307)(12 421 127 306)(13 420 128 305)(14 419 129 304)(15 418 130 303)(16 417 131 302)(17 416 132 301)(18 415 133 300)(19 414 134 299)(20 413 135 298)(21 412 136 297)(22 411 137 296)(23 410 138 295)(24 409 139 294)(25 408 140 293)(26 407 141 292)(27 406 142 291)(28 405 143 290)(29 404 144 289)(30 403 145 288)(31 402 146 287)(32 401 147 286)(33 400 148 285)(34 399 149 284)(35 398 150 283)(36 397 151 282)(37 396 152 281)(38 395 153 280)(39 394 154 279)(40 393 155 278)(41 392 156 277)(42 391 157 276)(43 390 158 275)(44 389 159 274)(45 388 160 273)(46 387 161 272)(47 386 162 271)(48 385 163 270)(49 384 164 269)(50 383 165 268)(51 382 166 267)(52 381 167 266)(53 380 168 265)(54 379 169 264)(55 378 170 263)(56 377 171 262)(57 376 172 261)(58 375 173 260)(59 374 174 259)(60 373 175 258)(61 372 176 257)(62 371 177 256)(63 370 178 255)(64 369 179 254)(65 368 180 253)(66 367 181 252)(67 366 182 251)(68 365 183 250)(69 364 184 249)(70 363 185 248)(71 362 186 247)(72 361 187 246)(73 360 188 245)(74 359 189 244)(75 358 190 243)(76 357 191 242)(77 356 192 241)(78 355 193 240)(79 354 194 239)(80 353 195 238)(81 352 196 237)(82 351 197 236)(83 350 198 235)(84 349 199 234)(85 348 200 233)(86 347 201 232)(87 346 202 231)(88 345 203 460)(89 344 204 459)(90 343 205 458)(91 342 206 457)(92 341 207 456)(93 340 208 455)(94 339 209 454)(95 338 210 453)(96 337 211 452)(97 336 212 451)(98 335 213 450)(99 334 214 449)(100 333 215 448)(101 332 216 447)(102 331 217 446)(103 330 218 445)(104 329 219 444)(105 328 220 443)(106 327 221 442)(107 326 222 441)(108 325 223 440)(109 324 224 439)(110 323 225 438)(111 322 226 437)(112 321 227 436)(113 320 228 435)(114 319 229 434)(115 318 230 433)

G:=sub<Sym(460)| (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)(231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460), (1,432,116,317)(2,431,117,316)(3,430,118,315)(4,429,119,314)(5,428,120,313)(6,427,121,312)(7,426,122,311)(8,425,123,310)(9,424,124,309)(10,423,125,308)(11,422,126,307)(12,421,127,306)(13,420,128,305)(14,419,129,304)(15,418,130,303)(16,417,131,302)(17,416,132,301)(18,415,133,300)(19,414,134,299)(20,413,135,298)(21,412,136,297)(22,411,137,296)(23,410,138,295)(24,409,139,294)(25,408,140,293)(26,407,141,292)(27,406,142,291)(28,405,143,290)(29,404,144,289)(30,403,145,288)(31,402,146,287)(32,401,147,286)(33,400,148,285)(34,399,149,284)(35,398,150,283)(36,397,151,282)(37,396,152,281)(38,395,153,280)(39,394,154,279)(40,393,155,278)(41,392,156,277)(42,391,157,276)(43,390,158,275)(44,389,159,274)(45,388,160,273)(46,387,161,272)(47,386,162,271)(48,385,163,270)(49,384,164,269)(50,383,165,268)(51,382,166,267)(52,381,167,266)(53,380,168,265)(54,379,169,264)(55,378,170,263)(56,377,171,262)(57,376,172,261)(58,375,173,260)(59,374,174,259)(60,373,175,258)(61,372,176,257)(62,371,177,256)(63,370,178,255)(64,369,179,254)(65,368,180,253)(66,367,181,252)(67,366,182,251)(68,365,183,250)(69,364,184,249)(70,363,185,248)(71,362,186,247)(72,361,187,246)(73,360,188,245)(74,359,189,244)(75,358,190,243)(76,357,191,242)(77,356,192,241)(78,355,193,240)(79,354,194,239)(80,353,195,238)(81,352,196,237)(82,351,197,236)(83,350,198,235)(84,349,199,234)(85,348,200,233)(86,347,201,232)(87,346,202,231)(88,345,203,460)(89,344,204,459)(90,343,205,458)(91,342,206,457)(92,341,207,456)(93,340,208,455)(94,339,209,454)(95,338,210,453)(96,337,211,452)(97,336,212,451)(98,335,213,450)(99,334,214,449)(100,333,215,448)(101,332,216,447)(102,331,217,446)(103,330,218,445)(104,329,219,444)(105,328,220,443)(106,327,221,442)(107,326,222,441)(108,325,223,440)(109,324,224,439)(110,323,225,438)(111,322,226,437)(112,321,227,436)(113,320,228,435)(114,319,229,434)(115,318,230,433)>;

