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G = C19×D11order 418 = 2·11·19

Direct product of C19 and D11

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C19×D11, C11⋊C38, C2093C2, SmallGroup(418,1)

Series: Derived Chief Lower central Upper central

C1C11 — C19×D11
C1C11C209 — C19×D11
C11 — C19×D11
C1C19

Generators and relations for C19×D11
 G = < a,b,c | a19=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C38

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 54 97 29 138 157 116 209 176 86 71)(2 55 98 30 139 158 117 191 177 87 72)(3 56 99 31 140 159 118 192 178 88 73)(4 57 100 32 141 160 119 193 179 89 74)(5 39 101 33 142 161 120 194 180 90 75)(6 40 102 34 143 162 121 195 181 91 76)(7 41 103 35 144 163 122 196 182 92 58)(8 42 104 36 145 164 123 197 183 93 59)(9 43 105 37 146 165 124 198 184 94 60)(10 44 106 38 147 166 125 199 185 95 61)(11 45 107 20 148 167 126 200 186 77 62)(12 46 108 21 149 168 127 201 187 78 63)(13 47 109 22 150 169 128 202 188 79 64)(14 48 110 23 151 170 129 203 189 80 65)(15 49 111 24 152 171 130 204 190 81 66)(16 50 112 25 134 153 131 205 172 82 67)(17 51 113 26 135 154 132 206 173 83 68)(18 52 114 27 136 155 133 207 174 84 69)(19 53 96 28 137 156 115 208 175 85 70)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 200)(21 201)(22 202)(23 203)(24 204)(25 205)(26 206)(27 207)(28 208)(29 209)(30 191)(31 192)(32 193)(33 194)(34 195)(35 196)(36 197)(37 198)(38 199)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(96 175)(97 176)(98 177)(99 178)(100 179)(101 180)(102 181)(103 182)(104 183)(105 184)(106 185)(107 186)(108 187)(109 188)(110 189)(111 190)(112 172)(113 173)(114 174)(115 137)(116 138)(117 139)(118 140)(119 141)(120 142)(121 143)(122 144)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 151)(130 152)(131 134)(132 135)(133 136)

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), (1,54,97,29,138,157,116,209,176,86,71)(2,55,98,30,139,158,117,191,177,87,72)(3,56,99,31,140,159,118,192,178,88,73)(4,57,100,32,141,160,119,193,179,89,74)(5,39,101,33,142,161,120,194,180,90,75)(6,40,102,34,143,162,121,195,181,91,76)(7,41,103,35,144,163,122,196,182,92,58)(8,42,104,36,145,164,123,197,183,93,59)(9,43,105,37,146,165,124,198,184,94,60)(10,44,106,38,147,166,125,199,185,95,61)(11,45,107,20,148,167,126,200,186,77,62)(12,46,108,21,149,168,127,201,187,78,63)(13,47,109,22,150,169,128,202,188,79,64)(14,48,110,23,151,170,129,203,189,80,65)(15,49,111,24,152,171,130,204,190,81,66)(16,50,112,25,134,153,131,205,172,82,67)(17,51,113,26,135,154,132,206,173,83,68)(18,52,114,27,136,155,133,207,174,84,69)(19,53,96,28,137,156,115,208,175,85,70), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,200)(21,201)(22,202)(23,203)(24,204)(25,205)(26,206)(27,207)(28,208)(29,209)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,197)(37,198)(38,199)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(96,175)(97,176)(98,177)(99,178)(100,179)(101,180)(102,181)(103,182)(104,183)(105,184)(106,185)(107,186)(108,187)(109,188)(110,189)(111,190)(112,172)(113,173)(114,174)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(121,143)(122,144)(123,145)(124,146)(125,147)(126,148)(127,149)(128,150)(129,151)(130,152)(131,134)(132,135)(133,136)>;

