direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×Dic17, C85⋊8C4, C17⋊2C20, C34.C10, C170.2C2, C10.2D17, C2.(C5×D17), SmallGroup(340,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C5×Dic17 |
Generators and relations for C5×Dic17
G = < a,b,c | a5=b34=1, c2=b17, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic17
►Regular action on 340 points
Generators in S340
(1 150 131 77 55)(2 151 132 78 56)(3 152 133 79 57)(4 153 134 80 58)(5 154 135 81 59)(6 155 136 82 60)(7 156 103 83 61)(8 157 104 84 62)(9 158 105 85 63)(10 159 106 86 64)(11 160 107 87 65)(12 161 108 88 66)(13 162 109 89 67)(14 163 110 90 68)(15 164 111 91 35)(16 165 112 92 36)(17 166 113 93 37)(18 167 114 94 38)(19 168 115 95 39)(20 169 116 96 40)(21 170 117 97 41)(22 137 118 98 42)(23 138 119 99 43)(24 139 120 100 44)(25 140 121 101 45)(26 141 122 102 46)(27 142 123 69 47)(28 143 124 70 48)(29 144 125 71 49)(30 145 126 72 50)(31 146 127 73 51)(32 147 128 74 52)(33 148 129 75 53)(34 149 130 76 54)(171 307 273 239 222)(172 308 274 240 223)(173 309 275 241 224)(174 310 276 242 225)(175 311 277 243 226)(176 312 278 244 227)(177 313 279 245 228)(178 314 280 246 229)(179 315 281 247 230)(180 316 282 248 231)(181 317 283 249 232)(182 318 284 250 233)(183 319 285 251 234)(184 320 286 252 235)(185 321 287 253 236)(186 322 288 254 237)(187 323 289 255 238)(188 324 290 256 205)(189 325 291 257 206)(190 326 292 258 207)(191 327 293 259 208)(192 328 294 260 209)(193 329 295 261 210)(194 330 296 262 211)(195 331 297 263 212)(196 332 298 264 213)(197 333 299 265 214)(198 334 300 266 215)(199 335 301 267 216)(200 336 302 268 217)(201 337 303 269 218)(202 338 304 270 219)(203 339 305 271 220)(204 340 306 272 221)
(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)
(1 188 18 171)(2 187 19 204)(3 186 20 203)(4 185 21 202)(5 184 22 201)(6 183 23 200)(7 182 24 199)(8 181 25 198)(9 180 26 197)(10 179 27 196)(11 178 28 195)(12 177 29 194)(13 176 30 193)(14 175 31 192)(15 174 32 191)(16 173 33 190)(17 172 34 189)(35 225 52 208)(36 224 53 207)(37 223 54 206)(38 222 55 205)(39 221 56 238)(40 220 57 237)(41 219 58 236)(42 218 59 235)(43 217 60 234)(44 216 61 233)(45 215 62 232)(46 214 63 231)(47 213 64 230)(48 212 65 229)(49 211 66 228)(50 210 67 227)(51 209 68 226)(69 264 86 247)(70 263 87 246)(71 262 88 245)(72 261 89 244)(73 260 90 243)(74 259 91 242)(75 258 92 241)(76 257 93 240)(77 256 94 239)(78 255 95 272)(79 254 96 271)(80 253 97 270)(81 252 98 269)(82 251 99 268)(83 250 100 267)(84 249 101 266)(85 248 102 265)(103 284 120 301)(104 283 121 300)(105 282 122 299)(106 281 123 298)(107 280 124 297)(108 279 125 296)(109 278 126 295)(110 277 127 294)(111 276 128 293)(112 275 129 292)(113 274 130 291)(114 273 131 290)(115 306 132 289)(116 305 133 288)(117 304 134 287)(118 303 135 286)(119 302 136 285)(137 337 154 320)(138 336 155 319)(139 335 156 318)(140 334 157 317)(141 333 158 316)(142 332 159 315)(143 331 160 314)(144 330 161 313)(145 329 162 312)(146 328 163 311)(147 327 164 310)(148 326 165 309)(149 325 166 308)(150 324 167 307)(151 323 168 340)(152 322 169 339)(153 321 170 338)
