direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic22, C33⋊3Q8, C44.1C6, C132.3C2, C6.13D22, C12.3D11, Dic11.C6, C66.13C22, C11⋊(C3×Q8), C4.(C3×D11), C22.1(C2×C6), C2.3(C6×D11), (C3×Dic11).2C2, SmallGroup(264,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic22
G = < a,b,c | a3=b44=1, c2=b22, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic22
►Regular action on 264 points
Generators in S264
(1 45 231)(2 46 232)(3 47 233)(4 48 234)(5 49 235)(6 50 236)(7 51 237)(8 52 238)(9 53 239)(10 54 240)(11 55 241)(12 56 242)(13 57 243)(14 58 244)(15 59 245)(16 60 246)(17 61 247)(18 62 248)(19 63 249)(20 64 250)(21 65 251)(22 66 252)(23 67 253)(24 68 254)(25 69 255)(26 70 256)(27 71 257)(28 72 258)(29 73 259)(30 74 260)(31 75 261)(32 76 262)(33 77 263)(34 78 264)(35 79 221)(36 80 222)(37 81 223)(38 82 224)(39 83 225)(40 84 226)(41 85 227)(42 86 228)(43 87 229)(44 88 230)(89 197 156)(90 198 157)(91 199 158)(92 200 159)(93 201 160)(94 202 161)(95 203 162)(96 204 163)(97 205 164)(98 206 165)(99 207 166)(100 208 167)(101 209 168)(102 210 169)(103 211 170)(104 212 171)(105 213 172)(106 214 173)(107 215 174)(108 216 175)(109 217 176)(110 218 133)(111 219 134)(112 220 135)(113 177 136)(114 178 137)(115 179 138)(116 180 139)(117 181 140)(118 182 141)(119 183 142)(120 184 143)(121 185 144)(122 186 145)(123 187 146)(124 188 147)(125 189 148)(126 190 149)(127 191 150)(128 192 151)(129 193 152)(130 194 153)(131 195 154)(132 196 155)
(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)
(1 94 23 116)(2 93 24 115)(3 92 25 114)(4 91 26 113)(5 90 27 112)(6 89 28 111)(7 132 29 110)(8 131 30 109)(9 130 31 108)(10 129 32 107)(11 128 33 106)(12 127 34 105)(13 126 35 104)(14 125 36 103)(15 124 37 102)(16 123 38 101)(17 122 39 100)(18 121 40 99)(19 120 41 98)(20 119 42 97)(21 118 43 96)(22 117 44 95)(45 202 67 180)(46 201 68 179)(47 200 69 178)(48 199 70 177)(49 198 71 220)(50 197 72 219)(51 196 73 218)(52 195 74 217)(53 194 75 216)(54 193 76 215)(55 192 77 214)(56 191 78 213)(57 190 79 212)(58 189 80 211)(59 188 81 210)(60 187 82 209)(61 186 83 208)(62 185 84 207)(63 184 85 206)(64 183 86 205)(65 182 87 204)(66 181 88 203)(133 237 155 259)(134 236 156 258)(135 235 157 257)(136 234 158 256)(137 233 159 255)(138 232 160 254)(139 231 161 253)(140 230 162 252)(141 229 163 251)(142 228 164 250)(143 227 165 249)(144 226 166 248)(145 225 167 247)(146 224 168 246)(147 223 169 245)(148 222 170 244)(149 221 171 243)(150 264 172 242)(151 263 173 241)(152 262 174 240)(153 261 175 239)(154 260 176 238)
G:=sub<Sym(264)| (1,45,231)(2,46,232)(3,47,233)(4,48,234)(5,49,235)(6,50,236)(7,51,237)(8,52,238)(9,53,239)(10,54,240)(11,55,241)(12,56,242)(13,57,243)(14,58,244)(15,59,245)(16,60,246)(17,61,247)(18,62,248)(19,63,249)(20,64,250)(21,65,251)(22,66,252)(23,67,253)(24,68,254)(25,69,255)(26,70,256)(27,71,257)(28,72,258)(29,73,259)(30,74,260)(31,75,261)(32,76,262)(33,77,263)(34,78,264)(35,79,221)(36,80,222)(37,81,223)(38,82,224)(39,83,225)(40,84,226)(41,85,227)(42,86,228)(43,87,229)(44,88,230)(89,197,156)(90,198,157)(91,199,158)(92,200,159)(93,201,160)(94,202,161)(95,203,162)(96,204,163)(97,205,164)(98,206,165)(99,207,166)(100,208,167)(101,209,168)(102,210,169)(103,211,170)(104,212,171)(105,213,172)(106,214,173)(107,215,174)(108,216,175)(109,217,176)(110,218,133)(111,219,134)(112,220,135)(113,177,136)(114,178,137)(115,179,138)(116,180,139)(117,181,140)(118,182