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G = D194order 388 = 22·97

Dihedral group

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

Aliases: D194, C2×D97, C194⋊C2, C97⋊C22, sometimes denoted D388 or Dih194 or Dih388, SmallGroup(388,4)

Series: Derived Chief Lower central Upper central

C1C97 — D194
C1C97D97 — D194
C97 — D194
C1C2

Generators and relations for D194
 G = < a,b | a194=b2=1, bab=a-1 >

97C2
97C2
97C22

Smallest permutation representation of D194
On 194 points
Generators in S194
(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)
(1 194)(2 193)(3 192)(4 191)(5 190)(6 189)(7 188)(8 187)(9 186)(10 185)(11 184)(12 183)(13 182)(14 181)(15 180)(16 179)(17 178)(18 177)(19 176)(20 175)(21 174)(22 173)(23 172)(24 171)(25 170)(26 169)(27 168)(28 167)(29 166)(30 165)(31 164)(32 163)(33 162)(34 161)(35 160)(36 159)(37 158)(38 157)(39 156)(40 155)(41 154)(42 153)(43 152)(44 151)(45 150)(46 149)(47 148)(48 147)(49 146)(50 145)(51 144)(52 143)(53 142)(54 141)(55 140)(56 139)(57 138)(58 137)(59 136)(60 135)(61 134)(62 133)(63 132)(64 131)(65 130)(66 129)(67 128)(68 127)(69 126)(70 125)(71 124)(72 123)(73 122)(74 121)(75 120)(76 119)(77 118)(78 117)(79 116)(80 115)(81 114)(82 113)(83 112)(84 111)(85 110)(86 109)(87 108)(88 107)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 99)(97 98)

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

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

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

100 conjugacy classes

class 1 2A2B2C97A···97AV194A···194AV
order122297···97194···194
size1197972···22···2

100 irreducible representations

dim11122
type+++++
imageC1C2C2D97D194
kernelD194D97C194C2C1
# reps1214848

Matrix representation of D194 in GL3(𝔽389) generated by

38800
0125375
01414
,
100
0125375
0338264
G:=sub<GL(3,GF(389))| [388,0,0,0,125,14,0,375,14],[1,0,0,0,125,338,0,375,264] >;

D194 in GAP, Magma, Sage, TeX

D_{194}
% in TeX

G:=Group("D194");
// GroupNames label

G:=SmallGroup(388,4);
// by ID

G=gap.SmallGroup(388,4);
# by ID

G:=PCGroup([3,-2,-2,-97,3458]);
// Polycyclic

G:=Group<a,b|a^194=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D194 in TeX

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