G:=sub<Sym(340)| (1,150,131,77,55)(2,151,132,78,56)(3,152,133,79,57)(4,153,134,80,58)(5,154,135,81,59)(6,155,136,82,60)(7,156,103,83,61)(8,157,104,84,62)(9,158,105,85,63)(10,159,106,86,64)(11,160,107,87,65)(12,161,108,88,66)(13,162,109,89,67)(14,163,110,90,68)(15,164,111,91,35)(16,165,112,92,36)(17,166,113,93,37)(18,167,114,94,38)(19,168,115,95,39)(20,169,116,96,40)(21,170,117,97,41)(22,137,118,98,42)(23,138,119,99,43)(24,139,120,100,44)(25,140,121,101,45)(26,141,122,102,46)(27,142,123,69,47)(28,143,124,70,48)(29,144,125,71,49)(30,145,126,72,50)(31,146,127,73,51)(32,147,128,74,52)(33,148,129,75,53)(34,149,130,76,54)(171,307,273,239,222)(172,308,274,240,223)(173,309,275,241,224)(174,310,276,242,225)(175,311,277,243,226)(176,312,278,244,227)(177,313,279,245,228)(178,314,280,246,229)(179,315,281,247,230)(180,316,282,248,231)(181,317,283,249,232)(182,318,284,250,233)(183,319,285,251,234)(184,320,286,252,235)(185,321,287,253,236)(186,322,288,254,237)(187,323,289,255,238)(188,324,290,256,205)(189,325,291,257,206)(190,326,292,258,207)(191,327,293,259,208)(192,328,294,260,209)(193,329,295,261,210)(194,330,296,262,211)(195,331,297,263,212)(196,332,298,264,213)(197,333,299,265,214)(198,334,300,266,215)(199,335,301,267,216)(200,336,302,268,217)(201,337,303,269,218)(202,338,304,270,219)(203,339,305,271,220)(204,340,306,272,221), (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), (1,188,18,171)(2,187,19,204)(3,186,20,203)(4,185,21,202)(5,184,22,201)(6,183,23,200)(7,182,24,199)(8,181,25,198)(9,180,26,197)(10,179,27,196)(11,178,28,195)(12,177,29,194)(13,176,30,193)(14,175,31,192)(15,174,32,191)(16,173,33,190)(17,172,34,189)(35,225,52,208)(36,224,53,207)(37,223,54,206)(38,222,55,205)(39,221,56,238)(40,220,57,237)(41,219,58,236)(42,218,59,235)(43,217,60,234)(44,216,61,233)(45,215,62,232)(46,214,63,231)(47,213,64,230)(48,212,65,229)(49,211,66,228)(50,210,67,227)(51,209,68,226)(69,264,86,247)(70,263,87,246)(71,262,88,245)(72,261,89,244)(73,260,90,243)(74,259,91,242)(75,258,92,241)(76,257,93,240)(77,256,94,239)(78,255,95,272)(79,254,96,271)(80,253,97,270)(81,252,98,269)(82,251,99,268)(83,250,100,267)(84,249,101,266)(85,248,102,265)(103,284,120,301)(104,283,121,300)(105,282,122,299)(106,281,123,298)(107,280,124,297)(108,279,125,296)(109,278,126,295)(110,277,127,294)(111,276,128,293)(112,275,129,292)(113,274,130,291)(114,273,131,290)(115,306,132,289)(116,305,133,288)(117,304,134,287)(118,303,135,286)(119,302,136,285)(137,337,154,320)(138,336,155,319)(139,335,156,318)(140,334,157,317)(141,333,158,316)(142,332,159,315)(143,331,160,314)(144,330,161,313)(145,329,162,312)(146,328,163,311)(147,327,164,310)(148,326,165,309)(149,325,166,308)(150,324,167,307)(151,323,168,340)(152,322,169,339)(153,321,170,338)>;
