direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic23, C23⋊C12, C69⋊2C4, C46.C6, C6.2D23, C138.2C2, C2.(C3×D23), SmallGroup(276,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C3×Dic23 |
Generators and relations for C3×Dic23
G = < a,b,c | a3=b46=1, c2=b23, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic23
►Regular action on 276 points
Generators in S276
(1 117 58)(2 118 59)(3 119 60)(4 120 61)(5 121 62)(6 122 63)(7 123 64)(8 124 65)(9 125 66)(10 126 67)(11 127 68)(12 128 69)(13 129 70)(14 130 71)(15 131 72)(16 132 73)(17 133 74)(18 134 75)(19 135 76)(20 136 77)(21 137 78)(22 138 79)(23 93 80)(24 94 81)(25 95 82)(26 96 83)(27 97 84)(28 98 85)(29 99 86)(30 100 87)(31 101 88)(32 102 89)(33 103 90)(34 104 91)(35 105 92)(36 106 47)(37 107 48)(38 108 49)(39 109 50)(40 110 51)(41 111 52)(42 112 53)(43 113 54)(44 114 55)(45 115 56)(46 116 57)(139 231 208)(140 232 209)(141 233 210)(142 234 211)(143 235 212)(144 236 213)(145 237 214)(146 238 215)(147 239 216)(148 240 217)(149 241 218)(150 242 219)(151 243 220)(152 244 221)(153 245 222)(154 246 223)(155 247 224)(156 248 225)(157 249 226)(158 250 227)(159 251 228)(160 252 229)(161 253 230)(162 254 185)(163 255 186)(164 256 187)(165 257 188)(166 258 189)(167 259 190)(168 260 191)(169 261 192)(170 262 193)(171 263 194)(172 264 195)(173 265 196)(174 266 197)(175 267 198)(176 268 199)(177 269 200)(178 270 201)(179 271 202)(180 272 203)(181 273 204)(182 274 205)(183 275 206)(184 276 207)
(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)
(1 162 24 139)(2 161 25 184)(3 160 26 183)(4 159 27 182)(5 158 28 181)(6 157 29 180)(7 156 30 179)(8 155 31 178)(9 154 32 177)(10 153 33 176)(11 152 34 175)(12 151 35 174)(13 150 36 173)(14 149 37 172)(15 148 38 171)(16 147 39 170)(17 146 40 169)(18 145 41 168)(19 144 42 167)(20 143 43 166)(21 142 44 165)(22 141 45 164)(23 140 46 163)(47 196 70 219)(48 195 71 218)(49 194 72 217)(50 193 73 216)(51 192 74 215)(52 191 75 214)(53 190 76 213)(54 189 77 212)(55 188 78 211)(56 187 79 210)(57 186 80 209)(58 185 81 208)(59 230 82 207)(60 229 83 206)(61 228 84 205)(62 227 85 204)(63 226 86 203)(64 225 87 202)(65 224 88 201)(66 223 89 200)(67 222 90 199)(68 221 91 198)(69 220 92 197)(93 232 116 255)(94 231 117 254)(95 276 118 253)(96 275 119 252)(97 274 120 251)(98 273 121 250)(99 272 122 249)(100 271 123 248)(101 270 124 247)(102 269 125 246)(103 268 126 245)(104 267 127 244)(105 266 128 243)(106 265 129 242)(107 264 130 241)(108 263 131 240)(109 262 132 239)(110 261 133 238)(111 260 134 237)(112 259 135 236)(113 258 136 235)(114 257 137 234)(115 256 138 233)
G:=sub<Sym(276)| (1,117,58)(2,118,59)(3,119,60)(4,120,61)(5,121,62)(6,122,63)(7,123,64)(8,124,65)(9,125,66)(10,126,67)(11,127,68)(12,128,69)(13,129,70)(14,130,71)(15,131,72)(16,132,73)(17,133,74)(18,134,75)(19,135,76)(20,136,77)(21,137,78)(22,138,79)(23,93,80)(24,94,81)(25,95,82)(26,96,83)(27,97,84)(28,98,85)(29,99,86)(30,100,87)(31,101,88)(32,102,89)(33,103,90)(34,104,91)(35,105,92)(36,106,47)(37,107,48)(38,108,49)(39,109,50)(40,110,51)(41,111,52)(42,112,53)(43,113,54)(44,114,55)(45,115,56)(46,116,57)(139,231,208)(140,232,209)(141,233,210)(142,234,211)(143,235,212)(144,236,213)(145,237,214)(146,238,215)(147,239,216)(148,240,217)(149,241,218)(150,242,219)(151,243,220)(152,244,221)(153,245,222)(154,246,223)(155,247,224)(156,248,225)(157,249,226)(158,250,227)(159,251,228)(160,252,229)(161,253,230)(162,254,185)(163,255,186)(164,256,187)(165,257,188)(166,258,189)(167,259,190)(168,260,191)(169,261,192)(170,262,193)(171,263,194)(172,