Copied to
clipboard

## G = D182order 364 = 22·7·13

### Dihedral group

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

 Derived series C1 — C91 — D182
 Chief series C1 — C13 — C91 — D91 — D182
 Lower central C91 — D182
 Upper central C1 — C2

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 2A 2B 2C 7A 7B 7C 13A ··· 13F 14A 14B 14C 26A ··· 26F 91A ··· 91AJ 182A ··· 182AJ order 1 2 2 2 7 7 7 13 ··· 13 14 14 14 26 ··· 26 91 ··· 91 182 ··· 182 size 1 1 91 91 2 2 2 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

94 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 D7 D13 D14 D26 D91 D182 kernel D182 D91 C182 C26 C14 C13 C7 C2 C1 # reps 1 2 1 3 6 3 6 36 36

Matrix representation of D182 in GL3(𝔽547) generated by

 546 0 0 0 83 236 0 311 133
,
 1 0 0 0 194 467 0 26 353
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

׿
×
𝔽