metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic85, C85⋊7C4, C2.D85, C34.D5, C10.D17, C17⋊2Dic5, C5⋊2Dic17, C170.1C2, SmallGroup(340,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C85 — Dic85 |
Generators and relations for Dic85
G = < a,b | a170=1, b2=a85, bab-1=a-1 >
Smallest permutation representation of Dic85
►Regular action on 340 points
Generators in S340
(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)
(1 308 86 223)(2 307 87 222)(3 306 88 221)(4 305 89 220)(5 304 90 219)(6 303 91 218)(7 302 92 217)(8 301 93 216)(9 300 94 215)(10 299 95 214)(11 298 96 213)(12 297 97 212)(13 296 98 211)(14 295 99 210)(15 294 100 209)(16 293 101 208)(17 292 102 207)(18 291 103 206)(19 290 104 205)(20 289 105 204)(21 288 106 203)(22 287 107 202)(23 286 108 201)(24 285 109 200)(25 284 110 199)(26 283 111 198)(27 282 112 197)(28 281 113 196)(29 280 114 195)(30 279 115 194)(31 278 116 193)(32 277 117 192)(33 276 118 191)(34 275 119 190)(35 274 120 189)(36 273 121 188)(37 272 122 187)(38 271 123 186)(39 270 124 185)(40 269 125 184)(41 268 126 183)(42 267 127 182)(43 266 128 181)(44 265 129 180)(45 264 130 179)(46 263 131 178)(47 262 132 177)(48 261 133 176)(49 260 134 175)(50 259 135 174)(51 258 136 173)(52 257 137 172)(53 256 138 171)(54 255 139 340)(55 254 140 339)(56 253 141 338)(57 252 142 337)(58 251 143 336)(59 250 144 335)(60 249 145 334)(61 248 146 333)(62 247 147 332)(63 246 148 331)(64 245 149 330)(65 244 150 329)(66 243 151 328)(67 242 152 327)(68 241 153 326)(69 240 154 325)(70 239 155 324)(71 238 156 323)(72 237 157 322)(73 236 158 321)(74 235 159 320)(75 234 160 319)(76 233 161 318)(77 232 162 317)(78 231 163 316)(79 230 164 315)(80 229 165 314)(81 228 166 313)(82 227 167 312)(83 226 168 311)(84 225 169 310)(85 224 170 309)
G:=sub<Sym(340)| (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), (1,308,86,223)(2,307,87,222)(3,306,88,221)(4,305,89,220)(5,304,90,219)(6,303,91,218)(7,302,92,217)(8,301,93,216)(9,300,94,215)(10,299,95,214)(11,298,96,213)(12,297,97,212)(13,296,98,211)(14,295,99,210)(15,294,100,209)(16,293,101,208)(17,292,102,207)(18,291,103,206)(19,290,104,205)(20,289,105,204)(21,288,106,203)(22,287,107,202)(23,286,108,201)(24,285,109,200)(25,284,110,199)(26,283,111,198)(27,282,112,197)(28,281,113,196)(29,280,114,195)(30,279,115,194)(31,278,116,193)(32,277,117,192)(33,276,118,191)(34,275,119,190)(35,274,120,189)(36,273,121,188)(37,272,122,187)(38,271,123,186)(39,270,124,185)(40,269,125,184)(41,268,126,183)(42,267,127,182)(43,266,128,181)(44,265,129,180)(45,264,130,179)(46,263,131,178)(47,262,132,177)(48,261,133,176)(49,260,134,175)(50,259,135,174)(51,258,136,173)(52,257,137,172)(53,256,138,171)(54,255,139,340)(55,254,140,339)(56,253,141,338)(57,252,142,337)(58,251,143,336)(59,250,144,335)(60,249,145,334)(61,248,146,333)(62,247,147,332)(63,246,148,331)(64,245,149,330)(65,244,150,329)(66,243,151,328)(67,242,152,327)(68,241,153,326)(69,240,154,325)(70,239,155,324)(71,238,156,323)(72,237,157,322)(73,236,158,321)(74,235,159,320)(75,234,160,319)(76,233,161,318)(77,232,162,317)(78,231,163,316)(79,230,164,315)(80,229,165,314)(81,228,166,313)(82,227,167,312)(83,226,168,311)(84,225,169,310)(85,224,170,309)>;
