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G = D176order 352 = 25·11

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D176, C111D16, C1761C2, D881C2, C161D11, C22.1D8, C2.3D88, C4.1D44, C8.13D22, C44.24D4, C88.14C22, sometimes denoted D352 or Dih176 or Dih352, SmallGroup(352,5)

Series: Derived Chief Lower central Upper central

C1C88 — D176
C1C11C22C44C88D88 — D176
C11C22C44C88 — D176
C1C2C4C8C16

Generators and relations for D176
 G = < a,b | a176=b2=1, bab=a-1 >

88C2
88C2
44C22
44C22
8D11
8D11
22D4
22D4
4D22
4D22
11D8
11D8
2D44
2D44
11D16

Smallest permutation representation of D176
On 176 points
Generators in S176
(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)
(1 176)(2 175)(3 174)(4 173)(5 172)(6 171)(7 170)(8 169)(9 168)(10 167)(11 166)(12 165)(13 164)(14 163)(15 162)(16 161)(17 160)(18 159)(19 158)(20 157)(21 156)(22 155)(23 154)(24 153)(25 152)(26 151)(27 150)(28 149)(29 148)(30 147)(31 146)(32 145)(33 144)(34 143)(35 142)(36 141)(37 140)(38 139)(39 138)(40 137)(41 136)(42 135)(43 134)(44 133)(45 132)(46 131)(47 130)(48 129)(49 128)(50 127)(51 126)(52 125)(53 124)(54 123)(55 122)(56 121)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 104)(74 103)(75 102)(76 101)(77 100)(78 99)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)

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

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

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

91 conjugacy classes

class 1 2A2B2C 4 8A8B11A···11E16A16B16C16D22A···22E44A···44J88A···88T176A···176AN
order122248811···111616161622···2244···4488···88176···176
size1188882222···222222···22···22···22···2

91 irreducible representations

dim11122222222
type+++++++++++
imageC1C2C2D4D8D11D16D22D44D88D176
kernelD176C176D88C44C22C16C11C8C4C2C1
# reps11212545102040

Matrix representation of D176 in GL2(𝔽353) generated by

341174
179199
,
341174
26912
G:=sub<GL(2,GF(353))| [341,179,174,199],[341,269,174,12] >;

D176 in GAP, Magma, Sage, TeX

D_{176}
% in TeX

G:=Group("D176");
// GroupNames label

G:=SmallGroup(352,5);
// by ID

G=gap.SmallGroup(352,5);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,73,79,218,122,579,69,11525]);
// Polycyclic

G:=Group<a,b|a^176=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D176 in TeX

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