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G = D198order 396 = 22·32·11

Dihedral group

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

Aliases: D198, C2×D99, C22⋊D9, C18⋊D11, C92D22, C3.D66, C112D18, C1981C2, C66.2S3, C992C22, C33.2D6, C6.2D33, sometimes denoted D396 or Dih198 or Dih396, SmallGroup(396,9)

Series: Derived Chief Lower central Upper central

C1C99 — D198
C1C3C33C99D99 — D198
C99 — D198
C1C2

Generators and relations for D198
 G = < a,b | a198=b2=1, bab=a-1 >

99C2
99C2
99C22
33S3
33S3
9D11
9D11
33D6
11D9
11D9
9D22
3D33
3D33
11D18
3D66

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 2A2B2C 3  6 9A9B9C11A···11E18A18B18C22A···22E33A···33J66A···66J99A···99AD198A···198AD
order12223699911···1118181822···2233···3366···6699···99198···198
size119999222222···22222···22···22···22···22···2

102 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D6D9D11D18D22D33D66D99D198
kernelD198D99C198C66C33C22C18C11C9C6C3C2C1
# reps12111353510103030

Matrix representation of D198 in GL3(𝔽199) generated by

19800
065195
0461
,
100
0461
065195
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

Export

Subgroup lattice of D198 in TeX

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