direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C21×D11, C11⋊C42, C77⋊6C6, C33⋊2C14, C231⋊5C2, SmallGroup(462,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C21×D11 |
Generators and relations for C21×D11
G = < a,b,c | a21=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C21×D11
►On 231 points
Generators in S231
(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)
(1 187 99 117 213 31 163 78 135 197 57)(2 188 100 118 214 32 164 79 136 198 58)(3 189 101 119 215 33 165 80 137 199 59)(4 169 102 120 216 34 166 81 138 200 60)(5 170 103 121 217 35 167 82 139 201 61)(6 171 104 122 218 36 168 83 140 202 62)(7 172 105 123 219 37 148 84 141 203 63)(8 173 85 124 220 38 149 64 142 204 43)(9 174 86 125 221 39 150 65 143 205 44)(10 175 87 126 222 40 151 66 144 206 45)(11 176 88 106 223 41 152 67 145 207 46)(12 177 89 107 224 42 153 68 146 208 47)(13 178 90 108 225 22 154 69 147 209 48)(14 179 91 109 226 23 155 70 127 210 49)(15 180 92 110 227 24 156 71 128 190 50)(16 181 93 111 228 25 157 72 129 191 51)(17 182 94 112 229 26 158 73 130 192 52)(18 183 95 113 230 27 159 74 131 193 53)(19 184 96 114 231 28 160 75 132 194 54)(20 185 97 115 211 29 161 76 133 195 55)(21 186 98 116 212 30 162 77 134 196 56)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(64 124)(65 125)(66 126)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 120)(82 121)(83 122)(84 123)(85 142)(86 143)(87 144)(88 145)(89 146)(90 147)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(148 219)(149 220)(150 221)(151 222)(152 223)(153 224)(154 225)(155 226)(156 227)(157 228)(158 229)(159 230)(160 231)(161 211)(162 212)(163 213)(164 214)(165 215)(166 216)(167 217)(168 218)(169 200)(170 201)(171 202)(172 203)(173 204)(174 205)(175 206)(176 207)(177 208)(178 209)(179 210)(180 190)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)
G:=sub<Sym(231)| (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), (1,187,99,117,213,31,163,78,135,197,57)(2,188,100,118,214,32,164,79,136,198,58)(3,189,101,119,215,33,165,80,137,199,59)(4,169,102,120,216,34,166,81,138,200,60)(5,170,103,121,217,35,167,82,139,201,61)(6,171,104,122,218,36,168,83,140,202,62)(7,172,105,123,219,37,148,84,141,203,63)(8,173,85,124,220,38,149,64,142,204,43)(9,174,86,125,221,39,150,65,143,205,44)(10,175,87,126,222,40,151,66,144,206,45)(11,176,88,106,223,41,152,67,145,207,46)(12,177,89,107,224,42,153,68,146,208,47)(13,178,90,108,225,22,154,69,147,209,48)(14,179,91,109,226,23,155,70,127,210,49)(15,180,92,110,227,24,156,71,128,190,50)(16,181,93,111,228,25,157,72,129,191,51)(17,182,94,112,229,26,158,73,130,192,52)(18,183,95,113,230,27,159,74,131,193,53)(19,184,96,114,231,28,160,75,132,194,54)(20,185,97,115,211,29,161,76,133,195,55)(21,186,98,116,212,30,162,77,134,196,56), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(64,124)(65,125)(66,126)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,121)(83,122)(84,123)(85,142)(86,143)(87,144)(88,145)(89,146)(90,147)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,225)(155,226)(156,227)(157,228)(158,229)(159,230)(160,231)(161,211)(162,212)(163,213)(164,214)(165,215)(166,216)(167,217)(168,218)(169,200)(170,201)(171,202)(172,203)(173,204)(174,205)(175,206)(176,207)(177,208)(178,209)(179,210)(180,190)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)>;
