direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D9×C22, C18⋊C22, C198⋊3C2, C66.6S3, C99⋊4C22, C33.3D6, C9⋊(C2×C22), C3.(S3×C22), C6.2(S3×C11), SmallGroup(396,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C22 |
Generators and relations for D9×C22
G = < a,b,c | a22=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
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)
(1 119 61 183 99 176 84 149 33)(2 120 62 184 100 155 85 150 34)(3 121 63 185 101 156 86 151 35)(4 122 64 186 102 157 87 152 36)(5 123 65 187 103 158 88 153 37)(6 124 66 188 104 159 67 154 38)(7 125 45 189 105 160 68 133 39)(8 126 46 190 106 161 69 134 40)(9 127 47 191 107 162 70 135 41)(10 128 48 192 108 163 71 136 42)(11 129 49 193 109 164 72 137 43)(12 130 50 194 110 165 73 138 44)(13 131 51 195 89 166 74 139 23)(14 132 52 196 90 167 75 140 24)(15 111 53 197 91 168 76 141 25)(16 112 54 198 92 169 77 142 26)(17 113 55 177 93 170 78 143 27)(18 114 56 178 94 171 79 144 28)(19 115 57 179 95 172 80 145 29)(20 116 58 180 96 173 81 146 30)(21 117 59 181 97 174 82 147 31)(22 118 60 182 98 175 83 148 32)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 67)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)(121 151)(122 152)(123 153)(124 154)(125 133)(126 134)(127 135)(128 136)(129 137)(130 138)(131 139)(132 140)(155 184)(156 185)(157 186)(158 187)(159 188)(160 189)(161 190)(162 191)(163 192)(164 193)(165 194)(166 195)(167 196)(168 197)(169 198)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(176 183)
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), (1,119,61,183,99,176,84,149,33)(2,120,62,184,100,155,85,150,34)(3,121,63,185,101,156,86,151,35)(4,122,64,186,102,157,87,152,36)(5,123,65,187,103,158,88,153,37)(6,124,66,188,104,159,67,154,38)(7,125,45,189,105,160,68,133,39)(8,126,46,190,106,161,69,134,40)(9,127,47,191,107,162,70,135,41)(10,128,48,192,108,163,71,136,42)(11,129,49,193,109,164,72,137,43)(12,130,50,194,110,165,73,138,44)(13,131,51,195,89,166,74,139,23)(14,132,52,196,90,167,75,140,24)(15,111,53,197,91,168,76,141,25)(16,112,54,198,92,169,77,142,26)(17,113,55,177,93,170,78,143,27)(18,114,56,178,94,171,79,144,28)(19,115,57,179,95,172,80,145,29)(20,116,58,180,96,173,81,146,30)(21,117,59,181,97,174,82,147,31)(22,118,60,182,98,175,83,148,32), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,67)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)(121,151)(122,152)(123,153)(124,154)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138)(131,139)(132,140)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,197)(169,198)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(176,183)>;
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), (1,119,61,183,99,176,84,149,33)(2,120,62,184,100,155,85,150,34)(3,121,63,185,101,156,86,151,35)(4,122,64,186,102,157,87,152,36)(5,123,65,187,103,158,88,153,37)(6,124,66,188,104,159,67,154,38)(7,125,45,189,105,160,68,133,39)(8,126,46,190,106,161,69,134,40)(9,127,47,191,107,162,70,135,41)(10,128,48,192,108,163,71,136,42)(11,129,49,193,109,164,72,137,43)(12,130,50,194,110,165,73,138,44)(13,131,51,195,89,166,74,139,23)(14,132,52,196,90,167,75,140,24)(15,111,53,197,91,168,76,141,25)(16,112,54,198,92,169,77,142,26)(17,113,55,177,93,170,78,143,27)(18,114,56,178,94,171,79,144,28)(19,115,57,179,95,172,80,145,29)(20,116,58,180,96,173,81,146,30)(21,117,59,181,97,174,82,147,31)(22,118,60,182,98,175,83,148,32), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,67)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)(121,151)(122,152)(123,153)(124,154)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138)(131,139)(132,140)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,197)(169,198)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(176,183) );
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)], [(1,119,61,183,99,176,84,149,33),(2,120,62,184,100,155,85,150,34),(3,121,63,185,101,156,86,151,35),(4,122,64,186,102,157,87,152,36),(5,123,65,187,103,158,88,153,37),(6,124,66,188,104,159,67,154,38),(7,125,45,189,105,160,68,133,39),(8,126,46,190,106,161,69,134,40),(9,127,47,191,107,162,70,135,41),(10,128,48,192,108,163,71,136,42),(11,129,49,193,109,164,72,137,43),(12,130,50,194,110,165,73,138,44),(13,131,51,195,89,166,74,139,23),(14,132,52,196,90,167,75,140,24),(15,111,53,197,91,168,76,141,25),(16,112,54,198,92,169,77,142,26),(17,113,55,177,93,170,78,143,27),(18,114,56,178,94,171,79,144,28),(19,115,57,179,95,172,80,145,29),(20,116,58,180,96,173,81,146,30),(21,117,59,181,97,174,82,147,31),(22,118,60,182,98,175,83,148,32)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(45,68),(46,69),(47,70),(48,71),(49,72),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,80),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,88),(66,67),(111,141),(112,142),(113,143),(114,144),(115,145),(116,146),(117,147),(118,148),(119,149),(120,150),(121,151),(122,152),(123,153),(124,154),(125,133),(126,134),(127,135),(128,136),(129,137),(130,138),(131,139),(132,140),(155,184),(156,185),(157,186),(158,187),(159,188),(160,189),(161,190),(162,191),(163,192),(164,193),(165,194),(166,195),(167,196),(168,197),(169,198),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(176,183)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 9A | 9B | 9C | 11A | ··· | 11J | 18A | 18B | 18C | 22A | ··· | 22J | 22K | ··· | 22AD | 33A | ··· | 33J | 66A | ··· | 66J | 99A | ··· | 99AD | 198A | ··· | 198AD |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 9 | 9 | 9 | 11 | ··· | 11 | 18 | 18 | 18 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 | 99 | ··· | 99 | 198 | ··· | 198 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | S3 | D6 | D9 | D18 | S3×C11 | S3×C22 | C11×D9 | D9×C22 |
| kernel | D9×C22 | C11×D9 | C198 | D18 | D9 | C18 | C66 | C33 | C22 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 10 | 20 | 10 | 1 | 1 | 3 | 3 | 10 | 10 | 30 | 30 |
Matrix representation of D9×C22 ►in GL2(𝔽199) generated by
| 181 | 0 |
| 0 | 181 |
| 57 | 108 |
| 91 | 148 |
| 57 | 148 |
| 91 | 142 |
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