direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×D19, C19⋊C22, C209⋊2C2, SmallGroup(418,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C11×D19 |
Generators and relations for C11×D19
G = < a,b,c | a11=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D19
►On 209 points
Generators in S209
(1 198 174 170 136 127 113 78 61 40 21)(2 199 175 171 137 128 114 79 62 41 22)(3 200 176 153 138 129 96 80 63 42 23)(4 201 177 154 139 130 97 81 64 43 24)(5 202 178 155 140 131 98 82 65 44 25)(6 203 179 156 141 132 99 83 66 45 26)(7 204 180 157 142 133 100 84 67 46 27)(8 205 181 158 143 115 101 85 68 47 28)(9 206 182 159 144 116 102 86 69 48 29)(10 207 183 160 145 117 103 87 70 49 30)(11 208 184 161 146 118 104 88 71 50 31)(12 209 185 162 147 119 105 89 72 51 32)(13 191 186 163 148 120 106 90 73 52 33)(14 192 187 164 149 121 107 91 74 53 34)(15 193 188 165 150 122 108 92 75 54 35)(16 194 189 166 151 123 109 93 76 55 36)(17 195 190 167 152 124 110 94 58 56 37)(18 196 172 168 134 125 111 95 59 57 38)(19 197 173 169 135 126 112 77 60 39 20)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 21)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(39 40)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 63)(59 62)(60 61)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(77 78)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(111 114)(112 113)(115 119)(116 118)(120 133)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(134 137)(135 136)(138 152)(139 151)(140 150)(141 149)(142 148)(143 147)(144 146)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)(168 171)(169 170)(172 175)(173 174)(176 190)(177 189)(178 188)(179 187)(180 186)(181 185)(182 184)(191 204)(192 203)(193 202)(194 201)(195 200)(196 199)(197 198)(205 209)(206 208)
G:=sub<Sym(209)| (1,198,174,170,136,127,113,78,61,40,21)(2,199,175,171,137,128,114,79,62,41,22)(3,200,176,153,138,129,96,80,63,42,23)(4,201,177,154,139,130,97,81,64,43,24)(5,202,178,155,140,131,98,82,65,44,25)(6,203,179,156,141,132,99,83,66,45,26)(7,204,180,157,142,133,100,84,67,46,27)(8,205,181,158,143,115,101,85,68,47,28)(9,206,182,159,144,116,102,86,69,48,29)(10,207,183,160,145,117,103,87,70,49,30)(11,208,184,161,146,118,104,88,71,50,31)(12,209,185,162,147,119,105,89,72,51,32)(13,191,186,163,148,120,106,90,73,52,33)(14,192,187,164,149,121,107,91,74,53,34)(15,193,188,165,150,122,108,92,75,54,35)(16,194,189,166,151,123,109,93,76,55,36)(17,195,190,167,152,124,110,94,58,56,37)(18,196,172,168,134,125,111,95,59,57,38)(19,197,173,169,135,126,112,77,60,39,20), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,21)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(39,40)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,63)(59,62)(60,61)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(77,78)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(111,114)(112,113)(115,119)(116,118)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(134,137)(135,136)(138,152)(139,151)(140,150)(141,149)(142,148)(143,147)(144,146)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(168,171)(169,170)(172,175)(173,174)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(191,204)(192,203)(193,202)(194,201)(195,200)(196,199)(197,198)(205,209)(206,208)>;
