direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C19×D11, C11⋊C38, C209⋊3C2, SmallGroup(418,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C19×D11 |
Generators and relations for C19×D11
G = < a,b,c | a19=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C19×D11
►On 209 points
Generators in S209
(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 34 90 133 165 188 52 195 67 142 106)(2 35 91 115 166 189 53 196 68 143 107)(3 36 92 116 167 190 54 197 69 144 108)(4 37 93 117 168 172 55 198 70 145 109)(5 38 94 118 169 173 56 199 71 146 110)(6 20 95 119 170 174 57 200 72 147 111)(7 21 77 120 171 175 39 201 73 148 112)(8 22 78 121 153 176 40 202 74 149 113)(9 23 79 122 154 177 41 203 75 150 114)(10 24 80 123 155 178 42 204 76 151 96)(11 25 81 124 156 179 43 205 58 152 97)(12 26 82 125 157 180 44 206 59 134 98)(13 27 83 126 158 181 45 207 60 135 99)(14 28 84 127 159 182 46 208 61 136 100)(15 29 85 128 160 183 47 209 62 137 101)(16 30 86 129 161 184 48 191 63 138 102)(17 31 87 130 162 185 49 192 64 139 103)(18 32 88 131 163 186 50 193 65 140 104)(19 33 89 132 164 187 51 194 66 141 105)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 147)(21 148)(22 149)(23 150)(24 151)(25 152)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 171)(40 153)(41 154)(42 155)(43 156)(44 157)(45 158)(46 159)(47 160)(48 161)(49 162)(50 163)(51 164)(52 165)(53 166)(54 167)(55 168)(56 169)(57 170)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)(73 77)(74 78)(75 79)(76 80)(115 196)(116 197)(117 198)(118 199)(119 200)(120 201)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 191)(130 192)(131 193)(132 194)(133 195)
G:=sub<Sym(209)| (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), (1,34,90,133,165,188,52,195,67,142,106)(2,35,91,115,166,189,53,196,68,143,107)(3,36,92,116,167,190,54,197,69,144,108)(4,37,93,117,168,172,55,198,70,145,109)(5,38,94,118,169,173,56,199,71,146,110)(6,20,95,119,170,174,57,200,72,147,111)(7,21,77,120,171,175,39,201,73,148,112)(8,22,78,121,153,176,40,202,74,149,113)(9,23,79,122,154,177,41,203,75,150,114)(10,24,80,123,155,178,42,204,76,151,96)(11,25,81,124,156,179,43,205,58,152,97)(12,26,82,125,157,180,44,206,59,134,98)(13,27,83,126,158,181,45,207,60,135,99)(14,28,84,127,159,182,46,208,61,136,100)(15,29,85,128,160,183,47,209,62,137,101)(16,30,86,129,161,184,48,191,63,138,102)(17,31,87,130,162,185,49,192,64,139,103)(18,32,88,131,163,186,50,193,65,140,104)(19,33,89,132,164,187,51,194,66,141,105), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,147)(21,148)(22,149)(23,150)(24,151)(25,152)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,171)(40,153)(41,154)(42,155)(43,156)(44,157)(45,158)(46,159)(47,160)(48,161)(49,162)(50,163)(51,164)(52,165)(53,166)(54,167)(55,168)(56,169)(57,170)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,77)(74,78)(75,79)(76,80)(115,196)(116,197)(117,198)(118,199)(119,200)(120,201)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,191)(130,192)(131,193)(132,194)(133,195)>;