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), (1,187,99,117,213,31,163,78,135,197,57)(2,188,100,118,214,32,164,79,136,198,58)(3,189,101,119,215,33,165,80,137,199,59)(4,169,102,120,216,34,166,81,138,200,60)(5,170,103,121,217,35,167,82,139,201,61)(6,171,104,122,218,36,168,83,140,202,62)(7,172,105,123,219,37,148,84,141,203,63)(8,173,85,124,220,38,149,64,142,204,43)(9,174,86,125,221,39,150,65,143,205,44)(10,175,87,126,222,40,151,66,144,206,45)(11,176,88,106,223,41,152,67,145,207,46)(12,177,89,107,224,42,153,68,146,208,47)(13,178,90,108,225,22,154,69,147,209,48)(14,179,91,109,226,23,155,70,127,210,49)(15,180,92,110,227,24,156,71,128,190,50)(16,181,93,111,228,25,157,72,129,191,51)(17,182,94,112,229,26,158,73,130,192,52)(18,183,95,113,230,27,159,74,131,193,53)(19,184,96,114,231,28,160,75,132,194,54)(20,185,97,115,211,29,161,76,133,195,55)(21,186,98,116,212,30,162,77,134,196,56), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(64,124)(65,125)(66,126)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,121)(83,122)(84,123)(85,142)(86,143)(87,144)(88,145)(89,146)(90,147)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,225)(155,226)(156,227)(157,228)(158,229)(159,230)(160,231)(161,211)(162,212)(163,213)(164,214)(165,215)(166,216)(167,217)(168,218)(169,200)(170,201)(171,202)(172,203)(173,204)(174,205)(175,206)(176,207)(177,208)(178,209)(179,210)(180,190)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199) );
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)], [(1,187,99,117,213,31,163,78,135,197,57),(2,188,100,118,214,32,164,79,136,198,58),(3,189,101,119,215,33,165,80,137,199,59),(4,169,102,120,216,34,166,81,138,200,60),(5,170,103,121,217,35,167,82,139,201,61),(6,171,104,122,218,36,168,83,140,202,62),(7,172,105,123,219,37,148,84,141,203,63),(8,173,85,124,220,38,149,64,142,204,43),(9,174,86,125,221,39,150,65,143,205,44),(10,175,87,126,222,40,151,66,144,206,45),(11,176,88,106,223,41,152,67,145,207,46),(12,177,89,107,224,42,153,68,146,208,47),(13,178,90,108,225,22,154,69,147,209,48),(14,179,91,109,226,23,155,70,127,210,49),(15,180,92,110,227,24,156,71,128,190,50),(16,181,93,111,228,25,157,72,129,191,51),(17,182,94,112,229,26,158,73,130,192,52),(18,183,95,113,230,27,159,74,131,193,53),(19,184,96,114,231,28,160,75,132,194,54),(20,185,97,115,211,29,161,76,133,195,55),(21,186,98,116,212,30,162,77,134,196,56)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(64,124),(65,125),(66,126),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,120),(82,121),(83,122),(84,123),(85,142),(86,143),(87,144),(88,145),(89,146),(90,147),(91,127),(92,128),(93,129),(94,130),(95,131),(96,132),(97,133),(98,134),(99,135),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(148,219),(149,220),(150,221),(151,222),(152,223),(153,224),(154,225),(155,226),(156,227),(157,228),(158,229),(159,230),(160,231),(161,211),(162,212),(163,213),(164,214),(165,215),(166,216),(167,217),(168,218),(169,200),(170,201),(171,202),(172,203),(173,204),(174,205),(175,206),(176,207),(177,208),(178,209),(179,210),(180,190),(181,191),(182,192),(183,193),(184,194),(185,195),(186,196),(187,197),(188,198),(189,199)]])
147 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7F | 11A | ··· | 11E | 14A | ··· | 14F | 21A | ··· | 21L | 33A | ··· | 33J | 42A | ··· | 42L | 77A | ··· | 77AD | 231A | ··· | 231BH |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 11 | ··· | 11 | 14 | ··· | 14 | 21 | ··· | 21 | 33 | ··· | 33 | 42 | ··· | 42 | 77 | ··· | 77 | 231 | ··· | 231 |
| size | 1 | 11 | 1 | 1 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 |
147 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | D11 | C3×D11 | C7×D11 | C21×D11 |
| kernel | C21×D11 | C231 | C7×D11 | C77 | C3×D11 | C33 | D11 | C11 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 5 | 10 | 30 | 60 |
Matrix representation of C21×D11 ►in GL2(𝔽43) generated by
| 25 | 0 |
| 0 | 25 |
| 4 | 17 |
| 5 | 0 |
| 0 | 17 |
| 38 | 0 |
G:=sub<GL(2,GF(43))| [25,0,0,25],[4,5,17,0],[0,38,17,0] >;
C21×D11 in GAP, Magma, Sage, TeX
C_{21}\times D_{11} % in TeX
G:=Group("C21xD11"); // GroupNames label
G:=SmallGroup(462,5);
// by ID
G=gap.SmallGroup(462,5);
# by ID
G:=PCGroup([4,-2,-3,-7,-11,6723]);
// Polycyclic
G:=Group<a,b,c|a^21=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export