metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic82, C41⋊Q8, C4.D41, C2.3D82, C164.1C2, C82.1C22, Dic41.1C2, SmallGroup(328,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic82
G = < a,b | a164=1, b2=a82, bab-1=a-1 >
Smallest permutation representation of Dic82
►Regular action on 328 points
Generators in S328
(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)
(1 279 83 197)(2 278 84 196)(3 277 85 195)(4 276 86 194)(5 275 87 193)(6 274 88 192)(7 273 89 191)(8 272 90 190)(9 271 91 189)(10 270 92 188)(11 269 93 187)(12 268 94 186)(13 267 95 185)(14 266 96 184)(15 265 97 183)(16 264 98 182)(17 263 99 181)(18 262 100 180)(19 261 101 179)(20 260 102 178)(21 259 103 177)(22 258 104 176)(23 257 105 175)(24 256 106 174)(25 255 107 173)(26 254 108 172)(27 253 109 171)(28 252 110 170)(29 251 111 169)(30 250 112 168)(31 249 113 167)(32 248 114 166)(33 247 115 165)(34 246 116 328)(35 245 117 327)(36 244 118 326)(37 243 119 325)(38 242 120 324)(39 241 121 323)(40 240 122 322)(41 239 123 321)(42 238 124 320)(43 237 125 319)(44 236 126 318)(45 235 127 317)(46 234 128 316)(47 233 129 315)(48 232 130 314)(49 231 131 313)(50 230 132 312)(51 229 133 311)(52 228 134 310)(53 227 135 309)(54 226 136 308)(55 225 137 307)(56 224 138 306)(57 223 139 305)(58 222 140 304)(59 221 141 303)(60 220 142 302)(61 219 143 301)(62 218 144 300)(63 217 145 299)(64 216 146 298)(65 215 147 297)(66 214 148 296)(67 213 149 295)(68 212 150 294)(69 211 151 293)(70 210 152 292)(71 209 153 291)(72 208 154 290)(73 207 155 289)(74 206 156 288)(75 205 157 287)(76 204 158 286)(77 203 159 285)(78 202 160 284)(79 201 161 283)(80 200 162 282)(81 199 163 281)(82 198 164 280)
G:=sub<Sym(328)| (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), (1,279,83,197)(2,278,84,196)(3,277,85,195)(4,276,86,194)(5,275,87,193)(6,274,88,192)(7,273,89,191)(8,272,90,190)(9,271,91,189)(10,270,92,188)(11,269,93,187)(12,268,94,186)(13,267,95,185)(14,266,96,184)(15,265,97,183)(16,264,98,182)(17,263,99,181)(18,262,100,180)(19,261,101,179)(20,260,102,178)(21,259,103,177)(22,258,104,176)(23,257,105,175)(24,256,106,174)(25,255,107,173)(26,254,108,172)(27,253,109,171)(28,252,110,170)(29,251,111,169)(30,250,112,168)(31,249,113,167)(32,248,114,166)(33,247,115,165)(34,246,116,328)(35,245,117,327)(36,244,118,326)(37,243,119,325)(38,242,120,324)(39,241,121,323)(40,240,122,322)(41,239,123,321)(42,238,124,320)(43,237,125,319)(44,236,126,318)(45,235,127,317)(46,234,128,316)(47,233,129,315)(48,232,130,314)(49,231,131,313)(50,230,132,312)(51,229,133,311)(52,228,134,310)(53,227,135,309)(54,226,136,308)(55,225,137,307)(56,224,138,306)(57,223,139,305)(58,222,140,304)(59,221,141,303)(60,220,142,302)(61,219,143,301)(62,218,144,300)(63,217,145,299)(64,216,146,298)(65,215,147,297)(66,214,148,296)(67,213,149,295)(68,212,150,294)(69,211,151,293)(70,210,152,292)(71,209,153,291)(72,208,154,290)(73,207,155,289)(74,206,156,288)(75,205,157,287)(76,204,158,286)(77,203,159,285)(78,202,160,284)(79,201,161,283)(80,200,162,282)(81,199,163,281)(82,198,164,280)>;
