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## G = Dic81order 324 = 22·34

### Dicyclic group

Aliases: Dic81, C81⋊C4, C2.D81, C162.C2, C54.1S3, C18.1D9, C6.1D27, C3.Dic27, C27.Dic3, C9.1Dic9, SmallGroup(324,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C81 — Dic81
 Chief series C1 — C3 — C9 — C27 — C81 — C162 — Dic81
 Lower central C81 — Dic81
 Upper central C1 — C2

Generators and relations for Dic81
G = < a,b | a162=1, b2=a81, bab-1=a-1 >

Smallest permutation representation of Dic81
Regular action on 324 points
Generators in S324
(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)
(1 183 82 264)(2 182 83 263)(3 181 84 262)(4 180 85 261)(5 179 86 260)(6 178 87 259)(7 177 88 258)(8 176 89 257)(9 175 90 256)(10 174 91 255)(11 173 92 254)(12 172 93 253)(13 171 94 252)(14 170 95 251)(15 169 96 250)(16 168 97 249)(17 167 98 248)(18 166 99 247)(19 165 100 246)(20 164 101 245)(21 163 102 244)(22 324 103 243)(23 323 104 242)(24 322 105 241)(25 321 106 240)(26 320 107 239)(27 319 108 238)(28 318 109 237)(29 317 110 236)(30 316 111 235)(31 315 112 234)(32 314 113 233)(33 313 114 232)(34 312 115 231)(35 311 116 230)(36 310 117 229)(37 309 118 228)(38 308 119 227)(39 307 120 226)(40 306 121 225)(41 305 122 224)(42 304 123 223)(43 303 124 222)(44 302 125 221)(45 301 126 220)(46 300 127 219)(47 299 128 218)(48 298 129 217)(49 297 130 216)(50 296 131 215)(51 295 132 214)(52 294 133 213)(53 293 134 212)(54 292 135 211)(55 291 136 210)(56 290 137 209)(57 289 138 208)(58 288 139 207)(59 287 140 206)(60 286 141 205)(61 285 142 204)(62 284 143 203)(63 283 144 202)(64 282 145 201)(65 281 146 200)(66 280 147 199)(67 279 148 198)(68 278 149 197)(69 277 150 196)(70 276 151 195)(71 275 152 194)(72 274 153 193)(73 273 154 192)(74 272 155 191)(75 271 156 190)(76 270 157 189)(77 269 158 188)(78 268 159 187)(79 267 160 186)(80 266 161 185)(81 265 162 184)

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

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

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

84 conjugacy classes

 class 1 2 3 4A 4B 6 9A 9B 9C 18A 18B 18C 27A ··· 27I 54A ··· 54I 81A ··· 81AA 162A ··· 162AA order 1 2 3 4 4 6 9 9 9 18 18 18 27 ··· 27 54 ··· 54 81 ··· 81 162 ··· 162 size 1 1 2 81 81 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 type + + + - + - + - + - image C1 C2 C4 S3 Dic3 D9 Dic9 D27 Dic27 D81 Dic81 kernel Dic81 C162 C81 C54 C27 C18 C9 C6 C3 C2 C1 # reps 1 1 2 1 1 3 3 9 9 27 27

Matrix representation of Dic81 in GL2(𝔽1297) generated by

 188 431 866 1054
,
 256 248 1289 1041
G:=sub<GL(2,GF(1297))| [188,866,431,1054],[256,1289,248,1041] >;

Dic81 in GAP, Magma, Sage, TeX

{\rm Dic}_{81}
% in TeX

G:=Group("Dic81");
// GroupNames label

G:=SmallGroup(324,1);
// by ID

G=gap.SmallGroup(324,1);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,362,284,1443,381,5404,208,7781]);
// Polycyclic

G:=Group<a,b|a^162=1,b^2=a^81,b*a*b^-1=a^-1>;
// generators/relations

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