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G = C5×Dic17order 340 = 22·5·17

Direct product of C5 and Dic17

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

Aliases: C5×Dic17, C858C4, C172C20, C34.C10, C170.2C2, C10.2D17, C2.(C5×D17), SmallGroup(340,2)

Series: Derived Chief Lower central Upper central

C1C17 — C5×Dic17
C1C17C34C170 — C5×Dic17
C17 — C5×Dic17
C1C10

Generators and relations for C5×Dic17
 G = < a,b,c | a5=b34=1, c2=b17, ab=ba, ac=ca, cbc-1=b-1 >

17C4
17C20

Smallest permutation representation of C5×Dic17
Regular action on 340 points
Generators in S340
(1 161 112 88 48)(2 162 113 89 49)(3 163 114 90 50)(4 164 115 91 51)(5 165 116 92 52)(6 166 117 93 53)(7 167 118 94 54)(8 168 119 95 55)(9 169 120 96 56)(10 170 121 97 57)(11 137 122 98 58)(12 138 123 99 59)(13 139 124 100 60)(14 140 125 101 61)(15 141 126 102 62)(16 142 127 69 63)(17 143 128 70 64)(18 144 129 71 65)(19 145 130 72 66)(20 146 131 73 67)(21 147 132 74 68)(22 148 133 75 35)(23 149 134 76 36)(24 150 135 77 37)(25 151 136 78 38)(26 152 103 79 39)(27 153 104 80 40)(28 154 105 81 41)(29 155 106 82 42)(30 156 107 83 43)(31 157 108 84 44)(32 158 109 85 45)(33 159 110 86 46)(34 160 111 87 47)(171 307 290 256 205)(172 308 291 257 206)(173 309 292 258 207)(174 310 293 259 208)(175 311 294 260 209)(176 312 295 261 210)(177 313 296 262 211)(178 314 297 263 212)(179 315 298 264 213)(180 316 299 265 214)(181 317 300 266 215)(182 318 301 267 216)(183 319 302 268 217)(184 320 303 269 218)(185 321 304 270 219)(186 322 305 271 220)(187 323 306 272 221)(188 324 273 239 222)(189 325 274 240 223)(190 326 275 241 224)(191 327 276 242 225)(192 328 277 243 226)(193 329 278 244 227)(194 330 279 245 228)(195 331 280 246 229)(196 332 281 247 230)(197 333 282 248 231)(198 334 283 249 232)(199 335 284 250 233)(200 336 285 251 234)(201 337 286 252 235)(202 338 287 253 236)(203 339 288 254 237)(204 340 289 255 238)
(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)(35 235 52 218)(36 234 53 217)(37 233 54 216)(38 232 55 215)(39 231 56 214)(40 230 57 213)(41 229 58 212)(42 228 59 211)(43 227 60 210)(44 226 61 209)(45 225 62 208)(46 224 63 207)(47 223 64 206)(48 222 65 205)(49 221 66 238)(50 220 67 237)(51 219 68 236)(69 258 86 241)(70 257 87 240)(71 256 88 239)(72 255 89 272)(73 254 90 271)(74 253 91 270)(75 252 92 269)(76 251 93 268)(77 250 94 267)(78 249 95 266)(79 248 96 265)(80 247 97 264)(81 246 98 263)(82 245 99 262)(83 244 100 261)(84 243 101 260)(85 242 102 259)(103 282 120 299)(104 281 121 298)(105 280 122 297)(106 279 123 296)(107 278 124 295)(108 277 125 294)(109 276 126 293)(110 275 127 292)(111 274 128 291)(112 273 129 290)(113 306 130 289)(114 305 131 288)(115 304 132 287)(116 303 133 286)(117 302 134 285)(118 301 135 284)(119 300 136 283)(137 314 154 331)(138 313 155 330)(139 312 156 329)(140 311 157 328)(141 310 158 327)(142 309 159 326)(143 308 160 325)(144 307 161 324)(145 340 162 323)(146 339 163 322)(147 338 164 321)(148 337 165 320)(149 336 166 319)(150 335 167 318)(151 334 168 317)(152 333 169 316)(153 332 170 315)

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

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

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

100 conjugacy classes

class 1  2 4A4B5A5B5C5D10A10B10C10D17A···17H20A···20H34A···34H85A···85AF170A···170AF
order124455551010101017···1720···2034···3485···85170···170
size111717111111112···217···172···22···22···2

100 irreducible representations

dim1111112222
type+++-
imageC1C2C4C5C10C20D17Dic17C5×D17C5×Dic17
kernelC5×Dic17C170C85Dic17C34C17C10C5C2C1
# reps112448883232

Matrix representation of C5×Dic17 in GL2(𝔽1021) generated by

9950
0995
,
11020
580442
,
592866
252429
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

Export

Subgroup lattice of C5×Dic17 in TeX

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