G:=Group( (1,150,131,77,55)(2,151,132,78,56)(3,152,133,79,57)(4,153,134,80,58)(5,154,135,81,59)(6,155,136,82,60)(7,156,103,83,61)(8,157,104,84,62)(9,158,105,85,63)(10,159,106,86,64)(11,160,107,87,65)(12,161,108,88,66)(13,162,109,89,67)(14,163,110,90,68)(15,164,111,91,35)(16,165,112,92,36)(17,166,113,93,37)(18,167,114,94,38)(19,168,115,95,39)(20,169,116,96,40)(21,170,117,97,41)(22,137,118,98,42)(23,138,119,99,43)(24,139,120,100,44)(25,140,121,101,45)(26,141,122,102,46)(27,142,123,69,47)(28,143,124,70,48)(29,144,125,71,49)(30,145,126,72,50)(31,146,127,73,51)(32,147,128,74,52)(33,148,129,75,53)(34,149,130,76,54)(171,307,273,239,222)(172,308,274,240,223)(173,309,275,241,224)(174,310,276,242,225)(175,311,277,243,226)(176,312,278,244,227)(177,313,279,245,228)(178,314,280,246,229)(179,315,281,247,230)(180,316,282,248,231)(181,317,283,249,232)(182,318,284,250,233)(183,319,285,251,234)(184,320,286,252,235)(185,321,287,253,236)(186,322,288,254,237)(187,323,289,255,238)(188,324,290,256,205)(189,325,291,257,206)(190,326,292,258,207)(191,327,293,259,208)(192,328,294,260,209)(193,329,295,261,210)(194,330,296,262,211)(195,331,297,263,212)(196,332,298,264,213)(197,333,299,265,214)(198,334,300,266,215)(199,335,301,267,216)(200,336,302,268,217)(201,337,303,269,218)(202,338,304,270,219)(203,339,305,271,220)(204,340,306,272,221), (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), (1,188,18,171)(2,187,19,204)(3,186,20,203)(4,185,21,202)(5,184,22,201)(6,183,23,200)(7,182,24,199)(8,181,25,198)(9,180,26,197)(10,179,27,196)(11,178,28,195)(12,177,29,194)(13,176,30,193)(14,175,31,192)(15,174,32,191)(16,173,33,190)(17,172,34,189)(35,225,52,208)(36,224,53,207)(37,223,54,206)(38,222,55,205)(39,221,56,238)(40,220,57,237)(41,219,58,236)(42,218,59,235)(43,217,60,234)(44,216,61,233)(45,215,62,232)(46,214,63,231)(47,213,64,230)(48,212,65,229)(49,211,66,228)(50,210,67,227)(51,209,68,226)(69,264,86,247)(70,263,87,246)(71,262,88,245)(72,261,89,244)(73,260,90,243)(74,259,91,242)(75,258,92,241)(76,257,93,240)(77,256,94,239)(78,255,95,272)(79,254,96,271)(80,253,97,270)(81,252,98,269)(82,251,99,268)(83,250,100,267)(84,249,101,266)(85,248,102,265)(103,284,120,301)(104,283,121,300)(105,282,122,299)(106,281,123,298)(107,280,124,297)(108,279,125,296)(109,278,126,295)(110,277,127,294)(111,276,128,293)(112,275,129,292)(113,274,130,291)(114,273,131,290)(115,306,132,289)(116,305,133,288)(117,304,134,287)(118,303,135,286)(119,302,136,285)(137,337,154,320)(138,336,155,319)(139,335,156,318)(140,334,157,317)(141,333,158,316)(142,332,159,315)(143,331,160,314)(144,330,161,313)(145,329,162,312)(146,328,163,311)(147,327,164,310)(148,326,165,309)(149,325,166,308)(150,324,167,307)(151,323,168,340)(152,322,169,339)(153,321,170,338) );
G=PermutationGroup([[(1,150,131,77,55),(2,151,132,78,56),(3,152,133,79,57),(4,153,134,80,58),(5,154,135,81,59),(6,155,136,82,60),(7,156,103,83,61),(8,157,104,84,62),(9,158,105,85,63),(10,159,106,86,64),(11,160,107,87,65),(12,161,108,88,66),(13,162,109,89,67),(14,163,110,90,68),(15,164,111,91,35),(16,165,112,92,36),(17,166,113,93,37),(18,167,114,94,38),(19,168,115,95,39),(20,169,116,96,40),(21,170,117,97,41),(22,137,118,98,42),(23,138,119,99,43),(24,139,120,100,44),(25,140,121,101,45),(26,141,122,102,46),(27,142,123,69,47),(28,143,124,70,48),(29,144,125,71,49),(30,145,126,72,50),(31,146,127,73,51),(32,147,128,74,52),(33,148,129,75,53),(34,149,130,76,54),(171,307,273,239,222),(172,308,274,240,223),(173,309,275,241,224),(174,310,276,242,225),(175,311,277,243,226),(176,312,278,244,227),(177,313,279,245,228),(178,314,280,246,229),(179,