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G = Dic102order 408 = 23·3·17

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic102, C4.D51, C512Q8, C68.1S3, C34.8D6, C6.8D34, C172Dic6, C32Dic34, C204.1C2, C12.1D17, C2.3D102, C102.8C22, Dic51.1C2, SmallGroup(408,25)

Series: Derived Chief Lower central Upper central

C1C102 — Dic102
C1C17C51C102Dic51 — Dic102
C51C102 — Dic102
C1C2C4

Generators and relations for Dic102
 G = < a,b | a204=1, b2=a102, bab-1=a-1 >

51C4
51C4
51Q8
17Dic3
17Dic3
3Dic17
3Dic17
17Dic6
3Dic34

Smallest permutation representation of Dic102
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)
(1 254 103 356)(2 253 104 355)(3 252 105 354)(4 251 106 353)(5 250 107 352)(6 249 108 351)(7 248 109 350)(8 247 110 349)(9 246 111 348)(10 245 112 347)(11 244 113 346)(12 243 114 345)(13 242 115 344)(14 241 116 343)(15 240 117 342)(16 239 118 341)(17 238 119 340)(18 237 120 339)(19 236 121 338)(20 235 122 337)(21 234 123 336)(22 233 124 335)(23 232 125 334)(24 231 126 333)(25 230 127 332)(26 229 128 331)(27 228 129 330)(28 227 130 329)(29 226 131 328)(30 225 132 327)(31 224 133 326)(32 223 134 325)(33 222 135 324)(34 221 136 323)(35 220 137 322)(36 219 138 321)(37 218 139 320)(38 217 140 319)(39 216 141 318)(40 215 142 317)(41 214 143 316)(42 213 144 315)(43 212 145 314)(44 211 146 313)(45 210 147 312)(46 209 148 311)(47 208 149 310)(48 207 150 309)(49 206 151 308)(50 205 152 307)(51 408 153 306)(52 407 154 305)(53 406 155 304)(54 405 156 303)(55 404 157 302)(56 403 158 301)(57 402 159 300)(58 401 160 299)(59 400 161 298)(60 399 162 297)(61 398 163 296)(62 397 164 295)(63 396 165 294)(64 395 166 293)(65 394 167 292)(66 393 168 291)(67 392 169 290)(68 391 170 289)(69 390 171 288)(70 389 172 287)(71 388 173 286)(72 387 174 285)(73 386 175 284)(74 385 176 283)(75 384 177 282)(76 383 178 281)(77 382 179 280)(78 381 180 279)(79 380 181 278)(80 379 182 277)(81 378 183 276)(82 377 184 275)(83 376 185 274)(84 375 186 273)(85 374 187 272)(86 373 188 271)(87 372 189 270)(88 371 190 269)(89 370 191 268)(90 369 192 267)(91 368 193 266)(92 367 194 265)(93 366 195 264)(94 365 196 263)(95 364 197 262)(96 363 198 261)(97 362 199 260)(98 361 200 259)(99 360 201 258)(100 359 202 257)(101 358 203 256)(102 357 204 255)

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

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

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

105 conjugacy classes

class 1  2  3 4A4B4C 6 12A12B17A···17H34A···34H51A···51P68A···68P102A···102P204A···204AF
order1234446121217···1734···3451···5168···68102···102204···204
size11221021022222···22···22···22···22···22···2

105 irreducible representations

dim1112222222222
type++++-+-+++-+-
imageC1C2C2S3Q8D6Dic6D17D34D51Dic34D102Dic102
kernelDic102Dic51C204C68C51C34C17C12C6C4C3C2C1
# reps12111128816161632

Matrix representation of Dic102 in GL2(𝔽409) generated by

281257
34277
,
109238
36300
G:=sub<GL(2,GF(409))| [281,342,257,77],[109,36,238,300] >;

Dic102 in GAP, Magma, Sage, TeX

{\rm Dic}_{102}
% in TeX

G:=Group("Dic102");
// GroupNames label

G:=SmallGroup(408,25);
// by ID

G=gap.SmallGroup(408,25);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,20,61,26,323,9604]);
// Polycyclic

G:=Group<a,b|a^204=1,b^2=a^102,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic102 in TeX

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