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G = C17×Dic5order 340 = 22·5·17

Direct product of C17 and Dic5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C17×Dic5, C859C4, C52C68, C10.C34, C34.2D5, C170.3C2, C2.(D5×C17), SmallGroup(340,1)

Series: Derived Chief Lower central Upper central

C1C5 — C17×Dic5
C1C5C10C170 — C17×Dic5
C5 — C17×Dic5
C1C34

Generators and relations for C17×Dic5
 G = < a,b,c | a17=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

5C4
5C68

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

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

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

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

136 conjugacy classes

class 1  2 4A4B5A5B10A10B17A···17P34A···34P68A···68AF85A···85AF170A···170AF
order124455101017···1734···3468···6885···85170···170
size115522221···11···15···52···22···2

136 irreducible representations

dim1111112222
type+++-
imageC1C2C4C17C34C68D5Dic5D5×C17C17×Dic5
kernelC17×Dic5C170C85Dic5C10C5C34C17C2C1
# reps112161632223232

Matrix representation of C17×Dic5 in GL3(𝔽1021) generated by

100
0810
0081
,
102000
010201
0456564
,
37400
0767714
0566254
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

Export

Subgroup lattice of C17×Dic5 in TeX

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