direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×D21, C77⋊3S3, C33⋊3D7, C231⋊4C2, C21⋊1C22, C7⋊(S3×C11), C3⋊(C11×D7), SmallGroup(462,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C11×D21 |
Generators and relations for C11×D21
G = < a,b,c | a11=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D21
►On 231 points
Generators in S231
(1 224 208 173 160 128 111 86 70 45 38)(2 225 209 174 161 129 112 87 71 46 39)(3 226 210 175 162 130 113 88 72 47 40)(4 227 190 176 163 131 114 89 73 48 41)(5 228 191 177 164 132 115 90 74 49 42)(6 229 192 178 165 133 116 91 75 50 22)(7 230 193 179 166 134 117 92 76 51 23)(8 231 194 180 167 135 118 93 77 52 24)(9 211 195 181 168 136 119 94 78 53 25)(10 212 196 182 148 137 120 95 79 54 26)(11 213 197 183 149 138 121 96 80 55 27)(12 214 198 184 150 139 122 97 81 56 28)(13 215 199 185 151 140 123 98 82 57 29)(14 216 200 186 152 141 124 99 83 58 30)(15 217 201 187 153 142 125 100 84 59 31)(16 218 202 188 154 143 126 101 64 60 32)(17 219 203 189 155 144 106 102 65 61 33)(18 220 204 169 156 145 107 103 66 62 34)(19 221 205 170 157 146 108 104 67 63 35)(20 222 206 171 158 147 109 105 68 43 36)(21 223 207 172 159 127 110 85 69 44 37)
(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)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 32)(23 31)(24 30)(25 29)(26 28)(33 42)(34 41)(35 40)(36 39)(37 38)(43 46)(44 45)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(76 84)(77 83)(78 82)(79 81)(85 86)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(106 115)(107 114)(108 113)(109 112)(110 111)(116 126)(117 125)(118 124)(119 123)(120 122)(127 128)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)(148 150)(151 168)(152 167)(153 166)(154 165)(155 164)(156 163)(157 162)(158 161)(159 160)(169 176)(170 175)(171 174)(172 173)(177 189)(178 188)(179 187)(180 186)(181 185)(182 184)(190 204)(191 203)(192 202)(193 201)(194 200)(195 199)(196 198)(205 210)(206 209)(207 208)(211 215)(212 214)(216 231)(217 230)(218 229)(219 228)(220 227)(221 226)(222 225)(223 224)
G:=sub<Sym(231)| (1,224,208,173,160,128,111,86,70,45,38)(2,225,209,174,161,129,112,87,71,46,39)(3,226,210,175,162,130,113,88,72,47,40)(4,227,190,176,163,131,114,89,73,48,41)(5,228,191,177,164,132,115,90,74,49,42)(6,229,192,178,165,133,116,91,75,50,22)(7,230,193,179,166,134,117,92,76,51,23)(8,231,194,180,167,135,118,93,77,52,24)(9,211,195,181,168,136,119,94,78,53,25)(10,212,196,182,148,137,120,95,79,54,26)(11,213,197,183,149,138,121,96,80,55,27)(12,214,198,184,150,139,122,97,81,56,28)(13,215,199,185,151,140,123,98,82,57,29)(14,216,200,186,152,141,124,99,83,58,30)(15,217,201,187,153,142,125,100,84,59,31)(16,218,202,188,154,143,126,101,64,60,32)(17,219,203,189,155,144,106,102,65,61,33)(18,220,204,169,156,145,107,103,66,62,34)(19,221,205,170,157,146,108,104,67,63,35)(20,222,206,171,158,147,109,105,68,43,36)(21,223,207,172,159,127,110,85,69,44,37), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,32)(23,31)(24,30)(25,29)(26,28)(33,42)(34,41)(35,40)(36,39)(37,38)(43,46)(44,45)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(76,84)(77,83)(78,82)(79,81)(85,86)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(106,115)(107,114)(108,113)(109,112)(110,111)(116,126)(117,125)(118,124)(119,123)(120,122)(127,128)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(148,150)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(169,176)(170,175)(171,174)(172,173)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)(205,210)(206,209)(207,208)(211,215)(212,214)(216,231)(217,230)(218,229)(219,228)(220,227)(221,226)(222,225)(223,224)>;
