direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17×Dic5, C85⋊9C4, C5⋊2C68, C10.C34, C34.2D5, C170.3C2, C2.(D5×C17), SmallGroup(340,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C17×Dic5 |
Generators and relations for C17×Dic5
G = < a,b,c | a17=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C17×Dic5
►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)
(1 280 110 240 101 195 270 322 329 302)(2 281 111 241 102 196 271 323 330 303)(3 282 112 242 86 197 272 307 331 304)(4 283 113 243 87 198 256 308 332 305)(5 284 114 244 88 199 257 309 333 306)(6 285 115 245 89 200 258 310 334 290)(7 286 116 246 90 201 259 311 335 291)(8 287 117 247 91 202 260 312 336 292)(9 288 118 248 92 203 261 313 337 293)(10 289 119 249 93 204 262 314 338 294)(11 273 103 250 94 188 263 315 339 295)(12 274 104 251 95 189 264 316 340 296)(13 275 105 252 96 190 265 317 324 297)(14 276 106 253 97 191 266 318 325 298)(15 277 107 254 98 192 267 319 326 299)(16 278 108 255 99 193 268 320 327 300)(17 279 109 239 100 194 269 321 328 301)(18 67 219 82 123 226 142 39 171 167)(19 68 220 83 124 227 143 40 172 168)(20 52 221 84 125 228 144 41 173 169)(21 53 205 85 126 229 145 42 174 170)(22 54 206 69 127 230 146 43 175 154)(23 55 207 70 128 231 147 44 176 155)(24 56 208 71 129 232 148 45 177 156)(25 57 209 72 130 233 149 46 178 157)(26 58 210 73 131 234 150 47 179 158)(27 59 211 74 132 235 151 48 180 159)(28 60 212 75 133 236 152 49 181 160)(29 61 213 76 134 237 153 50 182 161)(30 62 214 77 135 238 137 51 183 162)(31 63 215 78 136 222 138 35 184 163)(32 64 216 79 120 223 139 36 185 164)(33 65 217 80 121 224 140 37 186 165)(34 66 218 81 122 225 141 38 187 166)
(1 41 195 221)(2 42 196 205)(3 43 197 206)(4 44 198 207)(5 45 199 208)(6 46 200 209)(7 47 201 210)(8 48 202 211)(9 49 203 212)(10 50 204 213)(11 51 188 214)(12 35 189 215)(13 36 190 216)(14 37 191 217)(15 38 192 218)(16 39 193 219)(17 40 194 220)(18 108 226 320)(19 109 227 321)(20 110 228 322)(21 111 229 323)(22 112 230 307)(23 113 231 308)(24 114 232 309)(25 115 233 310)(26 116 234 311)(27 117 235 312)(28 118 236 313)(29 119 237 314)(30 103 238 315)(31 104 222 316)(32 105 223 317)(33 106 224 318)(34 107 225 319)(52 280 144 270)(53 281 145 271)(54 282 146 272)(55 283 147 256)(56 284 148 257)(57 285 149 258)(58 286 150 259)(59 287 151 260)(60 288 152 261)(61 289 153 262)(62 273 137 263)(63 274 138 264)(64 275 139 265)(65 276 140 266)(66 277 141 267)(67 278 142 268)(68 279 143 269)(69 304 175 86)(70 305 176 87)(71 306 177 88)(72 290 178 89)(73 291 179 90)(74 292 180 91)(75 293 181 92)(76 294 182 93)(77 295 183 94)(78 296 184 95)(79 297 185 96)(80 298 186 97)(81 299 187 98)(82 300 171 99)(83 301 172 100)(84 302 173 101)(85 303 174 102)(120 324 164 252)(121 325 165 253)(122 326 166 254)(123 327 167 255)(124 328 168 239)(125 329 169 240)(126 330 170 241)(127 331 154 242)(128 332 155 243)(129 333 156 244)(130 334 157 245)(131 335 158 246)(132 336 159 247)(133 337 160 248)(134 338 161 249)(135 339 162 250)(136 340 163 251)
