direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D198, C2×D99, C22⋊D9, C18⋊D11, C9⋊2D22, C3.D66, C11⋊2D18, C198⋊1C2, C66.2S3, C99⋊2C22, C33.2D6, C6.2D33, sometimes denoted D396 or Dih198 or Dih396, SmallGroup(396,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C99 — D198 |
Generators and relations for D198
G = < a,b | a198=b2=1, bab=a-1 >
Smallest permutation representation of D198
►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)
(1 198)(2 197)(3 196)(4 195)(5 194)(6 193)(7 192)(8 191)(9 190)(10 189)(11 188)(12 187)(13 186)(14 185)(15 184)(16 183)(17 182)(18 181)(19 180)(20 179)(21 178)(22 177)(23 176)(24 175)(25 174)(26 173)(27 172)(28 171)(29 170)(30 169)(31 168)(32 167)(33 166)(34 165)(35 164)(36 163)(37 162)(38 161)(39 160)(40 159)(41 158)(42 157)(43 156)(44 155)(45 154)(46 153)(47 152)(48 151)(49 150)(50 149)(51 148)(52 147)(53 146)(54 145)(55 144)(56 143)(57 142)(58 141)(59 140)(60 139)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)(73 126)(74 125)(75 124)(76 123)(77 122)(78 121)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)
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), (1,198)(2,197)(3,196)(4,195)(5,194)(6,193)(7,192)(8,191)(9,190)(10,189)(11,188)(12,187)(13,186)(14,185)(15,184)(16,183)(17,182)(18,181)(19,180)(20,179)(21,178)(22,177)(23,176)(24,175)(25,174)(26,173)(27,172)(28,171)(29,170)(30,169)(31,168)(32,167)(33,166)(34,165)(35,164)(36,163)(37,162)(38,161)(39,160)(40,159)(41,158)(42,157)(43,156)(44,155)(45,154)(46,153)(47,152)(48,151)(49,150)(50,149)(51,148)(52,147)(53,146)(54,145)(55,144)(56,143)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)>;
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), (1,198)(2,197)(3,196)(4,195)(5,194)(6,193)(7,192)(8,191)(9,190)(10,189)(11,188)(12,187)(13,186)(14,185)(15,184)(16,183)(17,182)(18,181)(19,180)(20,179)(21,178)(22,177)(23,176)(24,175)(25,174)(26,173)(27,172)(28,171)(29,170)(30,169)(31,168)(32,167)(33,166)(34,165)(35,164)(36,163)(37,162)(38,161)(39,160)(40,159)(41,158)(42,157)(43,156)(44,155)(45,154)(46,153)(47,152)(48,151)(49,150)(50,149)(51,148)(52,147)(53,146)(54,145)(55,144)(56,143)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100) );
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)], [(1,198),(2,197),(3,196),(4,195),(5,194),(6,193),(7,192),(8,191),(9,190),(10,189),(11,188),(12,187),(13,186),(14,185),(15,184),(16,183),(17,182),(18,181),(19,180),(20,179),(21,178),(22,177),(23,176),(24,175),(25,174),(26,173),(27,172),(28,171),(29,170),(30,169),(31,168),(32,167),(33,166),(34,165),(35,164),(36,163),(37,162),(38,161),(39,160),(40,159),(41,158),(42,157),(43,156),(44,155),(45,154),(46,153),(47,152),(48,151),(49,150),(50,149),(51,148),(52,147),(53,146),(54,145),(55,144),(56,143),(57,142),(58,141),(59,140),(60,139),(61,138),(62,137),(63,136),(64,135),(65,134),(66,133),(67,132),(68,131),(69,130),(70,129),(71,128),(72,127),(73,126),(74,125),(75,124),(76,123),(77,122),(78,121),(79,120),(80,119),(81,118),(82,117),(83,116),(84,115),(85,114),(86,113),(87,112),(88,111),(89,110),(90,109),(91,108),(92,107),(93,106),(94,105),(95,104),(96,103),(97,102),(98,101),(99,100)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 9A | 9B | 9C | 11A | ··· | 11E | 18A | 18B | 18C | 22A | ··· | 22E | 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 | 33 | ··· | 33 | 66 | ··· | 66 | 99 | ··· | 99 | 198 | ··· | 198 |
| size | 1 | 1 | 99 | 99 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D9 | D11 | D18 | D22 | D33 | D66 | D99 | D198 |
| kernel | D198 | D99 | C198 | C66 | C33 | C22 | C18 | C11 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 3 | 5 | 3 | 5 | 10 | 10 | 30 | 30 |
Matrix representation of D198 ►in GL3(𝔽199) generated by
| 198 | 0 | 0 |
| 0 | 65 | 195 |
| 0 | 4 | 61 |
| 1 | 0 | 0 |
| 0 | 4 | 61 |
| 0 | 65 | 195 |
G:=sub<GL(3,GF(199))| [198,0,0,0,65,4,0,195,61],[1,0,0,0,4,65,0,61,195] >;
D198 in GAP, Magma, Sage, TeX
D_{198} % in TeX
G:=Group("D198"); // GroupNames label
G:=SmallGroup(396,9);
// by ID
G=gap.SmallGroup(396,9);
# by ID
G:=PCGroup([5,-2,-2,-3,-11,-3,2102,1002,2403,6604]);
// Polycyclic
G:=Group<a,b|a^198=b^2=1,b*a*b=a^-1>;
// generators/relations
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