G:=Group( (1,224,208,173,160,128,111,86,70,45,38)(2,225,209,174,161,129,112,87,71,46,39)(3,226,210,175,162,130,113,88,72,47,40)(4,227,190,176,163,131,114,89,73,48,41)(5,228,191,177,164,132,115,90,74,49,42)(6,229,192,178,165,133,116,91,75,50,22)(7,230,193,179,166,134,117,92,76,51,23)(8,231,194,180,167,135,118,93,77,52,24)(9,211,195,181,168,136,119,94,78,53,25)(10,212,196,182,148,137,120,95,79,54,26)(11,213,197,183,149,138,121,96,80,55,27)(12,214,198,184,150,139,122,97,81,56,28)(13,215,199,185,151,140,123,98,82,57,29)(14,216,200,186,152,141,124,99,83,58,30)(15,217,201,187,153,142,125,100,84,59,31)(16,218,202,188,154,143,126,101,64,60,32)(17,219,203,189,155,144,106,102,65,61,33)(18,220,204,169,156,145,107,103,66,62,34)(19,221,205,170,157,146,108,104,67,63,35)(20,222,206,171,158,147,109,105,68,43,36)(21,223,207,172,159,127,110,85,69,44,37), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,32)(23,31)(24,30)(25,29)(26,28)(33,42)(34,41)(35,40)(36,39)(37,38)(43,46)(44,45)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(76,84)(77,83)(78,82)(79,81)(85,86)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(106,115)(107,114)(108,113)(109,112)(110,111)(116,126)(117,125)(118,124)(119,123)(120,122)(127,128)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(148,150)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(169,176)(170,175)(171,174)(172,173)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)(205,210)(206,209)(207,208)(211,215)(212,214)(216,231)(217,230)(218,229)(219,228)(220,227)(221,226)(222,225)(223,224) );
G=PermutationGroup([[(1,224,208,173,160,128,111,86,70,45,38),(2,225,209,174,161,129,112,87,71,46,39),(3,226,210,175,162,130,113,88,72,47,40),(4,227,190,176,163,131,114,89,73,48,41),(5,228,191,177,164,132,115,90,74,49,42),(6,229,192,178,165,133,116,91,75,50,22),(7,230,193,179,166,134,117,92,76,51,23),(8,231,194,180,167,135,118,93,77,52,24),(9,211,195,181,168,136,119,94,78,53,25),(10,212,196,182,148,137,120,95,79,54,26),(11,213,197,183,149,138,121,96,80,55,27),(12,214,198,184,150,139,122,97,81,56,28),(13,215,199,185,151,140,123,98,82,57,29),(14,216,200,186,152,141,124,99,83,58,30),(15,217,201,187,153,142,125,100,84,59,31),(16,218,202,188,154,143,126,101,64,60,32),(17,219,203,189,155,144,106,102,65,61,33),(18,220,204,169,156,145,107,103,66,62,34),(19,221,205,170,157,146,108,104,67,63,35),(20,222,206,171,158,147,109,105,68,43,36),(21,223,207,172,159,127,110,85,69,44,37)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,32),(23,31),(24,30),(25,29),(26,28),(33,42),(34,41),(35,40),(36,39),(37,38),(43,46),(44,45),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(76,84),(77,83),(78,82),(79,81),(85,86),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(106,115),(107,114),(108,113),(109,112),(110,111),(116,126),(117,125),(118,124),(119,123),(120,122),(127,128),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,140),(137,139),(148,150),(151,168),(152,167),(153,166),(154,165),(155,164),(156,163),(157,162),(158,161),(159,160),(169,176),(170,175),(171,174),(172,173),(177,189),(178,188),(179,187),(180,186),(181,185),(182,184),(190,204),(191,203),(192,202),(193,201),(194,200),(195,199),(196,198),(205,210),(206,209),(207,208),(211,215),(212,214),(216,231),(217,230),(218,229),(219,228),(220,227),(221,226),(222,225),(223,224)]])
132 conjugacy classes
| class | 1 | 2 | 3 | 7A | 7B | 7C | 11A | ··· | 11J | 21A | ··· | 21F | 22A | ··· | 22J | 33A | ··· | 33J | 77A | ··· | 77AD | 231A | ··· | 231BH |
| order | 1 | 2 | 3 | 7 | 7 | 7 | 11 | ··· | 11 | 21 | ··· | 21 | 22 | ··· | 22 | 33 | ··· | 33 | 77 | ··· | 77 | 231 | ··· | 231 |
| size | 1 | 21 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 21 | ··· | 21 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C11 | C22 | S3 | D7 | D21 | S3×C11 | C11×D7 | C11×D21 |
| kernel | C11×D21 | C231 | D21 | C21 | C77 | C33 | C11 | C7 | C3 | C1 |
| # reps | 1 | 1 | 10 | 10 | 1 | 3 | 6 | 10 | 30 | 60 |
Matrix representation of C11×D21 ►in GL2(𝔽463) generated by
| 362 | 0 |
| 0 | 362 |
| 274 | 329 |
| 134 | 95 |
| 95 | 171 |
| 329 | 368 |
G:=sub<GL(2,GF(463))| [362,0,0,362],[274,134,329,95],[95,329,171,368] >;
C11×D21 in GAP, Magma, Sage, TeX
C_{11}\times D_{21} % in TeX
G:=Group("C11xD21"); // GroupNames label
G:=SmallGroup(462,10);
// by ID
G=gap.SmallGroup(462,10);
# by ID
G:=PCGroup([4,-2,-11,-3,-7,530,6339]);
// Polycyclic
G:=Group<a,b,c|a^11=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export