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), (1,54,97,29,138,157,116,209,176,86,71)(2,55,98,30,139,158,117,191,177,87,72)(3,56,99,31,140,159,118,192,178,88,73)(4,57,100,32,141,160,119,193,179,89,74)(5,39,101,33,142,161,120,194,180,90,75)(6,40,102,34,143,162,121,195,181,91,76)(7,41,103,35,144,163,122,196,182,92,58)(8,42,104,36,145,164,123,197,183,93,59)(9,43,105,37,146,165,124,198,184,94,60)(10,44,106,38,147,166,125,199,185,95,61)(11,45,107,20,148,167,126,200,186,77,62)(12,46,108,21,149,168,127,201,187,78,63)(13,47,109,22,150,169,128,202,188,79,64)(14,48,110,23,151,170,129,203,189,80,65)(15,49,111,24,152,171,130,204,190,81,66)(16,50,112,25,134,153,131,205,172,82,67)(17,51,113,26,135,154,132,206,173,83,68)(18,52,114,27,136,155,133,207,174,84,69)(19,53,96,28,137,156,115,208,175,85,70), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,200)(21,201)(22,202)(23,203)(24,204)(25,205)(26,206)(27,207)(28,208)(29,209)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,197)(37,198)(38,199)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(96,175)(97,176)(98,177)(99,178)(100,179)(101,180)(102,181)(103,182)(104,183)(105,184)(106,185)(107,186)(108,187)(109,188)(110,189)(111,190)(112,172)(113,173)(114,174)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(121,143)(122,144)(123,145)(124,146)(125,147)(126,148)(127,149)(128,150)(129,151)(130,152)(131,134)(132,135)(133,136) );

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)], [(1,54,97,29,138,157,116,209,176,86,71),(2,55,98,30,139,158,117,191,177,87,72),(3,56,99,31,140,159,118,192,178,88,73),(4,57,100,32,141,160,119,193,179,89,74),(5,39,101,33,142,161,120,194,180,90,75),(6,40,102,34,143,162,121,195,181,91,76),(7,41,103,35,144,163,122,196,182,92,58),(8,42,104,36,145,164,123,197,183,93,59),(9,43,105,37,146,165,124,198,184,94,60),(10,44,106,38,147,166,125,199,185,95,61),(11,45,107,20,148,167,126,200,186,77,62),(12,46,108,21,149,168,127,201,187,78,63),(13,47,109,22,150,169,128,202,188,79,64),(14,48,110,23,151,170,129,203,189,80,65),(15,49,111,24,152,171,130,204,190,81,66),(16,50,112,25,134,153,131,205,172,82,67),(17,51,113,26,135,154,132,206,173,83,68),(18,52,114,27,136,155,133,207,174,84,69),(19,53,96,28,137,156,115,208,175,85,70)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,200),(21,201),(22,202),(23,203),(24,204),(25,205),(26,206),(27,207),(28,208),(29,209),(30,191),(31,192),(32,193),(33,194),(34,195),(35,196),(36,197),(37,198),(38,199),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(96,175),(97,176),(98,177),(99,178),(100,179),(101,180),(102,181),(103,182),(104,183),(105,184),(106,185),(107,186),(108,187),(109,188),(110,189),(111,190),(112,172),(113,173),(114,174),(115,137),(116,138),(117,139),(118,140),(119,141),(120,142),(121,143),(122,144),(123,145),(124,146),(125,147),(126,148),(127,149),(128,150),(129,151),(130,152),(131,134),(132,135),(133,136)])

133 conjugacy classes

class 1  2 11A···11E19A···19R38A···38R209A···209CL
order1211···1119···1938···38209···209
size1112···21···111···112···2

133 irreducible representations

dim111122
type+++
imageC1C2C19C38D11C19×D11
kernelC19×D11C209D11C11C19C1
# reps111818590

Matrix representation of C19×D11 in GL2(𝔽419) generated by

3790
0379
,
4181
36751
,
4180
3671
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

Subgroup lattice of C19×D11 in TeX

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