direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×Dic11, C11⋊C28, C77⋊2C4, C22.C14, C154.2C2, C14.2D11, C2.(C7×D11), SmallGroup(308,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C7×Dic11 |
Generators and relations for C7×Dic11
G = < a,b,c | a7=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×Dic11
►Regular action on 308 points
Generators in S308
(1 141 114 89 84 52 43)(2 142 115 90 85 53 44)(3 143 116 91 86 54 23)(4 144 117 92 87 55 24)(5 145 118 93 88 56 25)(6 146 119 94 67 57 26)(7 147 120 95 68 58 27)(8 148 121 96 69 59 28)(9 149 122 97 70 60 29)(10 150 123 98 71 61 30)(11 151 124 99 72 62 31)(12 152 125 100 73 63 32)(13 153 126 101 74 64 33)(14 154 127 102 75 65 34)(15 133 128 103 76 66 35)(16 134 129 104 77 45 36)(17 135 130 105 78 46 37)(18 136 131 106 79 47 38)(19 137 132 107 80 48 39)(20 138 111 108 81 49 40)(21 139 112 109 82 50 41)(22 140 113 110 83 51 42)(155 298 276 254 221 210 188)(156 299 277 255 222 211 189)(157 300 278 256 223 212 190)(158 301 279 257 224 213 191)(159 302 280 258 225 214 192)(160 303 281 259 226 215 193)(161 304 282 260 227 216 194)(162 305 283 261 228 217 195)(163 306 284 262 229 218 196)(164 307 285 263 230 219 197)(165 308 286 264 231 220 198)(166 287 265 243 232 199 177)(167 288 266 244 233 200 178)(168 289 267 245 234 201 179)(169 290 268 246 235 202 180)(170 291 269 247 236 203 181)(171 292 270 248 237 204 182)(172 293 271 249 238 205 183)(173 294 272 250 239 206 184)(174 295 273 251 240 207 185)(175 296 274 252 241 208 186)(176 297 275 253 242 209 187)
(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)
(1 155 12 166)(2 176 13 165)(3 175 14 164)(4 174 15 163)(5 173 16 162)(6 172 17 161)(7 171 18 160)(8 170 19 159)(9 169 20 158)(10 168 21 157)(11 167 22 156)(23 186 34 197)(24 185 35 196)(25 184 36 195)(26 183 37 194)(27 182 38 193)(28 181 39 192)(29 180 40 191)(30 179 41 190)(31 178 42 189)(32 177 43 188)(33 198 44 187)(45 217 56 206)(46 216 57 205)(47 215 58 204)(48 214 59 203)(49 213 60 202)(50 212 61 201)(51 211 62 200)(52 210 63 199)(53 209 64 220)(54 208 65 219)(55 207 66 218)(67 238 78 227)(68 237 79 226)(69 236 80 225)(70 235 81 224)(71 234 82 223)(72 233 83 222)(73 232 84 221)(74 231 85 242)(75 230 86 241)(76 229 87 240)(77 228 88 239)(89 254 100 243)(90 253 101 264)(91 252 102 263)(92 251 103 262)(93 250 104 261)(94 249 105 260)(95 248 106 259)(96 247 107 258)(97 246 108 257)(98 245 109 256)(99 244 110 255)(111 279 122 268)(112 278 123 267)(113 277 124 266)(114 276 125 265)(115 275 126 286)(116 274 127 285)(117 273 128 284)(118 272 129 283)(119 271 130 282)(120 270 131 281)(121 269 132 280)(133 306 144 295)(134 305 145 294)(135 304 146 293)(136 303 147 292)(137 302 148 291)(138 301 149 290)(139 300 150 289)(140 299 151 288)(141 298 152 287)(142 297 153 308)(143 296 154 307)
