direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×Dic7, C7⋊C44, C77⋊3C4, C14.C22, C22.2D7, C154.3C2, C2.(C11×D7), SmallGroup(308,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C11×Dic7 |
Generators and relations for C11×Dic7
G = < a,b,c | a11=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C11×Dic7
►Regular action on 308 points
Generators in S308
(1 144 139 113 112 94 80 57 56 29 15)(2 145 140 114 99 95 81 58 43 30 16)(3 146 127 115 100 96 82 59 44 31 17)(4 147 128 116 101 97 83 60 45 32 18)(5 148 129 117 102 98 84 61 46 33 19)(6 149 130 118 103 85 71 62 47 34 20)(7 150 131 119 104 86 72 63 48 35 21)(8 151 132 120 105 87 73 64 49 36 22)(9 152 133 121 106 88 74 65 50 37 23)(10 153 134 122 107 89 75 66 51 38 24)(11 154 135 123 108 90 76 67 52 39 25)(12 141 136 124 109 91 77 68 53 40 26)(13 142 137 125 110 92 78 69 54 41 27)(14 143 138 126 111 93 79 70 55 42 28)(155 295 288 267 253 239 232 218 197 190 176)(156 296 289 268 254 240 233 219 198 191 177)(157 297 290 269 255 241 234 220 199 192 178)(158 298 291 270 256 242 235 221 200 193 179)(159 299 292 271 257 243 236 222 201 194 180)(160 300 293 272 258 244 237 223 202 195 181)(161 301 294 273 259 245 238 224 203 196 182)(162 302 281 274 260 246 225 211 204 183 169)(163 303 282 275 261 247 226 212 205 184 170)(164 304 283 276 262 248 227 213 206 185 171)(165 305 284 277 263 249 228 214 207 186 172)(166 306 285 278 264 250 229 215 208 187 173)(167 307 286 279 265 251 230 216 209 188 174)(168 308 287 280 266 252 231 217 210 189 175)
(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)
(1 162 8 155)(2 161 9 168)(3 160 10 167)(4 159 11 166)(5 158 12 165)(6 157 13 164)(7 156 14 163)(15 169 22 176)(16 182 23 175)(17 181 24 174)(18 180 25 173)(19 179 26 172)(20 178 27 171)(21 177 28 170)(29 183 36 190)(30 196 37 189)(31 195 38 188)(32 194 39 187)(33 193 40 186)(34 192 41 185)(35 191 42 184)(43 203 50 210)(44 202 51 209)(45 201 52 208)(46 200 53 207)(47 199 54 206)(48 198 55 205)(49 197 56 204)(57 211 64 218)(58 224 65 217)(59 223 66 216)(60 222 67 215)(61 221 68 214)(62 220 69 213)(63 219 70 212)(71 234 78 227)(72 233 79 226)(73 232 80 225)(74 231 81 238)(75 230 82 237)(76 229 83 236)(77 228 84 235)(85 241 92 248)(86 240 93 247)(87 239 94 246)(88 252 95 245)(89 251 96 244)(90 250 97 243)(91 249 98 242)(99 259 106 266)(100 258 107 265)(101 257 108 264)(102 256 109 263)(103 255 110 262)(104 254 111 261)(105 253 112 260)(113 274 120 267)(114 273 121 280)(115 272 122 279)(116 271 123 278)(117 270 124 277)(118 269 125 276)(119 268 126 275)(127 293 134 286)(128 292 135 285)(129 291 136 284)(130 290 137 283)(131 289 138 282)(132 288 139 281)(133 287 140 294)(141 305 148 298)(142 304 149 297)(143 303 150 296)(144 302 151 295)(145 301 152 308)(146 300 153 307)(147 299 154 306)
G:=sub<Sym(308)| (1,144,139,113,112,94,80,57,56,29,15)(2,145,140,114,99,95,81,58,43,30,16)(3,146,127,115,100,96,82,59,44,31,17)(4,147,128,116,101,97,83,60,45,32,18)(5,148,129,117,102,98,84,61,46,33,19)(6,149,130,118,103,85,71,62,47,34,20)(7,150,131,119,104,86,72,63,48,35,21)(8,151,132,120,105,87,73,64,49,36,22)(9,152,133,121,106,88,74,65,50,37,23)(10,153,134,122,107,89,75,66,51,38,24)(11,154,135,123,108,90,76,67,52,39,25)(12,141,136,124,109,91,77,68,53,40,26)(13,142,137,125,110,92,78,69,54,41,27)(14,143,138,126,111,93,79,70,55,42,28)(155,295,288,267,253,239,232,218,197,190,176)(156,296,289,268,254,240,233,219,198,191,177)(157,297,290,269,255,241,234,220,199,192,178)(158,298,291,270,256,242,235,221,200,193,179)(159,299,292,271,257,243,236,222,201,194,180)(160,300,293,272,258,244,237,223,202,195,181)(161,301,294,273,259,245,238,224,203,196,182)(162,302,281,274,260,246,225,211,204,183,169)(163,303,282,275,261,247,226,212,205,184,170)(164,304,283,276,262,248,227,213,206,185,171)(165,305,284,277,