direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×C22, C6⋊C44, C66⋊3C4, C22.16D6, C66.21C22, C33⋊9(C2×C4), C3⋊2(C2×C44), (C2×C6).C22, C2.2(S3×C22), (C2×C66).3C2, (C2×C22).2S3, C6.4(C2×C22), C22.(S3×C11), SmallGroup(264,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C22 |
Generators and relations for Dic3×C22
G = < a,b,c | a22=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C22
►Regular action on 264 points
Generators in S264
(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)
(1 171 77 192 201 225)(2 172 78 193 202 226)(3 173 79 194 203 227)(4 174 80 195 204 228)(5 175 81 196 205 229)(6 176 82 197 206 230)(7 155 83 198 207 231)(8 156 84 177 208 232)(9 157 85 178 209 233)(10 158 86 179 210 234)(11 159 87 180 211 235)(12 160 88 181 212 236)(13 161 67 182 213 237)(14 162 68 183 214 238)(15 163 69 184 215 239)(16 164 70 185 216 240)(17 165 71 186 217 241)(18 166 72 187 218 242)(19 167 73 188 219 221)(20 168 74 189 220 222)(21 169 75 190 199 223)(22 170 76 191 200 224)(23 254 99 54 125 136)(24 255 100 55 126 137)(25 256 101 56 127 138)(26 257 102 57 128 139)(27 258 103 58 129 140)(28 259 104 59 130 141)(29 260 105 60 131 142)(30 261 106 61 132 143)(31 262 107 62 111 144)(32 263 108 63 112 145)(33 264 109 64 113 146)(34 243 110 65 114 147)(35 244 89 66 115 148)(36 245 90 45 116 149)(37 246 91 46 117 150)(38 247 92 47 118 151)(39 248 93 48 119 152)(40 249 94 49 120 153)(41 250 95 50 121 154)(42 251 96 51 122 133)(43 252 97 52 123 134)(44 253 98 53 124 135)
(1 61 192 30)(2 62 193 31)(3 63 194 32)(4 64 195 33)(5 65 196 34)(6 66 197 35)(7 45 198 36)(8 46 177 37)(9 47 178 38)(10 48 179 39)(11 49 180 40)(12 50 181 41)(13 51 182 42)(14 52 183 43)(15 53 184 44)(16 54 185 23)(17 55 186 24)(18 56 187 25)(19 57 188 26)(20 58 189 27)(21 59 190 28)(22 60 191 29)(67 251 237 122)(68 252 238 123)(69 253 239 124)(70 254 240 125)(71 255 241 126)(72 256 242 127)(73 257 221 128)(74 258 222 129)(75 259 223 130)(76 260 224 131)(77 261 225 132)(78 262 226 111)(79 263 227 112)(80 264 228 113)(81 243 229 114)(82 244 230 115)(83 245 231 116)(84 246 232 117)(85 247 233 118)(86 248 234 119)(87 249 235 120)(88 250 236 121)(89 206 148 176)(90 207 149 155)(91 208 150 156)(92 209 151 157)(93 210 152 158)(94 211 153 159)(95 212 154 160)(96 213 133 161)(97 214 134 162)(98 215 135 163)(99 216 136 164)(100 217 137 165)(101 218 138 166)(102 219 139 167)(103 220 140 168)(104 199 141 169)(105 200 142 170)(106 201 143 171)(107 202 144 172)(108 203 145 173)(109 204 146 174)(110 205 147 175)
G:=sub<Sym(264)| (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), (1,171,77,192,201,225)(2,172,78,193,202,226)(3,173,79,194,203,227)(4,174,80,195,204,228)(5,175,81,196,205,229)(6,176,82,197,206,230)(7,155,83,198,207,231)(8,156,84,177,208,232)(9,157,85,178,209,233)(10,158,86,179,210,234)(11,159,87,180,211,235)(12,160,88,181,212,236)(13,161,67,182,213,237)(14,162,68,183,214,238)(15,163,69,184,215,239)(16,164,70,185,216,240)(17,165,71,186,217,241)(18,166,72,187,218,242)(19,167,73,188,219,221)(20,168,74,189,220,222)(21,169,75,190,199,223)(22,170,76,191,200,224)(23,254,99,54,125,136)(24,255,100,55,126,137)(25,256,101,56,127,138)(26,257,102,57,128,139)(27,258,103,58,129,140)(28,259,104,59,130,141)(29,260,105,60,131,142)(30,261,106,61,132,143)(31,262,107,62,111,144)(32,263,108,63,112,145)(33,264,109,64,113,146)(34,243,110,65,114,147)(35,244,89,66,