direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C23, C3⋊C92, C69⋊3C4, C6.C46, C46.2S3, C138.3C2, C2.(S3×C23), SmallGroup(276,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C23 |
Generators and relations for Dic3×C23
G = < a,b,c | a23=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C23
►Regular action on 276 points
Generators in S276
(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)
(1 56 106 258 82 215)(2 57 107 259 83 216)(3 58 108 260 84 217)(4 59 109 261 85 218)(5 60 110 262 86 219)(6 61 111 263 87 220)(7 62 112 264 88 221)(8 63 113 265 89 222)(9 64 114 266 90 223)(10 65 115 267 91 224)(11 66 93 268 92 225)(12 67 94 269 70 226)(13 68 95 270 71 227)(14 69 96 271 72 228)(15 47 97 272 73 229)(16 48 98 273 74 230)(17 49 99 274 75 208)(18 50 100 275 76 209)(19 51 101 276 77 210)(20 52 102 254 78 211)(21 53 103 255 79 212)(22 54 104 256 80 213)(23 55 105 257 81 214)(24 169 118 244 205 153)(25 170 119 245 206 154)(26 171 120 246 207 155)(27 172 121 247 185 156)(28 173 122 248 186 157)(29 174 123 249 187 158)(30 175 124 250 188 159)(31 176 125 251 189 160)(32 177 126 252 190 161)(33 178 127 253 191 139)(34 179 128 231 192 140)(35 180 129 232 193 141)(36 181 130 233 194 142)(37 182 131 234 195 143)(38 183 132 235 196 144)(39 184 133 236 197 145)(40 162 134 237 198 146)(41 163 135 238 199 147)(42 164 136 239 200 148)(43 165 137 240 201 149)(44 166 138 241 202 150)(45 167 116 242 203 151)(46 168 117 243 204 152)
(1 123 258 158)(2 124 259 159)(3 125 260 160)(4 126 261 161)(5 127 262 139)(6 128 263 140)(7 129 264 141)(8 130 265 142)(9 131 266 143)(10 132 267 144)(11 133 268 145)(12 134 269 146)(13 135 270 147)(14 136 271 148)(15 137 272 149)(16 138 273 150)(17 116 274 151)(18 117 275 152)(19 118 276 153)(20 119 254 154)(21 120 255 155)(22 121 256 156)(23 122 257 157)(24 210 244 101)(25 211 245 102)(26 212 246 103)(27 213 247 104)(28 214 248 105)(29 215 249 106)(30 216 250 107)(31 217 251 108)(32 218 252 109)(33 219 253 110)(34 220 231 111)(35 221 232 112)(36 222 233 113)(37 223 234 114)(38 224 235 115)(39 225 236 93)(40 226 237 94)(41 227 238 95)(42 228 239 96)(43 229 240 97)(44 230 241 98)(45 208 242 99)(46 209 243 100)(47 165 73 201)(48 166 74 202)(49 167 75 203)(50 168 76 204)(51 169 77 205)(52 170 78 206)(53 171 79 207)(54 172 80 185)(55 173 81 186)(56 174 82 187)(57 175 83 188)(58 176 84 189)(59 177 85 190)(60 178 86 191)(61 179 87 192)(62 180 88 193)(63 181 89 194)(64 182 90 195)(65 183 91 196)(66 184 92 197)(67 162 70 198)(68 163 71 199)(69 164 72 200)
G:=sub<Sym(276)| (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), (1,56,106,258,82,215)(2,57,107,259,83,216)(3,58,108,260,84,217)(4,59,109,261,85,218)(5,60,110,262,86,219)(6,61,111,263,87,220)(7,62,112,264,88,221)(8,63,113,265,89,222)(9,64,114,266,90,223)(10,65,115,267,91,224)(11,66,93,268,92,225)(12,67,94,269,70,226)(13,68,95,270,71,227)(14,69,96,271,72,228)(15,47,97,272,73,229)(16,48,98,273,74,230)(17,49,99,274,75,208)(18,50,100,275,76,209)(19,51,101,276,77,210)(20,52,102,254,78,211)(21,53,103,255,79,212)(22,54,104,256,80,213)(23,55,105,257,81,214)(24,169,118,244,205,153)(25,170,119,245,206,154)(26,171,120,246,207,155)(27,172,121,247,185,156)(28,173,122,248,186,157)(29,174,123,249,187,158)(30,175,124,250,188,159)(31,176,125,251,189,160)(32,177,126,252,190,161)(33,178,127,253,191,139)(34,179,128,231,192,140)(35,180,129,232,193,141)(36,181,130,233,194,142)(