Copied to
clipboard

## G = Dic99order 396 = 22·32·11

### Dicyclic group

Aliases: Dic99, C991C4, C2.D99, C22.D9, C11⋊Dic9, C9⋊Dic11, C18.D11, C66.1S3, C6.1D33, C3.Dic33, C198.1C2, C33.1Dic3, SmallGroup(396,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C99 — Dic99
 Chief series C1 — C3 — C33 — C99 — C198 — Dic99
 Lower central C99 — Dic99
 Upper central C1 — C2

Generators and relations for Dic99
G = < a,b | a198=1, b2=a99, bab-1=a-1 >

Smallest permutation representation of Dic99
Regular action on 396 points
Generators in S396
(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)
(1 329 100 230)(2 328 101 229)(3 327 102 228)(4 326 103 227)(5 325 104 226)(6 324 105 225)(7 323 106 224)(8 322 107 223)(9 321 108 222)(10 320 109 221)(11 319 110 220)(12 318 111 219)(13 317 112 218)(14 316 113 217)(15 315 114 216)(16 314 115 215)(17 313 116 214)(18 312 117 213)(19 311 118 212)(20 310 119 211)(21 309 120 210)(22 308 121 209)(23 307 122 208)(24 306 123 207)(25 305 124 206)(26 304 125 205)(27 303 126 204)(28 302 127 203)(29 301 128 202)(30 300 129 201)(31 299 130 200)(32 298 131 199)(33 297 132 396)(34 296 133 395)(35 295 134 394)(36 294 135 393)(37 293 136 392)(38 292 137 391)(39 291 138 390)(40 290 139 389)(41 289 140 388)(42 288 141 387)(43 287 142 386)(44 286 143 385)(45 285 144 384)(46 284 145 383)(47 283 146 382)(48 282 147 381)(49 281 148 380)(50 280 149 379)(51 279 150 378)(52 278 151 377)(53 277 152 376)(54 276 153 375)(55 275 154 374)(56 274 155 373)(57 273 156 372)(58 272 157 371)(59 271 158 370)(60 270 159 369)(61 269 160 368)(62 268 161 367)(63 267 162 366)(64 266 163 365)(65 265 164 364)(66 264 165 363)(67 263 166 362)(68 262 167 361)(69 261 168 360)(70 260 169 359)(71 259 170 358)(72 258 171 357)(73 257 172 356)(74 256 173 355)(75 255 174 354)(76 254 175 353)(77 253 176 352)(78 252 177 351)(79 251 178 350)(80 250 179 349)(81 249 180 348)(82 248 181 347)(83 247 182 346)(84 246 183 345)(85 245 184 344)(86 244 185 343)(87 243 186 342)(88 242 187 341)(89 241 188 340)(90 240 189 339)(91 239 190 338)(92 238 191 337)(93 237 192 336)(94 236 193 335)(95 235 194 334)(96 234 195 333)(97 233 196 332)(98 232 197 331)(99 231 198 330)

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

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

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

102 conjugacy classes

 class 1 2 3 4A 4B 6 9A 9B 9C 11A ··· 11E 18A 18B 18C 22A ··· 22E 33A ··· 33J 66A ··· 66J 99A ··· 99AD 198A ··· 198AD order 1 2 3 4 4 6 9 9 9 11 ··· 11 18 18 18 22 ··· 22 33 ··· 33 66 ··· 66 99 ··· 99 198 ··· 198 size 1 1 2 99 99 2 2 2 2 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + - + + - - + - + - image C1 C2 C4 S3 Dic3 D9 D11 Dic9 Dic11 D33 Dic33 D99 Dic99 kernel Dic99 C198 C99 C66 C33 C22 C18 C11 C9 C6 C3 C2 C1 # reps 1 1 2 1 1 3 5 3 5 10 10 30 30

Matrix representation of Dic99 in GL3(𝔽397) generated by

 396 0 0 0 369 230 0 167 202
,
 334 0 0 0 126 336 0 65 271
G:=sub<GL(3,GF(397))| [396,0,0,0,369,167,0,230,202],[334,0,0,0,126,65,0,336,271] >;

Dic99 in GAP, Magma, Sage, TeX

{\rm Dic}_{99}
% in TeX

G:=Group("Dic99");
// GroupNames label

G:=SmallGroup(396,3);
// by ID

G=gap.SmallGroup(396,3);
# by ID

G:=PCGroup([5,-2,-2,-3,-11,-3,10,2102,1002,2403,6604]);
// Polycyclic

G:=Group<a,b|a^198=1,b^2=a^99,b*a*b^-1=a^-1>;
// generators/relations

Export

׿
×
𝔽