metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
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
| C81 — Dic81 |
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)
(1 164 82 245)(2 163 83 244)(3 324 84 243)(4 323 85 242)(5 322 86 241)(6 321 87 240)(7 320 88 239)(8 319 89 238)(9 318 90 237)(10 317 91 236)(11 316 92 235)(12 315 93 234)(13 314 94 233)(14 313 95 232)(15 312 96 231)(16 311 97 230)(17 310 98 229)(18 309 99 228)(19 308 100 227)(20 307 101 226)(21 306 102 225)(22 305 103 224)(23 304 104 223)(24 303 105 222)(25 302 106 221)(26 301 107 220)(27 300 108 219)(28 299 109 218)(29 298 110 217)(30 297 111 216)(31 296 112 215)(32 295 113 214)(33 294 114 213)(34 293 115 212)(35 292 116 211)(36 291 117 210)(37 290 118 209)(38 289 119 208)(39 288 120 207)(40 287 121 206)(41 286 122 205)(42 285 123 204)(43 284 124 203)(44 283 125 202)(45 282 126 201)(46 281 127 200)(47 280 128 199)(48 279 129 198)(49 278 130 197)(50 277 131 196)(51 276 132 195)(52 275 133 194)(53 274 134 193)(54 273 135 192)(55 272 136 191)(56 271 137 190)(57 270 138 189)(58 269 139 188)(59 268 140 187)(60 267 141 186)(61 266 142 185)(62 265 143 184)(63 264 144 183)(64 263 145 182)(65 262 146 181)(66 261 147 180)(67 260 148 179)(68 259 149 178)(69 258 150 177)(70 257 151 176)(71 256 152 175)(72 255 153 174)(73 254 154 173)(74 253 155 172)(75 252 156 171)(76 251 157 170)(77 250 158 169)(78 249 159 168)(79 248 160 167)(80 247 161 166)(81 246 162 165)
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), (1,164,82,245)(2,163,83,244)(3,324,84,243)(4,323,85,242)(5,322,86,241)(6,321,87,240)(7,320,88,239)(8,319,89,238)(9,318,90,237)(10,317,91,236)(11,316,92,235)(12,315,93,234)(13,314,94,233)(14,313,95,232)(15,312,96,231)(16,311,97,230)(17,310,98,229)(18,309,99,228)(19,308,100,227)(20,307,101,226)(21,306,102,225)(22,305,103,224)(23,304,104,223)(24,303,105,222)(25,302,106,221)(26,301,107,220)(27,300,108,219)(28,299,109,218)(29,298,110,217)(30,297,111,216)(31,296,112,215)(32,295,113,214)(33,294,114,213)(34,293,115,212)(35,292,116,211)(36,291,117,210)(37,290,118,209)(38,289,119,208)(39,288,120,207)(40,287,121,206)(41,286,122,205)(42,285,123,204)(43,284,124,203)(44,283,125,202)(45,282,126,201)(46,281,127,200)(47,280,128,199)(48,279,129,198)(49,278,130,197)(50,277,131,196)(51,276,132,195)(52,275,133,194)(53,274,134,193)(54,273,135,192)(55,272,136,191)(56,271,137,190)(57,270,138,189)(58,269,139,188)(59,268,140,187)(60,267,141,186)(61,266,142,185)(62,265,143,184)(63,264,144,183)(64,263,145,182)(65,262,146,181)(66,261,147,180)(67,260,148,179)(68,259,149,178)(69,258,150,177)(70,257,151,176)(71,256,152,175)(72,255,153,174)(73,254,154,173)(74,253,155,172)(75,252,156,171)(76,251,157,170)(77,250,158,169)(78,249,159,168)(79,248,160,167)(80,247,161,166)(81,246,162,165)>;
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), (1,164,82,245)(2,163,83,244)(3,324,84,243)(4,323,85,242)(5,322,86,241)(6,321,87,240)(7,320,88,239)(8,319,89,238)(9,318,90,237)(10,317,91,236)(11,316,92,235)(12,315,93,234)(13,314,94,233)(14,313,95,232)(15,312,96,231)(16,311,97,230)(17,310,98,229)(18,309,99,228)(19,308,100,227)(20,307,101,226)(21,306,102,225)(22,305,103,224)(23,304,104,223)(24,303,105,222)(25,302,106,221)(26,301,107,220)(27,300,108,219)(28,299,109,218)(29,298,110,217)(30,297,111,216)(31,296,112,215)(32,295,113,214)(33,294,114,213)(34,293,115,212)(35,292,116,211)(36,291,117,210)(37,290,118,209)(38,289,119,208)(39,288,120,207)(40,287,121,206)(41,286,122,205)(42,285,123,204)(43,284,124,203)(44,283,125,202)(45,282,126,201)(46,281,127,200)(47,280,128,199)(48,279,129,198)(49,278,130,197)(50,277,131,196)(51,276,132,195)(52,275,133,194)(53,274,134,193)(54,273,135,192)(55,272,136,191)(56,271,137,190)(57,270,138,189)(58,269,139,188)(59,268,140,187)(60,267,141,186)(61,266,142,185)(62,265,143,184)(63,264,144,183)(64,263,145,182)(65,262,146,181)(66,261,147,180)(67,260,148,179)(68,259,149,178)(69,258,150,177)(70,257,151,176)(71,256,152,175)(72,255,153,174)(73,254,154,173)(74,253,155,172)(75,252,156,171)(76,251,157,170)(77,250,158,169)(78,249,159,168)(79,248,160,167)(80,247,161,166)(81,246,162,165) );
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)], [(1,164,82,245),(2,163,83,244),(3,324,84,243),(4,323,85,242),(5,322,86,241),(6,321,87,240),(7,320,88,239),(8,319,89,238),(9,318,90,237),(10,317,91,236),(11,316,92,235),(12,315,93,234),(13,314,94,233),(14,313,95,232),(15,312,96,231),(16,311,97,230),(17,310,98,229),(18,309,99,228),(19,308,100,227),(20,307,101,226),(21,306,102,225),(22,305,103,224),(23,304,104,223),(24,303,105,222),(25,302,106,221),(26,301,107,220),(27,300,108,219),(28,299,109,218),(29,298,110,217),(30,297,111,216),(31,296,112,215),(32,295,113,214),(33,294,114,213),(34,293,115,212),(35,292,116,211),(36,291,117,210),(37,290,118,209),(38,289,119,208),(39,288,120,207),(40,287,121,206),(41,286,122,205),(42,285,123,204),(43,284,124,203),(44,283,125,202),(45,282,126,201),(46,281,127,200),(47,280,128,199),(48,279,129,198),(49,278,130,197),(50,277,131,196),(51,276,132,195),(52,275,133,194),(53,274,134,193),(54,273,135,192),(55,272,136,191),(56,271,137,190),(57,270,138,189),(58,269,139,188),(59,268,140,187),(60,267,141,186),(61,266,142,185),(62,265,143,184),(63,264,144,183),(64,263,145,182),(65,262,146,181),(66,261,147,180),(67,260,148,179),(68,259,149,178),(69,258,150,177),(70,257,151,176),(71,256,152,175),(72,255,153,174),(73,254,154,173),(74,253,155,172),(75,252,156,171),(76,251,157,170),(77,250,158,169),(78,249,159,168),(79,248,160,167),(80,247,161,166),(81,246,162,165)]])
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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