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G = D182order 364 = 22·7·13

Dihedral group

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

Aliases: D182, C2×D91, C26⋊D7, C14⋊D13, C72D26, C132D14, C1821C2, C912C22, sometimes denoted D364 or Dih182 or Dih364, SmallGroup(364,10)

Series: Derived Chief Lower central Upper central

C1C91 — D182
C1C13C91D91 — D182
C91 — D182
C1C2

Generators and relations for D182
 G = < a,b | a182=b2=1, bab=a-1 >

91C2
91C2
91C22
13D7
13D7
7D13
7D13
13D14
7D26

Smallest permutation representation of D182
On 182 points
Generators in S182
(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)
(1 182)(2 181)(3 180)(4 179)(5 178)(6 177)(7 176)(8 175)(9 174)(10 173)(11 172)(12 171)(13 170)(14 169)(15 168)(16 167)(17 166)(18 165)(19 164)(20 163)(21 162)(22 161)(23 160)(24 159)(25 158)(26 157)(27 156)(28 155)(29 154)(30 153)(31 152)(32 151)(33 150)(34 149)(35 148)(36 147)(37 146)(38 145)(39 144)(40 143)(41 142)(42 141)(43 140)(44 139)(45 138)(46 137)(47 136)(48 135)(49 134)(50 133)(51 132)(52 131)(53 130)(54 129)(55 128)(56 127)(57 126)(58 125)(59 124)(60 123)(61 122)(62 121)(63 120)(64 119)(65 118)(66 117)(67 116)(68 115)(69 114)(70 113)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)

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

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

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

94 conjugacy classes

class 1 2A2B2C7A7B7C13A···13F14A14B14C26A···26F91A···91AJ182A···182AJ
order122277713···1314141426···2691···91182···182
size1191912222···22222···22···22···2

94 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D7D13D14D26D91D182
kernelD182D91C182C26C14C13C7C2C1
# reps12136363636

Matrix representation of D182 in GL3(𝔽547) generated by

54600
083236
0311133
,
100
0194467
026353
G:=sub<GL(3,GF(547))| [546,0,0,0,83,311,0,236,133],[1,0,0,0,194,26,0,467,353] >;

D182 in GAP, Magma, Sage, TeX

D_{182}
% in TeX

G:=Group("D182");
// GroupNames label

G:=SmallGroup(364,10);
// by ID

G=gap.SmallGroup(364,10);
# by ID

G:=PCGroup([4,-2,-2,-7,-13,290,5379]);
// Polycyclic

G:=Group<a,b|a^182=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D182 in TeX

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