G:=Group( (1,198,174,170,136,127,113,78,61,40,21)(2,199,175,171,137,128,114,79,62,41,22)(3,200,176,153,138,129,96,80,63,42,23)(4,201,177,154,139,130,97,81,64,43,24)(5,202,178,155,140,131,98,82,65,44,25)(6,203,179,156,141,132,99,83,66,45,26)(7,204,180,157,142,133,100,84,67,46,27)(8,205,181,158,143,115,101,85,68,47,28)(9,206,182,159,144,116,102,86,69,48,29)(10,207,183,160,145,117,103,87,70,49,30)(11,208,184,161,146,118,104,88,71,50,31)(12,209,185,162,147,119,105,89,72,51,32)(13,191,186,163,148,120,106,90,73,52,33)(14,192,187,164,149,121,107,91,74,53,34)(15,193,188,165,150,122,108,92,75,54,35)(16,194,189,166,151,123,109,93,76,55,36)(17,195,190,167,152,124,110,94,58,56,37)(18,196,172,168,134,125,111,95,59,57,38)(19,197,173,169,135,126,112,77,60,39,20), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,21)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(39,40)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,63)(59,62)(60,61)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(77,78)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(111,114)(112,113)(115,119)(116,118)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(134,137)(135,136)(138,152)(139,151)(140,150)(141,149)(142,148)(143,147)(144,146)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(168,171)(169,170)(172,175)(173,174)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(191,204)(192,203)(193,202)(194,201)(195,200)(196,199)(197,198)(205,209)(206,208) );
G=PermutationGroup([[(1,198,174,170,136,127,113,78,61,40,21),(2,199,175,171,137,128,114,79,62,41,22),(3,200,176,153,138,129,96,80,63,42,23),(4,201,177,154,139,130,97,81,64,43,24),(5,202,178,155,140,131,98,82,65,44,25),(6,203,179,156,141,132,99,83,66,45,26),(7,204,180,157,142,133,100,84,67,46,27),(8,205,181,158,143,115,101,85,68,47,28),(9,206,182,159,144,116,102,86,69,48,29),(10,207,183,160,145,117,103,87,70,49,30),(11,208,184,161,146,118,104,88,71,50,31),(12,209,185,162,147,119,105,89,72,51,32),(13,191,186,163,148,120,106,90,73,52,33),(14,192,187,164,149,121,107,91,74,53,34),(15,193,188,165,150,122,108,92,75,54,35),(16,194,189,166,151,123,109,93,76,55,36),(17,195,190,167,152,124,110,94,58,56,37),(18,196,172,168,134,125,111,95,59,57,38),(19,197,173,169,135,126,112,77,60,39,20)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,21),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(39,40),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,63),(59,62),(60,61),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(77,78),(79,95),(80,94),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(96,110),(97,109),(98,108),(99,107),(100,106),(101,105),(102,104),(111,114),(112,113),(115,119),(116,118),(120,133),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(134,137),(135,136),(138,152),(139,151),(140,150),(141,149),(142,148),(143,147),(144,146),(153,167),(154,166),(155,165),(156,164),(157,163),(158,162),(159,161),(168,171),(169,170),(172,175),(173,174),(176,190),(177,189),(178,188),(179,187),(180,186),(181,185),(182,184),(191,204),(192,203),(193,202),(194,201),(195,200),(196,199),(197,198),(205,209),(206,208)]])
121 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11J | 19A | ··· | 19I | 22A | ··· | 22J | 209A | ··· | 209CL |
| order | 1 | 2 | 11 | ··· | 11 | 19 | ··· | 19 | 22 | ··· | 22 | 209 | ··· | 209 |
| size | 1 | 19 | 1 | ··· | 1 | 2 | ··· | 2 | 19 | ··· | 19 | 2 | ··· | 2 |
121 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C11 | C22 | D19 | C11×D19 |
| kernel | C11×D19 | C209 | D19 | C19 | C11 | C1 |
| # reps | 1 | 1 | 10 | 10 | 9 | 90 |
Matrix representation of C11×D19 ►in GL2(𝔽419) generated by
| 152 | 0 |
| 0 | 152 |
| 212 | 1 |
| 127 | 135 |
| 399 | 14 |
| 181 | 20 |
G:=sub<GL(2,GF(419))| [152,0,0,152],[212,127,1,135],[399,181,14,20] >;
C11×D19 in GAP, Magma, Sage, TeX
C_{11}\times D_{19} % in TeX
G:=Group("C11xD19"); // GroupNames label
G:=SmallGroup(418,2);
// by ID
G=gap.SmallGroup(418,2);
# by ID
G:=PCGroup([3,-2,-11,-19,3566]);
// Polycyclic
G:=Group<a,b,c|a^11=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export