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), (1,34,90,133,165,188,52,195,67,142,106)(2,35,91,115,166,189,53,196,68,143,107)(3,36,92,116,167,190,54,197,69,144,108)(4,37,93,117,168,172,55,198,70,145,109)(5,38,94,118,169,173,56,199,71,146,110)(6,20,95,119,170,174,57,200,72,147,111)(7,21,77,120,171,175,39,201,73,148,112)(8,22,78,121,153,176,40,202,74,149,113)(9,23,79,122,154,177,41,203,75,150,114)(10,24,80,123,155,178,42,204,76,151,96)(11,25,81,124,156,179,43,205,58,152,97)(12,26,82,125,157,180,44,206,59,134,98)(13,27,83,126,158,181,45,207,60,135,99)(14,28,84,127,159,182,46,208,61,136,100)(15,29,85,128,160,183,47,209,62,137,101)(16,30,86,129,161,184,48,191,63,138,102)(17,31,87,130,162,185,49,192,64,139,103)(18,32,88,131,163,186,50,193,65,140,104)(19,33,89,132,164,187,51,194,66,141,105), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,147)(21,148)(22,149)(23,150)(24,151)(25,152)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,171)(40,153)(41,154)(42,155)(43,156)(44,157)(45,158)(46,159)(47,160)(48,161)(49,162)(50,163)(51,164)(52,165)(53,166)(54,167)(55,168)(56,169)(57,170)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,77)(74,78)(75,79)(76,80)(115,196)(116,197)(117,198)(118,199)(119,200)(120,201)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,191)(130,192)(131,193)(132,194)(133,195) );
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)], [(1,34,90,133,165,188,52,195,67,142,106),(2,35,91,115,166,189,53,196,68,143,107),(3,36,92,116,167,190,54,197,69,144,108),(4,37,93,117,168,172,55,198,70,145,109),(5,38,94,118,169,173,56,199,71,146,110),(6,20,95,119,170,174,57,200,72,147,111),(7,21,77,120,171,175,39,201,73,148,112),(8,22,78,121,153,176,40,202,74,149,113),(9,23,79,122,154,177,41,203,75,150,114),(10,24,80,123,155,178,42,204,76,151,96),(11,25,81,124,156,179,43,205,58,152,97),(12,26,82,125,157,180,44,206,59,134,98),(13,27,83,126,158,181,45,207,60,135,99),(14,28,84,127,159,182,46,208,61,136,100),(15,29,85,128,160,183,47,209,62,137,101),(16,30,86,129,161,184,48,191,63,138,102),(17,31,87,130,162,185,49,192,64,139,103),(18,32,88,131,163,186,50,193,65,140,104),(19,33,89,132,164,187,51,194,66,141,105)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,147),(21,148),(22,149),(23,150),(24,151),(25,152),(26,134),(27,135),(28,136),(29,137),(30,138),(31,139),(32,140),(33,141),(34,142),(35,143),(36,144),(37,145),(38,146),(39,171),(40,153),(41,154),(42,155),(43,156),(44,157),(45,158),(46,159),(47,160),(48,161),(49,162),(50,163),(51,164),(52,165),(53,166),(54,167),(55,168),(56,169),(57,170),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95),(73,77),(74,78),(75,79),(76,80),(115,196),(116,197),(117,198),(118,199),(119,200),(120,201),(121,202),(122,203),(123,204),(124,205),(125,206),(126,207),(127,208),(128,209),(129,191),(130,192),(131,193),(132,194),(133,195)]])
133 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11E | 19A | ··· | 19R | 38A | ··· | 38R | 209A | ··· | 209CL |
| order | 1 | 2 | 11 | ··· | 11 | 19 | ··· | 19 | 38 | ··· | 38 | 209 | ··· | 209 |
| size | 1 | 11 | 2 | ··· | 2 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C19 | C38 | D11 | C19×D11 |
| kernel | C19×D11 | C209 | D11 | C11 | C19 | C1 |
| # reps | 1 | 1 | 18 | 18 | 5 | 90 |
Matrix representation of C19×D11 ►in GL2(𝔽419) generated by
| 379 | 0 |
| 0 | 379 |
| 418 | 1 |
| 367 | 51 |
| 418 | 0 |
| 367 | 1 |
G:=sub<GL(2,GF(419))| [379,0,0,379],[418,367,1,51],[418,367,0,1] >;
C19×D11 in GAP, Magma, Sage, TeX
C_{19}\times D_{11} % in TeX
G:=Group("C19xD11"); // GroupNames label
G:=SmallGroup(418,1);
// by ID
G=gap.SmallGroup(418,1);
# by ID
G:=PCGroup([3,-2,-19,-11,3422]);
// Polycyclic
G:=Group<a,b,c|a^19=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export