115,148)(36,245,90,45,116,149)(37,246,91,46,117,150)(38,247,92,47,118,151)(39,248,93,48,119,152)(40,249,94,49,120,153)(41,250,95,50,121,154)(42,251,96,51,122,133)(43,252,97,52,123,134)(44,253,98,53,124,135), (1,61,192,30)(2,62,193,31)(3,63,194,32)(4,64,195,33)(5,65,196,34)(6,66,197,35)(7,45,198,36)(8,46,177,37)(9,47,178,38)(10,48,179,39)(11,49,180,40)(12,50,181,41)(13,51,182,42)(14,52,183,43)(15,53,184,44)(16,54,185,23)(17,55,186,24)(18,56,187,25)(19,57,188,26)(20,58,189,27)(21,59,190,28)(22,60,191,29)(67,251,237,122)(68,252,238,123)(69,253,239,124)(70,254,240,125)(71,255,241,126)(72,256,242,127)(73,257,221,128)(74,258,222,129)(75,259,223,130)(76,260,224,131)(77,261,225,132)(78,262,226,111)(79,263,227,112)(80,264,228,113)(81,243,229,114)(82,244,230,115)(83,245,231,116)(84,246,232,117)(85,247,233,118)(86,248,234,119)(87,249,235,120)(88,250,236,121)(89,206,148,176)(90,207,149,155)(91,208,150,156)(92,209,151,157)(93,210,152,158)(94,211,153,159)(95,212,154,160)(96,213,133,161)(97,214,134,162)(98,215,135,163)(99,216,136,164)(100,217,137,165)(101,218,138,166)(102,219,139,167)(103,220,140,168)(104,199,141,169)(105,200,142,170)(106,201,143,171)(107,202,144,172)(108,203,145,173)(109,204,146,174)(110,205,147,175)>;
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), (1,171,77,192,201,225)(2,172,78,193,202,226)(3,173,79,194,203,227)(4,174,80,195,204,228)(5,175,81,196,205,229)(6,176,82,197,206,230)(7,155,83,198,207,231)(8,156,84,177,208,232)(9,157,85,178,209,233)(10,158,86,179,210,234)(11,159,87,180,211,235)(12,160,88,181,212,236)(13,161,67,182,213,237)(14,162,68,183,214,238)(15,163,69,184,215,239)(16,164,70,185,216,240)(17,165,71,186,217,241)(18,166,72,187,218,242)(19,167,73,188,219,221)(20,168,74,189,220,222)(21,169,75,190,199,223)(22,170,76,191,200,224)(23,254,99,54,125,136)(24,255,100,55,126,137)(25,256,101,56,127,138)(26,257,102,57,128,139)(27,258,103,58,129,140)(28,259,104,59,130,141)(29,260,105,60,131,142)(30,261,106,61,132,143)(31,262,107,62,111,144)(32,263,108,63,112,145)(33,264,109,64,113,146)(34,243,110,65,114,147)(35,244,89,66,115,148)(36,245,90,45,116,149)(37,246,91,46,117,150)(38,247,92,47,118,151)(39,248,93,48,119,152)(40,249,94,49,120,153)(41,250,95,50,121,154)(42,251,96,51,122,133)(43,252,97,52,123,134)(44,253,98,53,124,135), (1,61,192,30)(2,62,193,31)(3,63,194,32)(4,64,195,33)(5,65,196,34)(6,66,197,35)(7,45,198,36)(8,46,177,37)(9,47,178,38)(10,48,179,39)(11,49,180,40)(12,50,181,41)(13,51,182,42)(14,52,183,43)(15,53,184,44)(16,54,185,23)(17,55,186,24)(18,56,187,25)(19,57,188,26)(20,58,189,27)(21,59,190,28)(22,60,191,29)(67,251,237,122)(68,252,238,123)(69,253,239,124)(70,254,240,125)(71,255,241,126)(72,256,242,127)(73,257,221,128)(74,258,222,129)(75,259,223,130)(76,260,224,131)(77,261,225,132)(78,262,226,111)(79,263,227,112)(80,264,228,113)(81,243,229,114)(82,244,230,115)(83,245,231,116)(84,246,232,117)(85,247,233,118)(86,248,234,119)(87,249,235,120)(88,250,236,121)(89,206,148,176)(90,207,149,155)(91,208,150,156)(92,209,151,157)(93,210,152,158)(94,211,153,159)(95,212,154,160)(96,213,133,161)(97,214,134,162)(98,215,135,163)(99,216,136,164)(100,217,137,165)(101,218,138,166)(102,219,139,167)(103,220,140,168)(104,199,141,169)(105,200,142,170)(106,201,143,171)(107,202,144,172)(108,203,145,173)(109,204,146,174)(110,205,147,175) );