263,249,228,214,207,186,172)(166,306,285,278,264,250,229,215,208,187,173)(167,307,286,279,265,251,230,216,209,188,174)(168,308,287,280,266,252,231,217,210,189,175), (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), (1,162,8,155)(2,161,9,168)(3,160,10,167)(4,159,11,166)(5,158,12,165)(6,157,13,164)(7,156,14,163)(15,169,22,176)(16,182,23,175)(17,181,24,174)(18,180,25,173)(19,179,26,172)(20,178,27,171)(21,177,28,170)(29,183,36,190)(30,196,37,189)(31,195,38,188)(32,194,39,187)(33,193,40,186)(34,192,41,185)(35,191,42,184)(43,203,50,210)(44,202,51,209)(45,201,52,208)(46,200,53,207)(47,199,54,206)(48,198,55,205)(49,197,56,204)(57,211,64,218)(58,224,65,217)(59,223,66,216)(60,222,67,215)(61,221,68,214)(62,220,69,213)(63,219,70,212)(71,234,78,227)(72,233,79,226)(73,232,80,225)(74,231,81,238)(75,230,82,237)(76,229,83,236)(77,228,84,235)(85,241,92,248)(86,240,93,247)(87,239,94,246)(88,252,95,245)(89,251,96,244)(90,250,97,243)(91,249,98,242)(99,259,106,266)(100,258,107,265)(101,257,108,264)(102,256,109,263)(103,255,110,262)(104,254,111,261)(105,253,112,260)(113,274,120,267)(114,273,121,280)(115,272,122,279)(116,271,123,278)(117,270,124,277)(118,269,125,276)(119,268,126,275)(127,293,134,286)(128,292,135,285)(129,291,136,284)(130,290,137,283)(131,289,138,282)(132,288,139,281)(133,287,140,294)(141,305,148,298)(142,304,149,297)(143,303,150,296)(144,302,151,295)(145,301,152,308)(146,300,153,307)(147,299,154,306)>;
G:=Group( (1,144,139,113,112,94,80,57,56,29,15)(2,145,140,114,99,95,81,58,43,30,16)(3,146,127,115,100,96,82,59,44,31,17)(4,147,128,116,101,97,83,60,45,32,18)(5,148,129,117,102,98,84,61,46,33,19)(6,149,130,118,103,85,71,62,47,34,20)(7,150,131,119,104,86,72,63,48,35,21)(8,151,132,120,105,87,73,64,49,36,22)(9,152,133,121,106,88,74,65,50,37,23)(10,153,134,122,107,89,75,66,51,38,24)(11,154,135,123,108,90,76,67,52,39,25)(12,141,136,124,109,91,77,68,53,40,26)(13,142,137,125,110,92,78,69,54,41,27)(14,143,138,126,111,93,79,70,55,42,28)(155,295,288,267,253,239,232,218,197,190,176)(156,296,289,268,254,240,233,219,198,191,177)(157,297,290,269,255,241,234,220,199,192,178)(158,298,291,270,256,242,235,221,200,193,179)(159,299,292,271,257,243,236,222,201,194,180)(160,300,293,272,258,244,237,223,202,195,181)(161,301,294,273,259,245,238,224,203,196,182)(162,302,281,274,260,246,225,211,204,183,169)(163,303,282,275,261,247,226,212,205,184,170)(164,304,283,276,262,248,227,213,206,185,171)(165,305,284,277,263,249,228,214,207,186,172)(166,306,285,278,264,250,229,215,208,187,173)(167,307,286,279,265,251,230,216,209,188,174)(168,308,287,280,266,252,231,217,210,189,175), (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), (1,162,8,155)(2,161,9,168)(3,160,10,167)(4,159,11,166)(5,158,12,165)(6,157,13,164)(7,156,14,163)(15,169,22,176)(16,182,23,175)(17,181,24,174)(18,180,25,173)(19,179,26,172)(20,178,27,171)(21,177,28,170)(29,183,36,190)(30,196,37,189)(31,195,38,188)(32,194,39,187)(33,193,40,186)(34,192,41,185)(35,191,42,184)(43,203,50,210)(44,202,51,209)(45,201,52,208)(46,200,53,207)(47,199,54,206)(48,198,55,205)(49,197,56,204)(57,211,64,218)(58,224,65,217)(59,223,66,216)(60,222,67,215)(61,221,68,214)(62,220,69,213)(63,219,70,212)(71,234,78,227)(72,233,79,226)(73,232,80,225)(74,231,81,238)(75,230,82,237)(76,229,83,236)(77,228,84,235)(85,241,92,248)(86,240,93,247)(87,239,94,246)(88,252,95,245)(89,251,96,244)(90,250,97,243)(91,249,98,242)(99,259,106,266)(100,258,107,265)(101,257,108,264)(102,256,109,263)(103,255,110,262)(104,254,111,261)(105,253,112,260)(113,274,120,267)(114,273,121,280)(115,272,122,279)(116,271,123,278)(117,270,124,277)(118,269,125,276)(119,268,126,275)(127,293,134,286)(128,292,135,285)(129,291,136,284)(130,290,137,283)(131,289,138,282)(132,288,139,281)(133,287,140,294)(141,305,148,298)(142,304,149,297)(143,303,150,296)(144,302,151,295)(145,301,152,308)(146,300,153,307)(147,299,154,306) );