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)(231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460), (1,432,116,317)(2,431,117,316)(3,430,118,315)(4,429,119,314)(5,428,120,313)(6,427,121,312)(7,426,122,311)(8,425,123,310)(9,424,124,309)(10,423,125,308)(11,422,126,307)(12,421,127,306)(13,420,128,305)(14,419,129,304)(15,418,130,303)(16,417,131,302)(17,416,132,301)(18,415,133,300)(19,414,134,299)(20,413,135,298)(21,412,136,297)(22,411,137,296)(23,410,138,295)(24,409,139,294)(25,408,140,293)(26,407,141,292)(27,406,142,291)(28,405,143,290)(29,404,144,289)(30,403,145,288)(31,402,146,287)(32,401,147,286)(33,400,148,285)(34,399,149,284)(35,398,150,283)(36,397,151,282)(37,396,152,281)(38,395,153,280)(39,394,154,279)(40,393,155,278)(41,392,156,277)(42,391,157,276)(43,390,158,275)(44,389,159,274)(45,388,160,273)(46,387,161,272)(47,386,162,271)(48,385,163,270)(49,384,164,269)(50,383,165,268)(51,382,166,267)(52,381,167,266)(53,380,168,265)(54,379,169,264)(55,378,170,263)(56,377,171,262)(57,376,172,261)(58,375,173,260)(59,374,174,259)(60,373,175,258)(61,372,176,257)(62,371,177,256)(63,370,178,255)(64,369,179,254)(65,368,180,253)(66,367,181,252)(67,366,182,251)(68,365,183,250)(69,364,184,249)(70,363,185,248)(71,362,186,247)(72,361,187,246)(73,360,188,245)(74,359,189,244)(75,358,190,243)(76,357,191,242)(77,356,192,241)(78,355,193,240)(79,354,194,239)(80,353,195,238)(81,352,196,237)(82,351,197,236)(83,350,198,235)(84,349,199,234)(85,348,200,233)(86,347,201,232)(87,346,202,231)(88,345,203,460)(89,344,204,459)(90,343,205,458)(91,342,206,457)(92,341,207,456)(93,340,208,455)(94,339,209,454)(95,338,210,453)(96,337,211,452)(97,336,212,451)(98,335,213,450)(99,334,214,449)(100,333,215,448)(101,332,216,447)(102,331,217,446)(103,330,218,445)(104,329,219,444)(105,328,220,443)(106,327,221,442)(107,326,222,441)(108,325,223,440)(109,324,224,439)(110,323,225,438)(111,322,226,437)(112,321,227,436)(113,320,228,435)(114,319,229,434)(115,318,230,433) );

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),(231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460)], [(1,432,116,317),(2,431,117,316),(3,430,118,315),(4,429,119,314),(5,428,120,313),(6,427,121,312),(7,426,122,311),(8,425,123,310),(9,424,124,309),(10,423,125,308),(11,422,126,307),(12,421,127,306),(13,420,128,305),(14,419,129,304),(15,418,130,303),(16,417,131,302),(17,416,132,301),(18,415,133,300),(19,414,134,299),(20,413,135,298),(21,412,136,297),(22,411,137,296),(23,410,138,295),(24,409,139,294),(25,408,140,293),(26,407,141,292),(27,406,142,291),(28,405,143,290),(29,404,144,289),(30,403,145,288),(31,402,146,287),(32,401,147,286),(33,400,148,285),(34,399,149,284),(35,398,150,283),(36,397,151,282),(37,396,152,281),(38,395,153,280),(39,394,154,279),(40,393,155,278),(41,392,156,277),(42,391,157,276),(43,390,158,275),(44,389,159,274),(45,388,160,273),(46,387,161,272),(47,386,162,271),(48,385,163,270),(49,384,164,269),(50,383,165,268),(51,382,166,267),(52,381,167,266),(53,380,168,265),(54,379,169,264),(55,378,170,263),(56,377,171,262),(57,376,172,261),(58,375,173,260),(59,374,174,259),(60,373,175,258),(61,372,176,257),(62,371,177,256),(63,370,178,255),(64,369,179,254),(65,368,180,253),(66,367,181,252),(67,366,182,251),(68,365,183,250),(69,364,184,249),(70,363,185,248),(71,362,186,247),(72,361,187,246),(73,360,188,245),(74,359,189,244),(75,358,190,243),(76,357,191,242),(77,356,192,241),(78,355,193,240),(79,354,194,239),(80,353,195,238),(81,352,196,237),(82,351,197,236),(83,350,198,235),(84,349,199,234),(85,348,200,233),(86,347,201,232),(87,346,202,231),(88,345,203,460),(89,344,204,459),(90,343,205,458),(91,342,206,457),(92,341,207,456),(93,340,208,455),(94,339,209,454),(95,338,210,453),(96,337,211,452),(97,336,212,451),(98,335,213,450),(99,334,214,449),(100,333,215,448),(101,332,216,447),(102,331,217,446),(103,330,218,445),(104,329,219,444),(105,328,220,443),(106,327,221,442),(107,326,222,441),(108,325,223,440),(109,324,224,439),(110,323,225,438),(111,322,226,437),(112,321,227,436),(113,320,228,435),(114,319,229,434),(115,318,230,433)])

118 conjugacy classes

class 1  2 4A4B5A5B10A10B23A···23K46A···46K115A···115AR230A···230AR
order124455101023···2346···46115···115230···230
size1111511522222···22···22···22···2

118 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4D5Dic5D23Dic23D115Dic115
kernelDic115C230C115C46C23C10C5C2C1
# reps1122211114444

Matrix representation of Dic115 in GL3(𝔽461) generated by

46000
0332275
0186211
,
41300
043221
042129
G:=sub<GL(3,GF(461))| [460,0,0,0,332,186,0,275,211],[413,0,0,0,432,421,0,21,29] >;

Dic115 in GAP, Magma, Sage, TeX

{\rm Dic}_{115}
% in TeX

G:=Group("Dic115");
// GroupNames label

G:=SmallGroup(460,3);
// by ID

G=gap.SmallGroup(460,3);
# by ID

G:=PCGroup([4,-2,-2,-5,-23,8,194,7043]);
// Polycyclic

G:=Group<a,b|a^230=1,b^2=a^115,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic115 in TeX

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