metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C176⋊2C2, C16⋊2D11, C2.4D88, C22.2D8, C4.2D44, C11⋊1SD32, D88.1C2, C8.14D22, C44.25D4, Dic44⋊1C2, C88.15C22, SmallGroup(352,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C176⋊C2
G = < a,b | a176=b2=1, bab=a87 >
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
Export