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)], [(1,171,77,192,201,225),(2,172,78,193,202,226),(3,173,79,194,203,227),(4,174,80,195,204,228),(5,175,81,196,205,229),(6,176,82,197,206,230),(7,155,83,198,207,231),(8,156,84,177,208,232),(9,157,85,178,209,233),(10,158,86,179,210,234),(11,159,87,180,211,235),(12,160,88,181,212,236),(13,161,67,182,213,237),(14,162,68,183,214,238),(15,163,69,184,215,239),(16,164,70,185,216,240),(17,165,71,186,217,241),(18,166,72,187,218,242),(19,167,73,188,219,221),(20,168,74,189,220,222),(21,169,75,190,199,223),(22,170,76,191,200,224),(23,254,99,54,125,136),(24,255,100,55,126,137),(25,256,101,56,127,138),(26,257,102,57,128,139),(27,258,103,58,129,140),(28,259,104,59,130,141),(29,260,105,60,131,142),(30,261,106,61,132,143),(31,262,107,62,111,144),(32,263,108,63,112,145),(33,264,109,64,113,146),(34,243,110,65,114,147),(35,244,89,66,115,148),(36,245,90,45,116,149),(37,246,91,46,117,150),(38,247,92,47,118,151),(39,248,93,48,119,152),(40,249,94,49,120,153),(41,250,95,50,121,154),(42,251,96,51,122,133),(43,252,97,52,123,134),(44,253,98,53,124,135)], [(1,61,192,30),(2,62,193,31),(3,63,194,32),(4,64,195,33),(5,65,196,34),(6,66,197,35),(7,45,198,36),(8,46,177,37),(9,47,178,38),(10,48,179,39),(11,49,180,40),(12,50,181,41),(13,51,182,42),(14,52,183,43),(15,53,184,44),(16,54,185,23),(17,55,186,24),(18,56,187,25),(19,57,188,26),(20,58,189,27),(21,59,190,28),(22,60,191,29),(67,251,237,122),(68,252,238,123),(69,253,239,124),(70,254,240,125),(71,255,241,126),(72,256,242,127),(73,257,221,128),(74,258,222,129),(75,259,223,130),(76,260,224,131),(77,261,225,132),(78,262,226,111),(79,263,227,112),(80,264,228,113),(81,243,229,114),(82,244,230,115),(83,245,231,116),(84,246,232,117),(85,247,233,118),(86,248,234,119),(87,249,235,120),(88,250,236,121),(89,206,148,176),(90,207,149,155),(91,208,150,156),(92,209,151,157),(93,210,152,158),(94,211,153,159),(95,212,154,160),(96,213,133,161),(97,214,134,162),(98,215,135,163),(99,216,136,164),(100,217,137,165),(101,218,138,166),(102,219,139,167),(103,220,140,168),(104,199,141,169),(105,200,142,170),(106,201,143,171),(107,202,144,172),(108,203,145,173),(109,204,146,174),(110,205,147,175)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 11A | ··· | 11J | 22A | ··· | 22AD | 33A | ··· | 33J | 44A | ··· | 44AN | 66A | ··· | 66AD |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C11 | C22 | C22 | C44 | S3 | Dic3 | D6 | S3×C11 | C11×Dic3 | S3×C22 |
| kernel | Dic3×C22 | C11×Dic3 | C2×C66 | C66 | C2×Dic3 | Dic3 | C2×C6 | C6 | C2×C22 | C22 | C22 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 10 | 20 | 10 | 40 | 1 | 2 | 1 | 10 | 20 | 10 |
Matrix representation of Dic3×C22 ►in GL4(𝔽397) generated by
| 396 | 0 | 0 | 0 |
| 0 | 396 | 0 | 0 |
| 0 | 0 | 333 | 0 |
| 0 | 0 | 0 | 333 |
| 396 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 396 | 1 |
| 0 | 0 | 396 | 0 |
| 63 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 105 | 309 |
| 0 | 0 | 17 | 292 |
G:=sub<GL(4,GF(397))| [396,0,0,0,0,396,0,0,0,0,333,0,0,0,0,333],[396,0,0,0,0,1,0,0,0,0,396,396,0,0,1,0],[63,0,0,0,0,1,0,0,0,0,105,17,0,0,309,292] >;
Dic3×C22 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{22} % in TeX
G:=Group("Dic3xC22"); // GroupNames label
G:=SmallGroup(264,21);
// by ID
G=gap.SmallGroup(264,21);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-3,220,4404]);
// Polycyclic
G:=Group<a,b,c|a^22=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export