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G = Dic101order 404 = 22·101

Dicyclic group

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

Aliases: Dic101, C1012C4, C202.C2, C2.D101, SmallGroup(404,1)

Series: Derived Chief Lower central Upper central

C1C101 — Dic101
C1C101C202 — Dic101
C101 — Dic101
C1C2

Generators and relations for Dic101
 G = < a,b | a202=1, b2=a101, bab-1=a-1 >

101C4

Smallest permutation representation of Dic101
Regular action on 404 points
Generators in S404
(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)
(1 325 102 224)(2 324 103 223)(3 323 104 222)(4 322 105 221)(5 321 106 220)(6 320 107 219)(7 319 108 218)(8 318 109 217)(9 317 110 216)(10 316 111 215)(11 315 112 214)(12 314 113 213)(13 313 114 212)(14 312 115 211)(15 311 116 210)(16 310 117 209)(17 309 118 208)(18 308 119 207)(19 307 120 206)(20 306 121 205)(21 305 122 204)(22 304 123 203)(23 303 124 404)(24 302 125 403)(25 301 126 402)(26 300 127 401)(27 299 128 400)(28 298 129 399)(29 297 130 398)(30 296 131 397)(31 295 132 396)(32 294 133 395)(33 293 134 394)(34 292 135 393)(35 291 136 392)(36 290 137 391)(37 289 138 390)(38 288 139 389)(39 287 140 388)(40 286 141 387)(41 285 142 386)(42 284 143 385)(43 283 144 384)(44 282 145 383)(45 281 146 382)(46 280 147 381)(47 279 148 380)(48 278 149 379)(49 277 150 378)(50 276 151 377)(51 275 152 376)(52 274 153 375)(53 273 154 374)(54 272 155 373)(55 271 156 372)(56 270 157 371)(57 269 158 370)(58 268 159 369)(59 267 160 368)(60 266 161 367)(61 265 162 366)(62 264 163 365)(63 263 164 364)(64 262 165 363)(65 261 166 362)(66 260 167 361)(67 259 168 360)(68 258 169 359)(69 257 170 358)(70 256 171 357)(71 255 172 356)(72 254 173 355)(73 253 174 354)(74 252 175 353)(75 251 176 352)(76 250 177 351)(77 249 178 350)(78 248 179 349)(79 247 180 348)(80 246 181 347)(81 245 182 346)(82 244 183 345)(83 243 184 344)(84 242 185 343)(85 241 186 342)(86 240 187 341)(87 239 188 340)(88 238 189 339)(89 237 190 338)(90 236 191 337)(91 235 192 336)(92 234 193 335)(93 233 194 334)(94 232 195 333)(95 231 196 332)(96 230 197 331)(97 229 198 330)(98 228 199 329)(99 227 200 328)(100 226 201 327)(101 225 202 326)

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

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

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

104 conjugacy classes

class 1  2 4A4B101A···101AX202A···202AX
order1244101···101202···202
size111011012···22···2

104 irreducible representations

dim11122
type+++-
imageC1C2C4D101Dic101
kernelDic101C202C101C2C1
# reps1125050

Matrix representation of Dic101 in GL2(𝔽809) generated by

2201
8080
,
323385
517486
G:=sub<GL(2,GF(809))| [220,808,1,0],[323,517,385,486] >;

Dic101 in GAP, Magma, Sage, TeX

{\rm Dic}_{101}
% in TeX

G:=Group("Dic101");
// GroupNames label

G:=SmallGroup(404,1);
// by ID

G=gap.SmallGroup(404,1);
# by ID

G:=PCGroup([3,-2,-2,-101,6,3602]);
// Polycyclic

G:=Group<a,b|a^202=1,b^2=a^101,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic101 in TeX

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