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G = C18×D11order 396 = 22·32·11

Direct product of C18 and D11

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

Aliases: C18×D11, C22⋊C18, C1982C2, C66.3C6, C993C22, C11⋊(C2×C18), C33.(C2×C6), C3.(C6×D11), (C6×D11).C3, (C3×D11).C6, C6.3(C3×D11), SmallGroup(396,7)

Series: Derived Chief Lower central Upper central

C1C11 — C18×D11
C1C11C33C99C9×D11 — C18×D11
C11 — C18×D11
C1C18

Generators and relations for C18×D11
 G = < a,b,c | a18=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C2
11C22
11C6
11C6
11C2×C6
11C18
11C18
11C2×C18

Smallest permutation representation of C18×D11
On 198 points
Generators in S198
(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)
(1 28 101 112 170 155 41 184 142 70 89)(2 29 102 113 171 156 42 185 143 71 90)(3 30 103 114 172 157 43 186 144 72 73)(4 31 104 115 173 158 44 187 127 55 74)(5 32 105 116 174 159 45 188 128 56 75)(6 33 106 117 175 160 46 189 129 57 76)(7 34 107 118 176 161 47 190 130 58 77)(8 35 108 119 177 162 48 191 131 59 78)(9 36 91 120 178 145 49 192 132 60 79)(10 19 92 121 179 146 50 193 133 61 80)(11 20 93 122 180 147 51 194 134 62 81)(12 21 94 123 163 148 52 195 135 63 82)(13 22 95 124 164 149 53 196 136 64 83)(14 23 96 125 165 150 54 197 137 65 84)(15 24 97 126 166 151 37 198 138 66 85)(16 25 98 109 167 152 38 181 139 67 86)(17 26 99 110 168 153 39 182 140 68 87)(18 27 100 111 169 154 40 183 141 69 88)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 70)(20 71)(21 72)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 175)(38 176)(39 177)(40 178)(41 179)(42 180)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 174)(91 141)(92 142)(93 143)(94 144)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 190)(110 191)(111 192)(112 193)(113 194)(114 195)(115 196)(116 197)(117 198)(118 181)(119 182)(120 183)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(145 154)(146 155)(147 156)(148 157)(149 158)(150 159)(151 160)(152 161)(153 162)

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

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

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

126 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F9A···9F11A···11E18A···18F18G···18R22A···22E33A···33J66A···66J99A···99AD198A···198AD
order1222336666669···911···1118···1818···1822···2233···3366···6699···99198···198
size1111111111111111111···12···21···111···112···22···22···22···22···2

126 irreducible representations

dim111111111222222
type+++++
imageC1C2C2C3C6C6C9C18C18D11D22C3×D11C6×D11C9×D11C18×D11
kernelC18×D11C9×D11C198C6×D11C3×D11C66D22D11C22C18C9C6C3C2C1
# reps12124261265510103030

Matrix representation of C18×D11 in GL3(𝔽199) generated by

10700
01750
00175
,
100
014297
019875
,
100
032109
018167
G:=sub<GL(3,GF(199))| [107,0,0,0,175,0,0,0,175],[1,0,0,0,142,198,0,97,75],[1,0,0,0,32,18,0,109,167] >;

C18×D11 in GAP, Magma, Sage, TeX

C_{18}\times D_{11}
% in TeX

G:=Group("C18xD11");
// GroupNames label

G:=SmallGroup(396,7);
// by ID

G=gap.SmallGroup(396,7);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-11,57,9004]);
// Polycyclic

G:=Group<a,b,c|a^18=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C18×D11 in TeX

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