264,195)(173,265,196)(174,266,197)(175,267,198)(176,268,199)(177,269,200)(178,270,201)(179,271,202)(180,272,203)(181,273,204)(182,274,205)(183,275,206)(184,276,207), (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), (1,162,24,139)(2,161,25,184)(3,160,26,183)(4,159,27,182)(5,158,28,181)(6,157,29,180)(7,156,30,179)(8,155,31,178)(9,154,32,177)(10,153,33,176)(11,152,34,175)(12,151,35,174)(13,150,36,173)(14,149,37,172)(15,148,38,171)(16,147,39,170)(17,146,40,169)(18,145,41,168)(19,144,42,167)(20,143,43,166)(21,142,44,165)(22,141,45,164)(23,140,46,163)(47,196,70,219)(48,195,71,218)(49,194,72,217)(50,193,73,216)(51,192,74,215)(52,191,75,214)(53,190,76,213)(54,189,77,212)(55,188,78,211)(56,187,79,210)(57,186,80,209)(58,185,81,208)(59,230,82,207)(60,229,83,206)(61,228,84,205)(62,227,85,204)(63,226,86,203)(64,225,87,202)(65,224,88,201)(66,223,89,200)(67,222,90,199)(68,221,91,198)(69,220,92,197)(93,232,116,255)(94,231,117,254)(95,276,118,253)(96,275,119,252)(97,274,120,251)(98,273,121,250)(99,272,122,249)(100,271,123,248)(101,270,124,247)(102,269,125,246)(103,268,126,245)(104,267,127,244)(105,266,128,243)(106,265,129,242)(107,264,130,241)(108,263,131,240)(109,262,132,239)(110,261,133,238)(111,260,134,237)(112,259,135,236)(113,258,136,235)(114,257,137,234)(115,256,138,233)>;
G:=Group( (1,117,58)(2,118,59)(3,119,60)(4,120,61)(5,121,62)(6,122,63)(7,123,64)(8,124,65)(9,125,66)(10,126,67)(11,127,68)(12,128,69)(13,129,70)(14,130,71)(15,131,72)(16,132,73)(17,133,74)(18,134,75)(19,135,76)(20,136,77)(21,137,78)(22,138,79)(23,93,80)(24,94,81)(25,95,82)(26,96,83)(27,97,84)(28,98,85)(29,99,86)(30,100,87)(31,101,88)(32,102,89)(33,103,90)(34,104,91)(35,105,92)(36,106,47)(37,107,48)(38,108,49)(39,109,50)(40,110,51)(41,111,52)(42,112,53)(43,113,54)(44,114,55)(45,115,56)(46,116,57)(139,231,208)(140,232,209)(141,233,210)(142,234,211)(143,235,212)(144,236,213)(145,237,214)(146,238,215)(147,239,216)(148,240,217)(149,241,218)(150,242,219)(151,243,220)(152,244,221)(153,245,222)(154,246,223)(155,247,224)(156,248,225)(157,249,226)(158,250,227)(159,251,228)(160,252,229)(161,253,230)(162,254,185)(163,255,186)(164,256,187)(165,257,188)(166,258,189)(167,259,190)(168,260,191)(169,261,192)(170,262,193)(171,263,194)(172,264,195)(173,265,196)(174,266,197)(175,267,198)(176,268,199)(177,269,200)(178,270,201)(179,271,202)(180,272,203)(181,273,204)(182,274,205)(183,275,206)(184,276,207), (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), (1,162,24,139)(2,161,25,184)(3,160,26,183)(4,159,27,182)(5,158,28,181)(6,157,29,180)(7,156,30,179)(8,155,31,178)(9,154,32,177)(10,153,33,176)(11,152,34,175)(12,151,35,174)(13,150,36,173)(14,149,37,172)(15,148,38,171)(16,147,39,170)(17,146,40,169)(18,145,41,168)(19,144,42,167)(20,143,43,166)(21,142,44,165)(22,141,45,164)(23,140,46,163)(47,196,70,219)(48,195,71,218)(49,194,72,217)(50,193,73,216)(51,192,74,215)(52,191,75,214)(53,190,76,213)(54,189,77,212)(55,188,78,211)(56,187,79,210)(57,186,80,209)(58,185,81,208)(59,230,82,207)(60,229,83,206)(61,228,84,205)(62,227,85,204)(63,226,86,203)(64,225,87,202)(65,224,88,201)(66,223,89,200)(67,222,90,199)(68,221,91,198)(69,220,92,197)(93,232,116,255)(94,231,117,254)(95,276,118,253)(96,275,119,252)(97,274,120,251)(98,273,121,250)(99,272,122,249)(100,271,123,248)(101,270,124,247)(102,269,125,246)(103,268,126,245)(104,267,127,244)(105,266,128,243)(106,265,129,242)(107,264,130,241)(108,263,131,240)(109,262,132,239)(110,261,133,238)(111,260,134,237)(112,259,135,236)(113,258,136,235)(114,257,137,234)(115,256,138,233) );
