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G = D9×C22order 396 = 22·32·11

Direct product of C22 and D9

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

Aliases: D9×C22, C18⋊C22, C1983C2, C66.6S3, C994C22, C33.3D6, C9⋊(C2×C22), C3.(S3×C22), C6.2(S3×C11), SmallGroup(396,8)

Series: Derived Chief Lower central Upper central

C1C9 — D9×C22
C1C3C9C99C11×D9 — D9×C22
C9 — D9×C22
C1C22

Generators and relations for D9×C22
 G = < a,b,c | a22=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
9C2
9C22
3S3
3S3
9C22
9C22
3D6
9C2×C22
3S3×C11
3S3×C11
3S3×C22

Smallest permutation representation of D9×C22
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 29 175 150 64 114 77 108 196)(2 30 176 151 65 115 78 109 197)(3 31 155 152 66 116 79 110 198)(4 32 156 153 45 117 80 89 177)(5 33 157 154 46 118 81 90 178)(6 34 158 133 47 119 82 91 179)(7 35 159 134 48 120 83 92 180)(8 36 160 135 49 121 84 93 181)(9 37 161 136 50 122 85 94 182)(10 38 162 137 51 123 86 95 183)(11 39 163 138 52 124 87 96 184)(12 40 164 139 53 125 88 97 185)(13 41 165 140 54 126 67 98 186)(14 42 166 141 55 127 68 99 187)(15 43 167 142 56 128 69 100 188)(16 44 168 143 57 129 70 101 189)(17 23 169 144 58 130 71 102 190)(18 24 170 145 59 131 72 103 191)(19 25 171 146 60 132 73 104 192)(20 26 172 147 61 111 74 105 193)(21 27 173 148 62 112 75 106 194)(22 28 174 149 63 113 76 107 195)
(1 196)(2 197)(3 198)(4 177)(5 178)(6 179)(7 180)(8 181)(9 182)(10 183)(11 184)(12 185)(13 186)(14 187)(15 188)(16 189)(17 190)(18 191)(19 192)(20 193)(21 194)(22 195)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(67 165)(68 166)(69 167)(70 168)(71 169)(72 170)(73 171)(74 172)(75 173)(76 174)(77 175)(78 176)(79 155)(80 156)(81 157)(82 158)(83 159)(84 160)(85 161)(86 162)(87 163)(88 164)(111 147)(112 148)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(127 141)(128 142)(129 143)(130 144)(131 145)(132 146)

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

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

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

132 conjugacy classes

class 1 2A2B2C 3  6 9A9B9C11A···11J18A18B18C22A···22J22K···22AD33A···33J66A···66J99A···99AD198A···198AD
order12223699911···1118181822···2222···2233···3366···6699···99198···198
size1199222221···12221···19···92···22···22···22···2

132 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C11C22C22S3D6D9D18S3×C11S3×C22C11×D9D9×C22
kernelD9×C22C11×D9C198D18D9C18C66C33C22C11C6C3C2C1
# reps121102010113310103030

Matrix representation of D9×C22 in GL2(𝔽199) generated by

1810
0181
,
57108
91148
,
57148
91142
G:=sub<GL(2,GF(199))| [181,0,0,181],[57,91,108,148],[57,91,148,142] >;

D9×C22 in GAP, Magma, Sage, TeX

D_9\times C_{22}
% in TeX

G:=Group("D9xC22");
// GroupNames label

G:=SmallGroup(396,8);
// by ID

G=gap.SmallGroup(396,8);
# by ID

G:=PCGroup([5,-2,-2,-11,-3,-3,4403,138,6604]);
// Polycyclic

G:=Group<a,b,c|a^22=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D9×C22 in TeX

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