G:=sub<Sym(308)| (1,141,114,89,84,52,43)(2,142,115,90,85,53,44)(3,143,116,91,86,54,23)(4,144,117,92,87,55,24)(5,145,118,93,88,56,25)(6,146,119,94,67,57,26)(7,147,120,95,68,58,27)(8,148,121,96,69,59,28)(9,149,122,97,70,60,29)(10,150,123,98,71,61,30)(11,151,124,99,72,62,31)(12,152,125,100,73,63,32)(13,153,126,101,74,64,33)(14,154,127,102,75,65,34)(15,133,128,103,76,66,35)(16,134,129,104,77,45,36)(17,135,130,105,78,46,37)(18,136,131,106,79,47,38)(19,137,132,107,80,48,39)(20,138,111,108,81,49,40)(21,139,112,109,82,50,41)(22,140,113,110,83,51,42)(155,298,276,254,221,210,188)(156,299,277,255,222,211,189)(157,300,278,256,223,212,190)(158,301,279,257,224,213,191)(159,302,280,258,225,214,192)(160,303,281,259,226,215,193)(161,304,282,260,227,216,194)(162,305,283,261,228,217,195)(163,306,284,262,229,218,196)(164,307,285,263,230,219,197)(165,308,286,264,231,220,198)(166,287,265,243,232,199,177)(167,288,266,244,233,200,178)(168,289,267,245,234,201,179)(169,290,268,246,235,202,180)(170,291,269,247,236,203,181)(171,292,270,248,237,204,182)(172,293,271,249,238,205,183)(173,294,272,250,239,206,184)(174,295,273,251,240,207,185)(175,296,274,252,241,208,186)(176,297,275,253,242,209,187), (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), (1,155,12,166)(2,176,13,165)(3,175,14,164)(4,174,15,163)(5,173,16,162)(6,172,17,161)(7,171,18,160)(8,170,19,159)(9,169,20,158)(10,168,21,157)(11,167,22,156)(23,186,34,197)(24,185,35,196)(25,184,36,195)(26,183,37,194)(27,182,38,193)(28,181,39,192)(29,180,40,191)(30,179,41,190)(31,178,42,189)(32,177,43,188)(33,198,44,187)(45,217,56,206)(46,216,57,205)(47,215,58,204)(48,214,59,203)(49,213,60,202)(50,212,61,201)(51,211,62,200)(52,210,63,199)(53,209,64,220)(54,208,65,219)(55,207,66,218)(67,238,78,227)(68,237,79,226)(69,236,80,225)(70,235,81,224)(71,234,82,223)(72,233,83,222)(73,232,84,221)(74,231,85,242)(75,230,86,241)(76,229,87,240)(77,228,88,239)(89,254,100,243)(90,253,101,264)(91,252,102,263)(92,251,103,262)(93,250,104,261)(94,249,105,260)(95,248,106,259)(96,247,107,258)(97,246,108,257)(98,245,109,256)(99,244,110,255)(111,279,122,268)(112,278,123,267)(113,277,124,266)(114,276,125,265)(115,275,126,286)(116,274,127,285)(117,273,128,284)(118,272,129,283)(119,271,130,282)(120,270,131,281)(121,269,132,280)(133,306,144,295)(134,305,145,294)(135,304,146,293)(136,303,147,292)(137,302,148,291)(138,301,149,290)(139,300,150,289)(140,299,151,288)(141,298,152,287)(142,297,153,308)(143,296,154,307)>;