37,182,131,234,195,143)(38,183,132,235,196,144)(39,184,133,236,197,145)(40,162,134,237,198,146)(41,163,135,238,199,147)(42,164,136,239,200,148)(43,165,137,240,201,149)(44,166,138,241,202,150)(45,167,116,242,203,151)(46,168,117,243,204,152), (1,123,258,158)(2,124,259,159)(3,125,260,160)(4,126,261,161)(5,127,262,139)(6,128,263,140)(7,129,264,141)(8,130,265,142)(9,131,266,143)(10,132,267,144)(11,133,268,145)(12,134,269,146)(13,135,270,147)(14,136,271,148)(15,137,272,149)(16,138,273,150)(17,116,274,151)(18,117,275,152)(19,118,276,153)(20,119,254,154)(21,120,255,155)(22,121,256,156)(23,122,257,157)(24,210,244,101)(25,211,245,102)(26,212,246,103)(27,213,247,104)(28,214,248,105)(29,215,249,106)(30,216,250,107)(31,217,251,108)(32,218,252,109)(33,219,253,110)(34,220,231,111)(35,221,232,112)(36,222,233,113)(37,223,234,114)(38,224,235,115)(39,225,236,93)(40,226,237,94)(41,227,238,95)(42,228,239,96)(43,229,240,97)(44,230,241,98)(45,208,242,99)(46,209,243,100)(47,165,73,201)(48,166,74,202)(49,167,75,203)(50,168,76,204)(51,169,77,205)(52,170,78,206)(53,171,79,207)(54,172,80,185)(55,173,81,186)(56,174,82,187)(57,175,83,188)(58,176,84,189)(59,177,85,190)(60,178,86,191)(61,179,87,192)(62,180,88,193)(63,181,89,194)(64,182,90,195)(65,183,91,196)(66,184,92,197)(67,162,70,198)(68,163,71,199)(69,164,72,200)>;
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), (1,56,106,258,82,215)(2,57,107,259,83,216)(3,58,108,260,84,217)(4,59,109,261,85,218)(5,60,110,262,86,219)(6,61,111,263,87,220)(7,62,112,264,88,221)(8,63,113,265,89,222)(9,64,114,266,90,223)(10,65,115,267,91,224)(11,66,93,268,92,225)(12,67,94,269,70,226)(13,68,95,270,71,227)(14,69,96,271,72,228)(15,47,97,272,73,229)(16,48,98,273,74,230)(17,49,99,274,75,208)(18,50,100,275,76,209)(19,51,101,276,77,210)(20,52,102,254,78,211)(21,53,103,255,79,212)(22,54,104,256,80,213)(23,55,105,257,81,214)(24,169,118,244,205,153)(25,170,119,245,206,154)(26,171,120,246,207,155)(27,172,121,247,185,156)(28,173,122,248,186,157)(29,174,123,249,187,158)(30,175,124,250,188,159)(31,176,125,251,189,160)(32,177,126,252,190,161)(33,178,127,253,191,139)(34,179,128,231,192,140)(35,180,129,232,193,141)(36,181,130,233,194,142)(37,182,131,234,195,143)(38,183,132,235,196,144)(39,184,133,236,197,145)(40,162,134,237,198,146)(41,163,135,238,199,147)(42,164,136,239,200,148)(43,165,137,240,201,149)(44,166,138,241,202,150)(45,167,116,242,203,151)(46,168,117,243,204,152), (1,123,258,158)(2,124,259,159)(3,125,260,160)(4,126,261,161)(5,127,262,139)(6,128,263,140)(7,129,264,141)(8,130,265,142)(9,131,266,143)(10,132,267,144)(11,133,268,145)(12,134,269,146)(13,135,270,147)(14,136,271,148)(15,137,272,149)(16,138,273,150)(17,116,274,151)(18,117,275,152)(19,118,276,153)(20,119,254,154)(21,120,255,155)(22,121,256,156)(23,122,257,157)(24,210,244,101)(25,211,245,102)(26,212,246,103)(27,213,247,104)(28,214,248,105)(29,215,249,106)(30,216,250,107)(31,217,251,108)(32,218,252,109)(33,219,253,110)(34,220,231,111)(35,221,232,112)(36,222,233,113)(37,223,234,114)(38,224,235,115)(39,225,236,93)(40,226,237,94)(41,227,238,95)(42,228,239,96)(43,229,240,97)(44,230,241,98)(45,208,242,99)(46,209,243,100)(47,165,73,201)(48,166,74,202)(49,167,75,203)(50,168,76,204)(51,169,77,205)(52,170,78,206)(53,171,79,207)(54,172,80,185)(55,173,81,186)(56,174,82,187)(57,175,83,188)(58,176,84,189)(59,177,85,190)(60,178,86,191)(61,179,87,192)(62,180,88,193)(63,181,89,194)(64,182,90,195)(65,183,91,196)(66,184,92,197)(67,162,70,198)(68,163,71,199)(69,164,72,200) );