,141)(119,183,142)(120,184,143)(121,185,144)(122,186,145)(123,187,146)(124,188,147)(125,189,148)(126,190,149)(127,191,150)(128,192,151)(129,193,152)(130,194,153)(131,195,154)(132,196,155), (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), (1,94,23,116)(2,93,24,115)(3,92,25,114)(4,91,26,113)(5,90,27,112)(6,89,28,111)(7,132,29,110)(8,131,30,109)(9,130,31,108)(10,129,32,107)(11,128,33,106)(12,127,34,105)(13,126,35,104)(14,125,36,103)(15,124,37,102)(16,123,38,101)(17,122,39,100)(18,121,40,99)(19,120,41,98)(20,119,42,97)(21,118,43,96)(22,117,44,95)(45,202,67,180)(46,201,68,179)(47,200,69,178)(48,199,70,177)(49,198,71,220)(50,197,72,219)(51,196,73,218)(52,195,74,217)(53,194,75,216)(54,193,76,215)(55,192,77,214)(56,191,78,213)(57,190,79,212)(58,189,80,211)(59,188,81,210)(60,187,82,209)(61,186,83,208)(62,185,84,207)(63,184,85,206)(64,183,86,205)(65,182,87,204)(66,181,88,203)(133,237,155,259)(134,236,156,258)(135,235,157,257)(136,234,158,256)(137,233,159,255)(138,232,160,254)(139,231,161,253)(140,230,162,252)(141,229,163,251)(142,228,164,250)(143,227,165,249)(144,226,166,248)(145,225,167,247)(146,224,168,246)(147,223,169,245)(148,222,170,244)(149,221,171,243)(150,264,172,242)(151,263,173,241)(152,262,174,240)(153,261,175,239)(154,260,176,238)>;
G:=Group( (1,45,231)(2,46,232)(3,47,233)(4,48,234)(5,49,235)(6,50,236)(7,51,237)(8,52,238)(9,53,239)(10,54,240)(11,55,241)(12,56,242)(13,57,243)(14,58,244)(15,59,245)(16,60,246)(17,61,247)(18,62,248)(19,63,249)(20,64,250)(21,65,251)(22,66,252)(23,67,253)(24,68,254)(25,69,255)(26,70,256)(27,71,257)(28,72,258)(29,73,259)(30,74,260)(31,75,261)(32,76,262)(33,77,263)(34,78,264)(35,79,221)(36,80,222)(37,81,223)(38,82,224)(39,83,225)(40,84,226)(41,85,227)(42,86,228)(43,87,229)(44,88,230)(89,197,156)(90,198,157)(91,199,158)(92,200,159)(93,201,160)(94,202,161)(95,203,162)(96,204,163)(97,205,164)(98,206,165)(99,207,166)(100,208,167)(101,209,168)(102,210,169)(103,211,170)(104,212,171)(105,213,172)(106,214,173)(107,215,174)(108,216,175)(109,217,176)(110,218,133)(111,219,134)(112,220,135)(113,177,136)(114,178,137)(115,179,138)(116,180,139)(117,181,140)(118,182,141)(119,183,142)(120,184,143)(121,185,144)(122,186,145)(123,187,146)(124,188,147)(125,189,148)(126,190,149)(127,191,150)(128,192,151)(129,193,152)(130,194,153)(131,195,154)(132,196,155), (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), (1,94,23,116)(2,93,24,115)(3,92,25,114)(4,91,26,113)(5,90,27,112)(6,89,28,111)(7,132,29,110)(8,131,30,109)(9,130,31,108)(10,129,32,107)(11,128,33,106)(12,127,34,105)(13,126,35,104)(14,125,36,103)(15,124,37,102)(16,123,38,101)(17,122,39,100)(18,121,40,99)(19,120,41,98)(20,119,42,97)(21,118,43,96)(22,117,44,95)(45,202,67,180)(46,201,68,179)(47,200,69,178)(48,199,70,177)(49,198,71,220)(50,197,72,219)(51,196,73,218)(52,195,74,217)(53,194,75,216)(54,193,76,215)(55,192,77,214)(56,191,78,213)(57,190,79,212)(58,189,80,211)(59,188,81,210)(60,187,82,209)(61,186,83,208)(62,185,84,207)(63,184,85,206)(64,183,86,205)(65,182,87,204)(66,181,88,203)(133,237,155,259)(134,236,156,258)(135,235,157,257)(136,234,158,256)(137,233,159,255)(138,232,160,254)(139,231,161,253)(140,230,162,252)(141,229,163,251)(142,228,164,250)(143,227,165,249)(144,226,166,248)(145,225,167,247)(146,224,168,246)(147,223,169,245)(148,222,170,244)(149,221,171,243)(150,264,172,242)(151,263,173,241)(152,262,174,240)(153,261,175,239)(154,260,176,238) );
