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## G = C176⋊C2order 352 = 25·11

### 2nd semidirect product of C176 and C2 acting faithfully

Aliases: C1762C2, C162D11, C2.4D88, C22.2D8, C4.2D44, C111SD32, D88.1C2, C8.14D22, C44.25D4, Dic441C2, C88.15C22, SmallGroup(352,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C88 — C176⋊C2
 Chief series C1 — C11 — C22 — C44 — C88 — D88 — C176⋊C2
 Lower central C11 — C22 — C44 — C88 — C176⋊C2
 Upper central C1 — C2 — C4 — C8 — C16

Generators and relations for C176⋊C2
G = < a,b | a176=b2=1, bab=a87 >

88C2
44C4
44C22
8D11
22Q8
22D4
4D22
11Q16
11D8
2D44
11SD32

Smallest permutation representation of C176⋊C2
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)
(2 88)(3 175)(4 86)(5 173)(6 84)(7 171)(8 82)(9 169)(10 80)(11 167)(12 78)(13 165)(14 76)(15 163)(16 74)(17 161)(18 72)(19 159)(20 70)(21 157)(22 68)(23 155)(24 66)(25 153)(26 64)(27 151)(28 62)(29 149)(30 60)(31 147)(32 58)(33 145)(34 56)(35 143)(36 54)(37 141)(38 52)(39 139)(40 50)(41 137)(42 48)(43 135)(44 46)(45 133)(47 131)(49 129)(51 127)(53 125)(55 123)(57 121)(59 119)(61 117)(63 115)(65 113)(67 111)(69 109)(71 107)(73 105)(75 103)(77 101)(79 99)(81 97)(83 95)(85 93)(87 91)(90 176)(92 174)(94 172)(96 170)(98 168)(100 166)(102 164)(104 162)(106 160)(108 158)(110 156)(112 154)(114 152)(116 150)(118 148)(120 146)(122 144)(124 142)(126 140)(128 138)(130 136)(132 134)

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

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

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

91 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 11A ··· 11E 16A 16B 16C 16D 22A ··· 22E 44A ··· 44J 88A ··· 88T 176A ··· 176AN order 1 2 2 4 4 8 8 11 ··· 11 16 16 16 16 22 ··· 22 44 ··· 44 88 ··· 88 176 ··· 176 size 1 1 88 2 88 2 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

91 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 D4 D8 D11 SD32 D22 D44 D88 C176⋊C2 kernel C176⋊C2 C176 D88 Dic44 C44 C22 C16 C11 C8 C4 C2 C1 # reps 1 1 1 1 1 2 5 4 5 10 20 40

Matrix representation of C176⋊C2 in GL2(𝔽353) generated by

 258 343 10 198
,
 1 0 6 352
G:=sub<GL(2,GF(353))| [258,10,343,198],[1,6,0,352] >;

C176⋊C2 in GAP, Magma, Sage, TeX

C_{176}\rtimes C_2
% in TeX

G:=Group("C176:C2");
// GroupNames label

G:=SmallGroup(352,6);
// by ID

G=gap.SmallGroup(352,6);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,73,79,506,50,579,69,11525]);
// Polycyclic

G:=Group<a,b|a^176=b^2=1,b*a*b=a^87>;
// generators/relations

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