metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D23, C184⋊2C2, C4.8D46, C46.1D4, C2.3D92, C23⋊1SD16, D92.1C2, Dic46⋊1C2, C92.8C22, SmallGroup(368,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C184⋊C2
G = < a,b | a184=b2=1, bab=a91 >
Smallest permutation representation of C184⋊C2
►On 184 points
Generators in S184
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184)
(2 92)(3 183)(4 90)(5 181)(6 88)(7 179)(8 86)(9 177)(10 84)(11 175)(12 82)(13 173)(14 80)(15 171)(16 78)(17 169)(18 76)(19 167)(20 74)(21 165)(22 72)(23 163)(24 70)(25 161)(26 68)(27 159)(28 66)(29 157)(30 64)(31 155)(32 62)(33 153)(34 60)(35 151)(36 58)(37 149)(38 56)(39 147)(40 54)(41 145)(42 52)(43 143)(44 50)(45 141)(46 48)(47 139)(49 137)(51 135)(53 133)(55 131)(57 129)(59 127)(61 125)(63 123)(65 121)(67 119)(69 117)(71 115)(73 113)(75 111)(77 109)(79 107)(81 105)(83 103)(85 101)(87 99)(89 97)(91 95)(94 184)(96 182)(98 180)(100 178)(102 176)(104 174)(106 172)(108 170)(110 168)(112 166)(114 164)(116 162)(118 160)(120 158)(122 156)(124 154)(126 152)(128 150)(130 148)(132 146)(134 144)(136 142)(138 140)
G:=sub<Sym(184)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184), (2,92)(3,183)(4,90)(5,181)(6,88)(7,179)(8,86)(9,177)(10,84)(11,175)(12,82)(13,173)(14,80)(15,171)(16,78)(17,169)(18,76)(19,167)(20,74)(21,165)(22,72)(23,163)(24,70)(25,161)(26,68)(27,159)(28,66)(29,157)(30,64)(31,155)(32,62)(33,153)(34,60)(35,151)(36,58)(37,149)(38,56)(39,147)(40,54)(41,145)(42,52)(43,143)(44,50)(45,141)(46,48)(47,139)(49,137)(51,135)(53,133)(55,131)(57,129)(59,127)(61,125)(63,123)(65,121)(67,119)(69,117)(71,115)(73,113)(75,111)(77,109)(79,107)(81,105)(83,103)(85,101)(87,99)(89,97)(91,95)(94,184)(96,182)(98,180)(100,178)(102,176)(104,174)(106,172)(108,170)(110,168)(112,166)(114,164)(116,162)(118,160)(120,158)(122,156)(124,154)(126,152)(128,150)(130,148)(132,146)(134,144)(136,142)(138,140)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184), (2,92)(3,183)(4,90)(5,181)(6,88)(7,179)(8,86)(9,177)(10,84)(11,175)(12,82)(13,173)(14,80)(15,171)(16,78)(17,169)(18,76)(19,167)(20,74)(21,165)(22,72)(23,163)(24,70)(25,161)(26,68)(27,159)(28,66)(29,157)(30,64)(31,155)(32,62)(33,153)(34,60)(35,151)(36,58)(37,149)(38,56)(39,147)(40,54)(41,145)(42,52)(43,143)(44,50)(45,141)(46,48)(47,139)(49,137)(51,135)(53,133)(55,131)(57,129)(59,127)(61,125)(63,123)(65,121)(67,119)(69,117)(71,115)(73,113)(75,111)(77,109)(79,107)(81,105)(83,103)(85,101)(87,99)(89,97)(91,95)(94,184)(96,182)(98,180)(100,178)(102,176)(104,174)(106,172)(108,170)(110,168)(112,166)(114,164)(116,162)(118,160)(120,158)(122,156)(124,154)(126,152)(128,150)(130,148)(132,146)(134,144)(136,142)(138,140) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)], [(2,92),(3,183),(4,90),(5,181),(6,88),(7,179),(8,86),(9,177),(10,84),(11,175),(12,82),(13,173),(14,80),(15,171),(16,78),(17,169),(18,76),(19,167),(20,74),(21,165),(22,72),(23,163),(24,70),(25,161),(26,68),(27,159),(28,66),(29,157),(30,64),(31,155),(32,62),(33,153),(34,60),(35,151),(36,58),(37,149),(38,56),(39,147),(40,54),(41,145),(42,52),(43,143),(44,50),(45,141),(46,48),(47,139),(49,137),(51,135),(53,133),(55,131),(57,129),(59,127),(61,125),(63,123),(65,121),(67,119),(69,117),(71,115),(73,113),(75,111),(77,109),(79,107),(81,105),(83,103),(85,101),(87,99),(89,97),(91,95),(94,184),(96,182),(98,180),(100,178),(102,176),(104,174),(106,172),(108,170),(110,168),(112,166),(114,164),(116,162),(118,160),(120,158),(122,156),(124,154),(126,152),(128,150),(130,148),(132,146),(134,144),(136,142),(138,140)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92V | 184A | ··· | 184AR |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 92 | 2 | 92 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D23 | D46 | D92 | C184⋊C2 |
| kernel | C184⋊C2 | C184 | Dic46 | D92 | C46 | C23 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 11 | 11 | 22 | 44 |
Matrix representation of C184⋊C2 ►in GL2(𝔽1289) generated by
| 10 | 858 |
| 459 | 841 |
| 338 | 755 |
| 1235 | 951 |
G:=sub<GL(2,GF(1289))| [10,459,858,841],[338,1235,755,951] >;
C184⋊C2 in GAP, Magma, Sage, TeX
C_{184}\rtimes C_2 % in TeX
G:=Group("C184:C2"); // GroupNames label
G:=SmallGroup(368,5);
// by ID
G=gap.SmallGroup(368,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,61,26,182,42,8804]);
// Polycyclic
G:=Group<a,b|a^184=b^2=1,b*a*b=a^91>;
// generators/relations
Export