G=PermutationGroup([[(1,117,58),(2,118,59),(3,119,60),(4,120,61),(5,121,62),(6,122,63),(7,123,64),(8,124,65),(9,125,66),(10,126,67),(11,127,68),(12,128,69),(13,129,70),(14,130,71),(15,131,72),(16,132,73),(17,133,74),(18,134,75),(19,135,76),(20,136,77),(21,137,78),(22,138,79),(23,93,80),(24,94,81),(25,95,82),(26,96,83),(27,97,84),(28,98,85),(29,99,86),(30,100,87),(31,101,88),(32,102,89),(33,103,90),(34,104,91),(35,105,92),(36,106,47),(37,107,48),(38,108,49),(39,109,50),(40,110,51),(41,111,52),(42,112,53),(43,113,54),(44,114,55),(45,115,56),(46,116,57),(139,231,208),(140,232,209),(141,233,210),(142,234,211),(143,235,212),(144,236,213),(145,237,214),(146,238,215),(147,239,216),(148,240,217),(149,241,218),(150,242,219),(151,243,220),(152,244,221),(153,245,222),(154,246,223),(155,247,224),(156,248,225),(157,249,226),(158,250,227),(159,251,228),(160,252,229),(161,253,230),(162,254,185),(163,255,186),(164,256,187),(165,257,188),(166,258,189),(167,259,190),(168,260,191),(169,261,192),(170,262,193),(171,263,194),(172,264,195),(173,265,196),(174,266,197),(175,267,198),(176,268,199),(177,269,200),(178,270,201),(179,271,202),(180,272,203),(181,273,204),(182,274,205),(183,275,206),(184,276,207)], [(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)], [(1,162,24,139),(2,161,25,184),(3,160,26,183),(4,159,27,182),(5,158,28,181),(6,157,29,180),(7,156,30,179),(8,155,31,178),(9,154,32,177),(10,153,33,176),(11,152,34,175),(12,151,35,174),(13,150,36,173),(14,149,37,172),(15,148,38,171),(16,147,39,170),(17,146,40,169),(18,145,41,168),(19,144,42,167),(20,143,43,166),(21,142,44,165),(22,141,45,164),(23,140,46,163),(47,196,70,219),(48,195,71,218),(49,194,72,217),(50,193,73,216),(51,192,74,215),(52,191,75,214),(53,190,76,213),(54,189,77,212),(55,188,78,211),(56,187,79,210),(57,186,80,209),(58,185,81,208),(59,230,82,207),(60,229,83,206),(61,228,84,205),(62,227,85,204),(63,226,86,203),(64,225,87,202),(65,224,88,201),(66,223,89,200),(67,222,90,199),(68,221,91,198),(69,220,92,197),(93,232,116,255),(94,231,117,254),(95,276,118,253),(96,275,119,252),(97,274,120,251),(98,273,121,250),(99,272,122,249),(100,271,123,248),(101,270,124,247),(102,269,125,246),(103,268,126,245),(104,267,127,244),(105,266,128,243),(106,265,129,242),(107,264,130,241),(108,263,131,240),(109,262,132,239),(110,261,133,238),(111,260,134,237),(112,259,135,236),(113,258,136,235),(114,257,137,234),(115,256,138,233)]])
78 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 23A | ··· | 23K | 46A | ··· | 46K | 69A | ··· | 69V | 138A | ··· | 138V |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 138 | ··· | 138 |
| size | 1 | 1 | 1 | 1 | 23 | 23 | 1 | 1 | 23 | 23 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D23 | Dic23 | C3×D23 | C3×Dic23 |
| kernel | C3×Dic23 | C138 | Dic23 | C69 | C46 | C23 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 11 | 11 | 22 | 22 |
Matrix representation of C3×Dic23 ►in GL3(𝔽277) generated by
| 1 | 0 | 0 |
| 0 | 116 | 0 |
| 0 | 0 | 116 |
| 276 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 276 | 65 |
| 60 | 0 | 0 |
| 0 | 12 | 69 |
| 0 | 18 | 265 |
G:=sub<GL(3,GF(277))| [1,0,0,0,116,0,0,0,116],[276,0,0,0,0,276,0,1,65],[60,0,0,0,12,18,0,69,265] >;
C3×Dic23 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{23} % in TeX
G:=Group("C3xDic23"); // GroupNames label
G:=SmallGroup(276,2);
// by ID
G=gap.SmallGroup(276,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-23,24,4227]);
// Polycyclic
G:=Group<a,b,c|a^3=b^46=1,c^2=b^23,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export