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), (1,280,110,240,101,195,270,322,329,302)(2,281,111,241,102,196,271,323,330,303)(3,282,112,242,86,197,272,307,331,304)(4,283,113,243,87,198,256,308,332,305)(5,284,114,244,88,199,257,309,333,306)(6,285,115,245,89,200,258,310,334,290)(7,286,116,246,90,201,259,311,335,291)(8,287,117,247,91,202,260,312,336,292)(9,288,118,248,92,203,261,313,337,293)(10,289,119,249,93,204,262,314,338,294)(11,273,103,250,94,188,263,315,339,295)(12,274,104,251,95,189,264,316,340,296)(13,275,105,252,96,190,265,317,324,297)(14,276,106,253,97,191,266,318,325,298)(15,277,107,254,98,192,267,319,326,299)(16,278,108,255,99,193,268,320,327,300)(17,279,109,239,100,194,269,321,328,301)(18,67,219,82,123,226,142,39,171,167)(19,68,220,83,124,227,143,40,172,168)(20,52,221,84,125,228,144,41,173,169)(21,53,205,85,126,229,145,42,174,170)(22,54,206,69,127,230,146,43,175,154)(23,55,207,70,128,231,147,44,176,155)(24,56,208,71,129,232,148,45,177,156)(25,57,209,72,130,233,149,46,178,157)(26,58,210,73,131,234,150,47,179,158)(27,59,211,74,132,235,151,48,180,159)(28,60,212,75,133,236,152,49,181,160)(29,61,213,76,134,237,153,50,182,161)(30,62,214,77,135,238,137,51,183,162)(31,63,215,78,136,222,138,35,184,163)(32,64,216,79,120,223,139,36,185,164)(33,65,217,80,121,224,140,37,186,165)(34,66,218,81,122,225,141,38,187,166), (1,41,195,221)(2,42,196,205)(3,43,197,206)(4,44,198,207)(5,45,199,208)(6,46,200,209)(7,47,201,210)(8,48,202,211)(9,49,203,212)(10,50,204,213)(11,51,188,214)(12,35,189,215)(13,36,190,216)(14,37,191,217)(15,38,192,218)(16,39,193,219)(17,40,194,220)(18,108,226,320)(19,109,227,321)(20,110,228,322)(21,111,229,323)(22,112,230,307)(23,113,231,308)(24,114,232,309)(25,115,233,310)(26,116,234,311)(27,117,235,312)(28,118,236,313)(29,119,237,314)(30,103,238,315)(31,104,222,316)(32,105,223,317)(33,106,224,318)(34,107,225,319)(52,280,144,270)(53,281,145,271)(54,282,146,272)(55,283,147,256)(56,284,148,257)(57,285,149,258)(58,286,150,259)(59,287,151,260)(60,288,152,261)(61,289,153,262)(62,273,137,263)(63,274,138,264)(64,275,139,265)(65,276,140,266)(66,277,141,267)(67,278,142,268)(68,279,143,269)(69,304,175,86)(70,305,176,87)(71,306,177,88)(72,290,178,89)(73,291,179,90)(74,292,180,91)(75,293,181,92)(76,294,182,93)(77,295,183,94)(78,296,184,95)(79,297,185,96)(80,298,186,97)(81,299,187,98)(82,300,171,99)(83,301,172,100)(84,302,173,101)(85,303,174,102)(120,324,164,252)(121,325,165,253)(122,326,166,254)(123,327,167,255)(124,328,168,239)(125,329,169,240)(126,330,170,241)(127,331,154,242)(128,332,155,243)(129,333,156,244)(130,334,157,245)(131,335,158,246)(132,336,159,247)(133,337,160,248)(134,338,161,249)(135,339,162,250)(136,340,163,251)>;
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), (1,280,110,240,101,195,270,322,329,302)(2,281,111,241,102,196,271,323,330,303)(3,282,112,242,86,197,272,307,331,304)(4,283,113,243,87,198,256,308,332,305)(5,284,114,244,88,199,257,309,333,306)(6,285,115,245,89,200,258,310,334,290)(7,286,116,246,90,201,259,311,335,291)(8,287,117,247,91,202,260,312,336,292)(9,288,118,248,92,203,261,313,337,293)(10,289,119,249,93,204,262,314,338,294)(11,273,103,250,94,188,263,315,339