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G = D197order 394 = 2·197

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D197, C197⋊C2, sometimes denoted D394 or Dih197 or Dih394, SmallGroup(394,1)

Series: Derived Chief Lower central Upper central

C1C197 — D197
C1C197 — D197
C197 — D197
C1

Generators and relations for D197
 G = < a,b | a197=b2=1, bab=a-1 >

197C2

Smallest permutation representation of D197
On 197 points: primitive
Generators in S197
(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)
(1 197)(2 196)(3 195)(4 194)(5 193)(6 192)(7 191)(8 190)(9 189)(10 188)(11 187)(12 186)(13 185)(14 184)(15 183)(16 182)(17 181)(18 180)(19 179)(20 178)(21 177)(22 176)(23 175)(24 174)(25 173)(26 172)(27 171)(28 170)(29 169)(30 168)(31 167)(32 166)(33 165)(34 164)(35 163)(36 162)(37 161)(38 160)(39 159)(40 158)(41 157)(42 156)(43 155)(44 154)(45 153)(46 152)(47 151)(48 150)(49 149)(50 148)(51 147)(52 146)(53 145)(54 144)(55 143)(56 142)(57 141)(58 140)(59 139)(60 138)(61 137)(62 136)(63 135)(64 134)(65 133)(66 132)(67 131)(68 130)(69 129)(70 128)(71 127)(72 126)(73 125)(74 124)(75 123)(76 122)(77 121)(78 120)(79 119)(80 118)(81 117)(82 116)(83 115)(84 114)(85 113)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)

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

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

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

100 conjugacy classes

class 1  2 197A···197CT
order12197···197
size11972···2

100 irreducible representations

dim112
type+++
imageC1C2D197
kernelD197C197C1
# reps1198

Matrix representation of D197 in GL2(𝔽3547) generated by

12593546
10
,
12593546
31182288
G:=sub<GL(2,GF(3547))| [1259,1,3546,0],[1259,3118,3546,2288] >;

D197 in GAP, Magma, Sage, TeX

D_{197}
% in TeX

G:=Group("D197");
// GroupNames label

G:=SmallGroup(394,1);
// by ID

G=gap.SmallGroup(394,1);
# by ID

G:=PCGroup([2,-2,-197,1569]);
// Polycyclic

G:=Group<a,b|a^197=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D197 in TeX

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