315,281,247,230),(180,316,282,248,231),(181,317,283,249,232),(182,318,284,250,233),(183,319,285,251,234),(184,320,286,252,235),(185,321,287,253,236),(186,322,288,254,237),(187,323,289,255,238),(188,324,290,256,205),(189,325,291,257,206),(190,326,292,258,207),(191,327,293,259,208),(192,328,294,260,209),(193,329,295,261,210),(194,330,296,262,211),(195,331,297,263,212),(196,332,298,264,213),(197,333,299,265,214),(198,334,300,266,215),(199,335,301,267,216),(200,336,302,268,217),(201,337,303,269,218),(202,338,304,270,219),(203,339,305,271,220),(204,340,306,272,221)], [(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)], [(1,188,18,171),(2,187,19,204),(3,186,20,203),(4,185,21,202),(5,184,22,201),(6,183,23,200),(7,182,24,199),(8,181,25,198),(9,180,26,197),(10,179,27,196),(11,178,28,195),(12,177,29,194),(13,176,30,193),(14,175,31,192),(15,174,32,191),(16,173,33,190),(17,172,34,189),(35,225,52,208),(36,224,53,207),(37,223,54,206),(38,222,55,205),(39,221,56,238),(40,220,57,237),(41,219,58,236),(42,218,59,235),(43,217,60,234),(44,216,61,233),(45,215,62,232),(46,214,63,231),(47,213,64,230),(48,212,65,229),(49,211,66,228),(50,210,67,227),(51,209,68,226),(69,264,86,247),(70,263,87,246),(71,262,88,245),(72,261,89,244),(73,260,90,243),(74,259,91,242),(75,258,92,241),(76,257,93,240),(77,256,94,239),(78,255,95,272),(79,254,96,271),(80,253,97,270),(81,252,98,269),(82,251,99,268),(83,250,100,267),(84,249,101,266),(85,248,102,265),(103,284,120,301),(104,283,121,300),(105,282,122,299),(106,281,123,298),(107,280,124,297),(108,279,125,296),(109,278,126,295),(110,277,127,294),(111,276,128,293),(112,275,129,292),(113,274,130,291),(114,273,131,290),(115,306,132,289),(116,305,133,288),(117,304,134,287),(118,303,135,286),(119,302,136,285),(137,337,154,320),(138,336,155,319),(139,335,156,318),(140,334,157,317),(141,333,158,316),(142,332,159,315),(143,331,160,314),(144,330,161,313),(145,329,162,312),(146,328,163,311),(147,327,164,310),(148,326,165,309),(149,325,166,308),(150,324,167,307),(151,323,168,340),(152,322,169,339),(153,321,170,338)]])
100 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 17A | ··· | 17H | 20A | ··· | 20H | 34A | ··· | 34H | 85A | ··· | 85AF | 170A | ··· | 170AF |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 17 | ··· | 17 | 20 | ··· | 20 | 34 | ··· | 34 | 85 | ··· | 85 | 170 | ··· | 170 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D17 | Dic17 | C5×D17 | C5×Dic17 |
| kernel | C5×Dic17 | C170 | C85 | Dic17 | C34 | C17 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 32 | 32 |
Matrix representation of C5×Dic17 ►in GL2(𝔽1021) generated by
| 995 | 0 |
| 0 | 995 |
| 1 | 1020 |
| 580 | 442 |
| 592 | 866 |
| 252 | 429 |
G:=sub<GL(2,GF(1021))| [995,0,0,995],[1,580,1020,442],[592,252,866,429] >;
C5×Dic17 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{17} % in TeX
G:=Group("C5xDic17"); // GroupNames label
G:=SmallGroup(340,2);
// by ID
G=gap.SmallGroup(340,2);
# by ID
G:=PCGroup([4,-2,-5,-2,-17,40,5123]);
// Polycyclic
G:=Group<a,b,c|a^5=b^34=1,c^2=b^17,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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