Copied to
clipboard

G = Dic85order 340 = 22·5·17

Dicyclic group

Aliases: Dic85, C857C4, C2.D85, C34.D5, C10.D17, C172Dic5, C52Dic17, C170.1C2, SmallGroup(340,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C85 — Dic85
 Chief series C1 — C17 — C85 — C170 — Dic85
 Lower central C85 — Dic85
 Upper central C1 — C2

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

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

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

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

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

Export

׿
×
𝔽