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), (1,279,83,197)(2,278,84,196)(3,277,85,195)(4,276,86,194)(5,275,87,193)(6,274,88,192)(7,273,89,191)(8,272,90,190)(9,271,91,189)(10,270,92,188)(11,269,93,187)(12,268,94,186)(13,267,95,185)(14,266,96,184)(15,265,97,183)(16,264,98,182)(17,263,99,181)(18,262,100,180)(19,261,101,179)(20,260,102,178)(21,259,103,177)(22,258,104,176)(23,257,105,175)(24,256,106,174)(25,255,107,173)(26,254,108,172)(27,253,109,171)(28,252,110,170)(29,251,111,169)(30,250,112,168)(31,249,113,167)(32,248,114,166)(33,247,115,165)(34,246,116,328)(35,245,117,327)(36,244,118,326)(37,243,119,325)(38,242,120,324)(39,241,121,323)(40,240,122,322)(41,239,123,321)(42,238,124,320)(43,237,125,319)(44,236,126,318)(45,235,127,317)(46,234,128,316)(47,233,129,315)(48,232,130,314)(49,231,131,313)(50,230,132,312)(51,229,133,311)(52,228,134,310)(53,227,135,309)(54,226,136,308)(55,225,137,307)(56,224,138,306)(57,223,139,305)(58,222,140,304)(59,221,141,303)(60,220,142,302)(61,219,143,301)(62,218,144,300)(63,217,145,299)(64,216,146,298)(65,215,147,297)(66,214,148,296)(67,213,149,295)(68,212,150,294)(69,211,151,293)(70,210,152,292)(71,209,153,291)(72,208,154,290)(73,207,155,289)(74,206,156,288)(75,205,157,287)(76,204,158,286)(77,203,159,285)(78,202,160,284)(79,201,161,283)(80,200,162,282)(81,199,163,281)(82,198,164,280) );
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)], [(1,279,83,197),(2,278,84,196),(3,277,85,195),(4,276,86,194),(5,275,87,193),(6,274,88,192),(7,273,89,191),(8,272,90,190),(9,271,91,189),(10,270,92,188),(11,269,93,187),(12,268,94,186),(13,267,95,185),(14,266,96,184),(15,265,97,183),(16,264,98,182),(17,263,99,181),(18,262,100,180),(19,261,101,179),(20,260,102,178),(21,259,103,177),(22,258,104,176),(23,257,105,175),(24,256,106,174),(25,255,107,173),(26,254,108,172),(27,253,109,171),(28,252,110,170),(29,251,111,169),(30,250,112,168),(31,249,113,167),(32,248,114,166),(33,247,115,165),(34,246,116,328),(35,245,117,327),(36,244,118,326),(37,243,119,325),(38,242,120,324),(39,241,121,323),(40,240,122,322),(41,239,123,321),(42,238,124,320),(43,237,125,319),(44,236,126,318),(45,235,127,317),(46,234,128,316),(47,233,129,315),(48,232,130,314),(49,231,131,313),(50,230,132,312),(51,229,133,311),(52,228,134,310),(53,227,135,309),(54,226,136,308),(55,225,137,307),(56,224,138,306),(57,223,139,305),(58,222,140,304),(59,221,141,303),(60,220,142,302),(61,219,143,301),(62,218,144,300),(63,217,145,299),(64,216,146,298),(65,215,147,297),(66,214,148,296),(67,213,149,295),(68,212,150,294),(69,211,151,293),(70,210,152,292),(71,209,153,291),(72,208,154,290),(73,207,155,289),(74,206,156,288),(75,205,157,287),(76,204,158,286),(77,203,159,285),(78,202,160,284),(79,201,161,283),(80,200,162,282),(81,199,163,281),(82,198,164,280)]])
85 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 41A | ··· | 41T | 82A | ··· | 82T | 164A | ··· | 164AN |
| order | 1 | 2 | 4 | 4 | 4 | 41 | ··· | 41 | 82 | ··· | 82 | 164 | ··· | 164 |
| size | 1 | 1 | 2 | 82 | 82 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D41 | D82 | Dic82 |
| kernel | Dic82 | Dic41 | C164 | C41 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 20 | 20 | 40 |
Matrix representation of Dic82 ►in GL2(𝔽821) generated by
| 5 | 323 |
| 761 | 65 |
| 362 | 553 |
| 780 | 459 |
G:=sub<GL(2,GF(821))| [5,761,323,65],[362,780,553,459] >;
Dic82 in GAP, Magma, Sage, TeX
{\rm Dic}_{82} % in TeX
G:=Group("Dic82"); // GroupNames label
G:=SmallGroup(328,4);
// by ID
G=gap.SmallGroup(328,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-41,16,49,21,5123]);
// Polycyclic
G:=Group<a,b|a^164=1,b^2=a^82,b*a*b^-1=a^-1>;
// generators/relations
Export