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)], [(1,56,106,258,82,215),(2,57,107,259,83,216),(3,58,108,260,84,217),(4,59,109,261,85,218),(5,60,110,262,86,219),(6,61,111,263,87,220),(7,62,112,264,88,221),(8,63,113,265,89,222),(9,64,114,266,90,223),(10,65,115,267,91,224),(11,66,93,268,92,225),(12,67,94,269,70,226),(13,68,95,270,71,227),(14,69,96,271,72,228),(15,47,97,272,73,229),(16,48,98,273,74,230),(17,49,99,274,75,208),(18,50,100,275,76,209),(19,51,101,276,77,210),(20,52,102,254,78,211),(21,53,103,255,79,212),(22,54,104,256,80,213),(23,55,105,257,81,214),(24,169,118,244,205,153),(25,170,119,245,206,154),(26,171,120,246,207,155),(27,172,121,247,185,156),(28,173,122,248,186,157),(29,174,123,249,187,158),(30,175,124,250,188,159),(31,176,125,251,189,160),(32,177,126,252,190,161),(33,178,127,253,191,139),(34,179,128,231,192,140),(35,180,129,232,193,141),(36,181,130,233,194,142),(37,182,131,234,195,143),(38,183,132,235,196,144),(39,184,133,236,197,145),(40,162,134,237,198,146),(41,163,135,238,199,147),(42,164,136,239,200,148),(43,165,137,240,201,149),(44,166,138,241,202,150),(45,167,116,242,203,151),(46,168,117,243,204,152)], [(1,123,258,158),(2,124,259,159),(3,125,260,160),(4,126,261,161),(5,127,262,139),(6,128,263,140),(7,129,264,141),(8,130,265,142),(9,131,266,143),(10,132,267,144),(11,133,268,145),(12,134,269,146),(13,135,270,147),(14,136,271,148),(15,137,272,149),(16,138,273,150),(17,116,274,151),(18,117,275,152),(19,118,276,153),(20,119,254,154),(21,120,255,155),(22,121,256,156),(23,122,257,157),(24,210,244,101),(25,211,245,102),(26,212,246,103),(27,213,247,104),(28,214,248,105),(29,215,249,106),(30,216,250,107),(31,217,251,108),(32,218,252,109),(33,219,253,110),(34,220,231,111),(35,221,232,112),(36,222,233,113),(37,223,234,114),(38,224,235,115),(39,225,236,93),(40,226,237,94),(41,227,238,95),(42,228,239,96),(43,229,240,97),(44,230,241,98),(45,208,242,99),(46,209,243,100),(47,165,73,201),(48,166,74,202),(49,167,75,203),(50,168,76,204),(51,169,77,205),(52,170,78,206),(53,171,79,207),(54,172,80,185),(55,173,81,186),(56,174,82,187),(57,175,83,188),(58,176,84,189),(59,177,85,190),(60,178,86,191),(61,179,87,192),(62,180,88,193),(63,181,89,194),(64,182,90,195),(65,183,91,196),(66,184,92,197),(67,162,70,198),(68,163,71,199),(69,164,72,200)]])
138 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 23A | ··· | 23V | 46A | ··· | 46V | 69A | ··· | 69V | 92A | ··· | 92AR | 138A | ··· | 138V |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 92 | ··· | 92 | 138 | ··· | 138 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
138 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C23 | C46 | C92 | S3 | Dic3 | S3×C23 | Dic3×C23 |
| kernel | Dic3×C23 | C138 | C69 | Dic3 | C6 | C3 | C46 | C23 | C2 | C1 |
| # reps | 1 | 1 | 2 | 22 | 22 | 44 | 1 | 1 | 22 | 22 |
Matrix representation of Dic3×C23 ►in GL2(𝔽47) generated by
| 37 | 0 |
| 0 | 37 |
| 17 | 15 |
| 10 | 31 |
| 0 | 5 |
| 28 | 0 |
G:=sub<GL(2,GF(47))| [37,0,0,37],[17,10,15,31],[0,28,5,0] >;
Dic3×C23 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{23} % in TeX
G:=Group("Dic3xC23"); // GroupNames label
G:=SmallGroup(276,1);
// by ID
G=gap.SmallGroup(276,1);
# by ID
G:=PCGroup([4,-2,-23,-2,-3,184,2947]);
// Polycyclic
G:=Group<a,b,c|a^23=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export