G:=Group( (1,141,114,89,84,52,43)(2,142,115,90,85,53,44)(3,143,116,91,86,54,23)(4,144,117,92,87,55,24)(5,145,118,93,88,56,25)(6,146,119,94,67,57,26)(7,147,120,95,68,58,27)(8,148,121,96,69,59,28)(9,149,122,97,70,60,29)(10,150,123,98,71,61,30)(11,151,124,99,72,62,31)(12,152,125,100,73,63,32)(13,153,126,101,74,64,33)(14,154,127,102,75,65,34)(15,133,128,103,76,66,35)(16,134,129,104,77,45,36)(17,135,130,105,78,46,37)(18,136,131,106,79,47,38)(19,137,132,107,80,48,39)(20,138,111,108,81,49,40)(21,139,112,109,82,50,41)(22,140,113,110,83,51,42)(155,298,276,254,221,210,188)(156,299,277,255,222,211,189)(157,300,278,256,223,212,190)(158,301,279,257,224,213,191)(159,302,280,258,225,214,192)(160,303,281,259,226,215,193)(161,304,282,260,227,216,194)(162,305,283,261,228,217,195)(163,306,284,262,229,218,196)(164,307,285,263,230,219,197)(165,308,286,264,231,220,198)(166,287,265,243,232,199,177)(167,288,266,244,233,200,178)(168,289,267,245,234,201,179)(169,290,268,246,235,202,180)(170,291,269,247,236,203,181)(171,292,270,248,237,204,182)(172,293,271,249,238,205,183)(173,294,272,250,239,206,184)(174,295,273,251,240,207,185)(175,296,274,252,241,208,186)(176,297,275,253,242,209,187), (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), (1,155,12,166)(2,176,13,165)(3,175,14,164)(4,174,15,163)(5,173,16,162)(6,172,17,161)(7,171,18,160)(8,170,19,159)(9,169,20,158)(10,168,21,157)(11,167,22,156)(23,186,34,197)(24,185,35,196)(25,184,36,195)(26,183,37,194)(27,182,38,193)(28,181,39,192)(29,180,40,191)(30,179,41,190)(31,178,42,189)(32,177,43,188)(33,198,44,187)(45,217,56,206)(46,216,57,205)(47,215,58,204)(48,214,59,203)(49,213,60,202)(50,212,61,201)(51,211,62,200)(52,210,63,199)(53,209,64,220)(54,208,65,219)(55,207,66,218)(67,238,78,227)(68,237,79,226)(69,236,80,225)(70,235,81,224)(71,234,82,223)(72,233,83,222)(73,232,84,221)(74,231,85,242)(75,230,86,241)(76,229,87,240)(77,228,88,239)(89,254,100,243)(90,253,101,264)(91,252,102,263)(92,251,103,262)(93,250,104,261)(94,249,105,260)(95,248,106,259)(96,247,107,258)(97,246,108,257)(98,245,109,256)(99,244,110,255)(111,279,122,268)(112,278,123,267)(113,277,124,266)(114,276,125,265)(115,275,126,286)(116,274,127,285)(117,273,128,284)(118,272,129,283)(119,271,130,282)(120,270,131,281)(121,269,132,280)(133,306,144,295)(134,305,145,294)(135,304,146,293)(136,303,147,292)(137,302,148,291)(138,301,149,290)(139,300,150,289)(140,299,151,288)(141,298,152,287)(142,297,153,308)(143,296,154,307) );
G=PermutationGroup([[(1,141,114,89,84,52,43),(2,142,115,90,85,53,44),(3,143,116,91,86,54,23),(4,144,117,92,87,55,24),(5,145,118,93,88,56,25),(6,146,119,94,67,57,26),(7,147,120,95,68,58,27),(8,148,121,96,69,59,28),(9,149,122,97,70,60,29),(10,150,123,98,71,61,30),(11,151,124,99,72,62,31),(12,152,125,100,73,63,32),(13,153,126,101,74,64,33),(14,154,127,102,75,65,34),(15,133,128,103,76,66,35),(16,134,129,104,77,45,36),(17,135,130,105,78,46,37),(18,136,131,106,79,47,38),(19,137,132,107,80,48,39),(20,138,111,108,81,49,40),(21,139,112,109,82,50,41),(22,140,113,110,83,51,42),(155,298,276,254,221,210,188),(156,299,277,255,222,211,189),(157,300,278,256,223,212,190),(158,301,279,257,224,213,191),(159,302,280,258,225,214,192),(160,303,281,259,226,215,193),(161,304,282,260,227,216,194),(162,305,283,261,228,