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), (1,308,86,223)(2,307,87,222)(3,306,88,221)(4,305,89,220)(5,304,90,219)(6,303,91,218)(7,302,92,217)(8,301,93,216)(9,300,94,215)(10,299,95,214)(11,298,96,213)(12,297,97,212)(13,296,98,211)(14,295,99,210)(15,294,100,209)(16,293,101,208)(17,292,102,207)(18,291,103,206)(19,290,104,205)(20,289,105,204)(21,288,106,203)(22,287,107,202)(23,286,108,201)(24,285,109,200)(25,284,110,199)(26,283,111,198)(27,282,112,197)(28,281,113,196)(29,280,114,195)(30,279,115,194)(31,278,116,193)(32,277,117,192)(33,276,118,191)(34,275,119,190)(35,274,120,189)(36,273,121,188)(37,272,122,187)(38,271,123,186)(39,270,124,185)(40,269,125,184)(41,268,126,183)(42,267,127,182)(43,266,128,181)(44,265,129,180)(45,264,130,179)(46,263,131,178)(47,262,132,177)(48,261,133,176)(49,260,134,175)(50,259,135,174)(51,258,136,173)(52,257,137,172)(53,256,138,171)(54,255,139,340)(55,254,140,339)(56,253,141,338)(57,252,142,337)(58,251,143,336)(59,250,144,335)(60,249,145,334)(61,248,146,333)(62,247,147,332)(63,246,148,331)(64,245,149,330)(65,244,150,329)(66,243,151,328)(67,242,152,327)(68,241,153,326)(69,240,154,325)(70,239,155,324)(71,238,156,323)(72,237,157,322)(73,236,158,321)(74,235,159,320)(75,234,160,319)(76,233,161,318)(77,232,162,317)(78,231,163,316)(79,230,164,315)(80,229,165,314)(81,228,166,313)(82,227,167,312)(83,226,168,311)(84,225,169,310)(85,224,170,309) );
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)], [(1,308,86,223),(2,307,87,222),(3,306,88,221),(4,305,89,220),(5,304,90,219),(6,303,91,218),(7,302,92,217),(8,301,93,216),(9,300,94,215),(10,299,95,214),(11,298,96,213),(12,297,97,212),(13,296,98,211),(14,295,99,210),(15,294,100,209),(16,293,101,208),(17,292,102,207),(18,291,103,206),(19,290,104,205),(20,289,105,204),(21,288,106,203),(22,287,107,202),(23,286,108,201),(24,285,109,200),(25,284,110,199),(26,283,111,198),(27,282,112,197),(28,281,113,196),(29,280,114,195),(30,279,115,194),(31,278,116,193),(32,277,117,192),(33,276,118,191),(34,275,119,190),(35,274,120,189),(36,273,121,188),(37,272,122,187),(38,271,123,186),(39,270,124,185),(40,269,125,184),(41,268,126,183),(42,267,127,182),(43,266,128,181),(44,265,129,180),(45,264,130,179),(46,263,131,178),(47,262,132,177),(48,261,133,176),(49,260,134,175),(50,259,135,174),(51,258,136,173),(52,257,137,172),(53,256,138,171),(54,255,139,340),(55,254,140,339),(56,253,141,338),(57,252,142,337),(58,251,143,336),(59,250,144,335),(60,249,145,334),(61,248,146,333),(62,247,147,332),(63,246,148,331),(64,245,149,330),(65,244,150,329),(66,243,151,328),(67,242,152,327),(68,241,153,326),(69,240,154,325),(70,239,155,324),(71,238,156,323),(72,237,157,322),(73,236,158,321),(74,235,159,320),(75,234,160,319),(76,233,161,318),(77,232,162,317),(78,231,163,316),(79,230,164,315),(80,229,165,314),(81,228,166,313),(82,227,167,312),(83,226,168,311),(84,225,169,310),(85,224,170,309)]])
88 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 17A | ··· | 17H | 34A | ··· | 34H | 85A | ··· | 85AF | 170A | ··· | 170AF |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 17 | ··· | 17 | 34 | ··· | 34 | 85 | ··· | 85 | 170 | ··· | 170 |
| size | 1 | 1 | 85 | 85 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | D5 | Dic5 | D17 | Dic17 | D85 | Dic85 |
| kernel | Dic85 | C170 | C85 | C34 | C17 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 32 | 32 |
Matrix representation of Dic85 ►in GL2(𝔽1021) generated by
| 602 | 383 |
| 638 | 238 |
| 962 | 59 |
| 339 | 59 |
G:=sub<GL(2,GF(1021))| [602,638,383,238],[962,339,59,59] >;
Dic85 in GAP, Magma, Sage, TeX
{\rm Dic}_{85} % in TeX
G:=Group("Dic85"); // GroupNames label
G:=SmallGroup(340,3);
// by ID
G=gap.SmallGroup(340,3);
# by ID
G:=PCGroup([4,-2,-2,-5,-17,8,194,5123]);
// Polycyclic
G:=Group<a,b|a^170=1,b^2=a^85,b*a*b^-1=a^-1>;
// generators/relations
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