,295)(12,274,104,251,95,189,264,316,340,296)(13,275,105,252,96,190,265,317,324,297)(14,276,106,253,97,191,266,318,325,298)(15,277,107,254,98,192,267,319,326,299)(16,278,108,255,99,193,268,320,327,300)(17,279,109,239,100,194,269,321,328,301)(18,67,219,82,123,226,142,39,171,167)(19,68,220,83,124,227,143,40,172,168)(20,52,221,84,125,228,144,41,173,169)(21,53,205,85,126,229,145,42,174,170)(22,54,206,69,127,230,146,43,175,154)(23,55,207,70,128,231,147,44,176,155)(24,56,208,71,129,232,148,45,177,156)(25,57,209,72,130,233,149,46,178,157)(26,58,210,73,131,234,150,47,179,158)(27,59,211,74,132,235,151,48,180,159)(28,60,212,75,133,236,152,49,181,160)(29,61,213,76,134,237,153,50,182,161)(30,62,214,77,135,238,137,51,183,162)(31,63,215,78,136,222,138,35,184,163)(32,64,216,79,120,223,139,36,185,164)(33,65,217,80,121,224,140,37,186,165)(34,66,218,81,122,225,141,38,187,166), (1,41,195,221)(2,42,196,205)(3,43,197,206)(4,44,198,207)(5,45,199,208)(6,46,200,209)(7,47,201,210)(8,48,202,211)(9,49,203,212)(10,50,204,213)(11,51,188,214)(12,35,189,215)(13,36,190,216)(14,37,191,217)(15,38,192,218)(16,39,193,219)(17,40,194,220)(18,108,226,320)(19,109,227,321)(20,110,228,322)(21,111,229,323)(22,112,230,307)(23,113,231,308)(24,114,232,309)(25,115,233,310)(26,116,234,311)(27,117,235,312)(28,118,236,313)(29,119,237,314)(30,103,238,315)(31,104,222,316)(32,105,223,317)(33,106,224,318)(34,107,225,319)(52,280,144,270)(53,281,145,271)(54,282,146,272)(55,283,147,256)(56,284,148,257)(57,285,149,258)(58,286,150,259)(59,287,151,260)(60,288,152,261)(61,289,153,262)(62,273,137,263)(63,274,138,264)(64,275,139,265)(65,276,140,266)(66,277,141,267)(67,278,142,268)(68,279,143,269)(69,304,175,86)(70,305,176,87)(71,306,177,88)(72,290,178,89)(73,291,179,90)(74,292,180,91)(75,293,181,92)(76,294,182,93)(77,295,183,94)(78,296,184,95)(79,297,185,96)(80,298,186,97)(81,299,187,98)(82,300,171,99)(83,301,172,100)(84,302,173,101)(85,303,174,102)(120,324,164,252)(121,325,165,253)(122,326,166,254)(123,327,167,255)(124,328,168,239)(125,329,169,240)(126,330,170,241)(127,331,154,242)(128,332,155,243)(129,333,156,244)(130,334,157,245)(131,335,158,246)(132,336,159,247)(133,337,160,248)(134,338,161,249)(135,339,162,250)(136,340,163,251) );
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)], [(1,280,110,240,101,195,270,322,329,302),(2,281,111,241,102,196,271,323,330,303),(3,282,112,242,86,197,272,307,331,304),(4,283,113,243,87,198,256,308,332,305),(5,284,114,244,88,199,257,309,333,306),(6,285,115,245,89,200,258,310,334,290),(7,286,116,246,90,201,259,311,335,291),(8,287,117,247,91,202,260,312,336,292),(9,288,118,248,92,203,261,313,337,293),(10,289,119,249,93,204,262,314,338,294),(11,273,103,250,94,188,263,315,339,295),(12,274,104,251,95,189,264,316,340,296),(13,275,105,252,96,190,265,317,324,297),(14,276,106,253,97,191,266,318,325,298),(15,277,107,254,98,192,267,319,326,299),(16,278,108,255,99,193,268,320,327,300),(17,279,109,239,100,194,269,321,328,301),(18,67,219,82,123,226,142,39,171,167),(19,68,220,83,124,227,143,40,172,168),(20,52,221,84,125,228,144,41,173,169),(21,53,205,85,126,229,145