217,195),(163,306,284,262,229,218,196),(164,307,285,263,230,219,197),(165,308,286,264,231,220,198),(166,287,265,243,232,199,177),(167,288,266,244,233,200,178),(168,289,267,245,234,201,179),(169,290,268,246,235,202,180),(170,291,269,247,236,203,181),(171,292,270,248,237,204,182),(172,293,271,249,238,205,183),(173,294,272,250,239,206,184),(174,295,273,251,240,207,185),(175,296,274,252,241,208,186),(176,297,275,253,242,209,187)], [(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)], [(1,155,12,166),(2,176,13,165),(3,175,14,164),(4,174,15,163),(5,173,16,162),(6,172,17,161),(7,171,18,160),(8,170,19,159),(9,169,20,158),(10,168,21,157),(11,167,22,156),(23,186,34,197),(24,185,35,196),(25,184,36,195),(26,183,37,194),(27,182,38,193),(28,181,39,192),(29,180,40,191),(30,179,41,190),(31,178,42,189),(32,177,43,188),(33,198,44,187),(45,217,56,206),(46,216,57,205),(47,215,58,204),(48,214,59,203),(49,213,60,202),(50,212,61,201),(51,211,62,200),(52,210,63,199),(53,209,64,220),(54,208,65,219),(55,207,66,218),(67,238,78,227),(68,237,79,226),(69,236,80,225),(70,235,81,224),(71,234,82,223),(72,233,83,222),(73,232,84,221),(74,231,85,242),(75,230,86,241),(76,229,87,240),(77,228,88,239),(89,254,100,243),(90,253,101,264),(91,252,102,263),(92,251,103,262),(93,250,104,261),(94,249,105,260),(95,248,106,259),(96,247,107,258),(97,246,108,257),(98,245,109,256),(99,244,110,255),(111,279,122,268),(112,278,123,267),(113,277,124,266),(114,276,125,265),(115,275,126,286),(116,274,127,285),(117,273,128,284),(118,272,129,283),(119,271,130,282),(120,270,131,281),(121,269,132,280),(133,306,144,295),(134,305,145,294),(135,304,146,293),(136,303,147,292),(137,302,148,291),(138,301,149,290),(139,300,150,289),(140,299,151,288),(141,298,152,287),(142,297,153,308),(143,296,154,307)]])
98 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7F | 11A | ··· | 11E | 14A | ··· | 14F | 22A | ··· | 22E | 28A | ··· | 28L | 77A | ··· | 77AD | 154A | ··· | 154AD |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 11 | ··· | 11 | 14 | ··· | 14 | 22 | ··· | 22 | 28 | ··· | 28 | 77 | ··· | 77 | 154 | ··· | 154 |
| size | 1 | 1 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C7 | C14 | C28 | D11 | Dic11 | C7×D11 | C7×Dic11 |
| kernel | C7×Dic11 | C154 | C77 | Dic11 | C22 | C11 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 5 | 5 | 30 | 30 |
Matrix representation of C7×Dic11 ►in GL2(𝔽43) generated by
| 16 | 0 |
| 0 | 16 |
| 9 | 7 |
| 28 | 41 |
| 0 | 22 |
| 41 | 0 |
G:=sub<GL(2,GF(43))| [16,0,0,16],[9,28,7,41],[0,41,22,0] >;
C7×Dic11 in GAP, Magma, Sage, TeX
C_7\times {\rm Dic}_{11} % in TeX
G:=Group("C7xDic11"); // GroupNames label
G:=SmallGroup(308,2);
// by ID
G=gap.SmallGroup(308,2);
# by ID
G:=PCGroup([4,-2,-7,-2,-11,56,4483]);
// Polycyclic
G:=Group<a,b,c|a^7=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export