G=PermutationGroup([[(1,45,231),(2,46,232),(3,47,233),(4,48,234),(5,49,235),(6,50,236),(7,51,237),(8,52,238),(9,53,239),(10,54,240),(11,55,241),(12,56,242),(13,57,243),(14,58,244),(15,59,245),(16,60,246),(17,61,247),(18,62,248),(19,63,249),(20,64,250),(21,65,251),(22,66,252),(23,67,253),(24,68,254),(25,69,255),(26,70,256),(27,71,257),(28,72,258),(29,73,259),(30,74,260),(31,75,261),(32,76,262),(33,77,263),(34,78,264),(35,79,221),(36,80,222),(37,81,223),(38,82,224),(39,83,225),(40,84,226),(41,85,227),(42,86,228),(43,87,229),(44,88,230),(89,197,156),(90,198,157),(91,199,158),(92,200,159),(93,201,160),(94,202,161),(95,203,162),(96,204,163),(97,205,164),(98,206,165),(99,207,166),(100,208,167),(101,209,168),(102,210,169),(103,211,170),(104,212,171),(105,213,172),(106,214,173),(107,215,174),(108,216,175),(109,217,176),(110,218,133),(111,219,134),(112,220,135),(113,177,136),(114,178,137),(115,179,138),(116,180,139),(117,181,140),(118,182,141),(119,183,142),(120,184,143),(121,185,144),(122,186,145),(123,187,146),(124,188,147),(125,189,148),(126,190,149),(127,191,150),(128,192,151),(129,193,152),(130,194,153),(131,195,154),(132,196,155)], [(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)], [(1,94,23,116),(2,93,24,115),(3,92,25,114),(4,91,26,113),(5,90,27,112),(6,89,28,111),(7,132,29,110),(8,131,30,109),(9,130,31,108),(10,129,32,107),(11,128,33,106),(12,127,34,105),(13,126,35,104),(14,125,36,103),(15,124,37,102),(16,123,38,101),(17,122,39,100),(18,121,40,99),(19,120,41,98),(20,119,42,97),(21,118,43,96),(22,117,44,95),(45,202,67,180),(46,201,68,179),(47,200,69,178),(48,199,70,177),(49,198,71,220),(50,197,72,219),(51,196,73,218),(52,195,74,217),(53,194,75,216),(54,193,76,215),(55,192,77,214),(56,191,78,213),(57,190,79,212),(58,189,80,211),(59,188,81,210),(60,187,82,209),(61,186,83,208),(62,185,84,207),(63,184,85,206),(64,183,86,205),(65,182,87,204),(66,181,88,203),(133,237,155,259),(134,236,156,258),(135,235,157,257),(136,234,158,256),(137,233,159,255),(138,232,160,254),(139,231,161,253),(140,230,162,252),(141,229,163,251),(142,228,164,250),(143,227,165,249),(144,226,166,248),(145,225,167,247),(146,224,168,246),(147,223,169,245),(148,222,170,244),(149,221,171,243),(150,264,172,242),(151,263,173,241),(152,262,174,240),(153,261,175,239),(154,260,176,238)]])
75 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 11A | ··· | 11E | 12A | 12B | 12C | 12D | 12E | 12F | 22A | ··· | 22E | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 12 | 12 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 1 | 1 | 2 | 22 | 22 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | D11 | C3×Q8 | D22 | C3×D11 | Dic22 | C6×D11 | C3×Dic22 |
| kernel | C3×Dic22 | C3×Dic11 | C132 | Dic22 | Dic11 | C44 | C33 | C12 | C11 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 5 | 2 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of C3×Dic22 ►in GL2(𝔽43) generated by
| 36 | 0 |
| 0 | 36 |
| 30 | 20 |
| 32 | 40 |
| 22 | 33 |
| 27 | 21 |
G:=sub<GL(2,GF(43))| [36,0,0,36],[30,32,20,40],[22,27,33,21] >;
C3×Dic22 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{22} % in TeX
G:=Group("C3xDic22"); // GroupNames label
G:=SmallGroup(264,13);
// by ID
G=gap.SmallGroup(264,13);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-11,60,141,66,6004]);
// Polycyclic
G:=Group<a,b,c|a^3=b^44=1,c^2=b^22,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export