,42,174,170),(22,54,206,69,127,230,146,43,175,154),(23,55,207,70,128,231,147,44,176,155),(24,56,208,71,129,232,148,45,177,156),(25,57,209,72,130,233,149,46,178,157),(26,58,210,73,131,234,150,47,179,158),(27,59,211,74,132,235,151,48,180,159),(28,60,212,75,133,236,152,49,181,160),(29,61,213,76,134,237,153,50,182,161),(30,62,214,77,135,238,137,51,183,162),(31,63,215,78,136,222,138,35,184,163),(32,64,216,79,120,223,139,36,185,164),(33,65,217,80,121,224,140,37,186,165),(34,66,218,81,122,225,141,38,187,166)], [(1,41,195,221),(2,42,196,205),(3,43,197,206),(4,44,198,207),(5,45,199,208),(6,46,200,209),(7,47,201,210),(8,48,202,211),(9,49,203,212),(10,50,204,213),(11,51,188,214),(12,35,189,215),(13,36,190,216),(14,37,191,217),(15,38,192,218),(16,39,193,219),(17,40,194,220),(18,108,226,320),(19,109,227,321),(20,110,228,322),(21,111,229,323),(22,112,230,307),(23,113,231,308),(24,114,232,309),(25,115,233,310),(26,116,234,311),(27,117,235,312),(28,118,236,313),(29,119,237,314),(30,103,238,315),(31,104,222,316),(32,105,223,317),(33,106,224,318),(34,107,225,319),(52,280,144,270),(53,281,145,271),(54,282,146,272),(55,283,147,256),(56,284,148,257),(57,285,149,258),(58,286,150,259),(59,287,151,260),(60,288,152,261),(61,289,153,262),(62,273,137,263),(63,274,138,264),(64,275,139,265),(65,276,140,266),(66,277,141,267),(67,278,142,268),(68,279,143,269),(69,304,175,86),(70,305,176,87),(71,306,177,88),(72,290,178,89),(73,291,179,90),(74,292,180,91),(75,293,181,92),(76,294,182,93),(77,295,183,94),(78,296,184,95),(79,297,185,96),(80,298,186,97),(81,299,187,98),(82,300,171,99),(83,301,172,100),(84,302,173,101),(85,303,174,102),(120,324,164,252),(121,325,165,253),(122,326,166,254),(123,327,167,255),(124,328,168,239),(125,329,169,240),(126,330,170,241),(127,331,154,242),(128,332,155,243),(129,333,156,244),(130,334,157,245),(131,335,158,246),(132,336,159,247),(133,337,160,248),(134,338,161,249),(135,339,162,250),(136,340,163,251)]])
136 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 17A | ··· | 17P | 34A | ··· | 34P | 68A | ··· | 68AF | 85A | ··· | 85AF | 170A | ··· | 170AF |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 85 | ··· | 85 | 170 | ··· | 170 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C17 | C34 | C68 | D5 | Dic5 | D5×C17 | C17×Dic5 |
| kernel | C17×Dic5 | C170 | C85 | Dic5 | C10 | C5 | C34 | C17 | C2 | C1 |
| # reps | 1 | 1 | 2 | 16 | 16 | 32 | 2 | 2 | 32 | 32 |
Matrix representation of C17×Dic5 ►in GL3(𝔽1021) generated by
| 1 | 0 | 0 |
| 0 | 81 | 0 |
| 0 | 0 | 81 |
| 1020 | 0 | 0 |
| 0 | 1020 | 1 |
| 0 | 456 | 564 |
| 374 | 0 | 0 |
| 0 | 767 | 714 |
| 0 | 566 | 254 |
G:=sub<GL(3,GF(1021))| [1,0,0,0,81,0,0,0,81],[1020,0,0,0,1020,456,0,1,564],[374,0,0,0,767,566,0,714,254] >;
C17×Dic5 in GAP, Magma, Sage, TeX
C_{17}\times {\rm Dic}_5 % in TeX
G:=Group("C17xDic5"); // GroupNames label
G:=SmallGroup(340,1);
// by ID
G=gap.SmallGroup(340,1);
# by ID
G:=PCGroup([4,-2,-17,-2,-5,136,4355]);
// Polycyclic
G:=Group<a,b,c|a^17=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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