G=PermutationGroup([[(1,144,139,113,112,94,80,57,56,29,15),(2,145,140,114,99,95,81,58,43,30,16),(3,146,127,115,100,96,82,59,44,31,17),(4,147,128,116,101,97,83,60,45,32,18),(5,148,129,117,102,98,84,61,46,33,19),(6,149,130,118,103,85,71,62,47,34,20),(7,150,131,119,104,86,72,63,48,35,21),(8,151,132,120,105,87,73,64,49,36,22),(9,152,133,121,106,88,74,65,50,37,23),(10,153,134,122,107,89,75,66,51,38,24),(11,154,135,123,108,90,76,67,52,39,25),(12,141,136,124,109,91,77,68,53,40,26),(13,142,137,125,110,92,78,69,54,41,27),(14,143,138,126,111,93,79,70,55,42,28),(155,295,288,267,253,239,232,218,197,190,176),(156,296,289,268,254,240,233,219,198,191,177),(157,297,290,269,255,241,234,220,199,192,178),(158,298,291,270,256,242,235,221,200,193,179),(159,299,292,271,257,243,236,222,201,194,180),(160,300,293,272,258,244,237,223,202,195,181),(161,301,294,273,259,245,238,224,203,196,182),(162,302,281,274,260,246,225,211,204,183,169),(163,303,282,275,261,247,226,212,205,184,170),(164,304,283,276,262,248,227,213,206,185,171),(165,305,284,277,263,249,228,214,207,186,172),(166,306,285,278,264,250,229,215,208,187,173),(167,307,286,279,265,251,230,216,209,188,174),(168,308,287,280,266,252,231,217,210,189,175)], [(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)], [(1,162,8,155),(2,161,9,168),(3,160,10,167),(4,159,11,166),(5,158,12,165),(6,157,13,164),(7,156,14,163),(15,169,22,176),(16,182,23,175),(17,181,24,174),(18,180,25,173),(19,179,26,172),(20,178,27,171),(21,177,28,170),(29,183,36,190),(30,196,37,189),(31,195,38,188),(32,194,39,187),(33,193,40,186),(34,192,41,185),(35,191,42,184),(43,203,50,210),(44,202,51,209),(45,201,52,208),(46,200,53,207),(47,199,54,206),(48,198,55,205),(49,197,56,204),(57,211,64,218),(58,224,65,217),(59,223,66,216),(60,222,67,215),(61,221,68,214),(62,220,69,213),(63,219,70,212),(71,234,78,227),(72,233,79,226),(73,232,80,225),(74,231,81,238),(75,230,82,237),(76,229,83,236),(77,228,84,235),(85,241,92,248),(86,240,93,247),(87,239,94,246),(88,252,95,245),(89,251,96,244),(90,250,97,243),(91,249,98,242),(99,259,106,266),(100,258,107,265),(101,257,108,264),(102,256,109,263),(103,255,110,262),(104,254,111,261),(105,253,112,260),(113,274,120,267),(114,273,121,280),(115,272,122,279),(116,271,123,278),(117,270,124,277),(118,269,125,276),(119,268,126,275),(127,293,134,286),(128,292,135,285),(129,291,136,284),(130,290,137,283),(131,289,138,282),(132,288,139,281),(133,287,140,294),(141,305,148,298),(142,304,149,297),(143,303,150,296),(144,302,151,295),(145,301,152,308),(146,300,153,307),(147,299,154,306)]])
110 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 11A | ··· | 11J | 14A | 14B | 14C | 22A | ··· | 22J | 44A | ··· | 44T | 77A | ··· | 77AD | 154A | ··· | 154AD |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 11 | ··· | 11 | 14 | 14 | 14 | 22 | ··· | 22 | 44 | ··· | 44 | 77 | ··· | 77 | 154 | ··· | 154 |
| size | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C11 | C22 | C44 | D7 | Dic7 | C11×D7 | C11×Dic7 |
| kernel | C11×Dic7 | C154 | C77 | Dic7 | C14 | C7 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 3 | 3 | 30 | 30 |
Matrix representation of C11×Dic7 ►in GL3(𝔽617) generated by
| 489 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 616 |
| 0 | 1 | 271 |
| 1 | 0 | 0 |
| 0 | 158 | 508 |
| 0 | 263 | 459 |
G:=sub<GL(3,GF(617))| [489,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,616,271],[1,0,0,0,158,263,0,508,459] >;
C11×Dic7 in GAP, Magma, Sage, TeX
C_{11}\times {\rm Dic}_7 % in TeX
G:=Group("C11xDic7"); // GroupNames label
G:=SmallGroup(308,1);
// by ID
G=gap.SmallGroup(308,1);
# by ID
G:=PCGroup([4,-2,-11,-2,-7,88,4227]);
// Polycyclic
G:=Group<a,b,c|a^11=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export