direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic23, C46⋊C4, C2.2D46, C22.D23, C46.4C22, C23⋊2(C2×C4), (C2×C46).C2, SmallGroup(184,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C2×Dic23 |
Generators and relations for C2×Dic23
G = < a,b,c | a2=b46=1, c2=b23, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic23
►Regular action on 184 points
Generators in S184
(1 88)(2 89)(3 90)(4 91)(5 92)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 161)(116 162)(117 163)(118 164)(119 165)(120 166)(121 167)(122 168)(123 169)(124 170)(125 171)(126 172)(127 173)(128 174)(129 175)(130 176)(131 177)(132 178)(133 179)(134 180)(135 181)(136 182)(137 183)(138 184)
(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)
(1 116 24 93)(2 115 25 138)(3 114 26 137)(4 113 27 136)(5 112 28 135)(6 111 29 134)(7 110 30 133)(8 109 31 132)(9 108 32 131)(10 107 33 130)(11 106 34 129)(12 105 35 128)(13 104 36 127)(14 103 37 126)(15 102 38 125)(16 101 39 124)(17 100 40 123)(18 99 41 122)(19 98 42 121)(20 97 43 120)(21 96 44 119)(22 95 45 118)(23 94 46 117)(47 157 70 180)(48 156 71 179)(49 155 72 178)(50 154 73 177)(51 153 74 176)(52 152 75 175)(53 151 76 174)(54 150 77 173)(55 149 78 172)(56 148 79 171)(57 147 80 170)(58 146 81 169)(59 145 82 168)(60 144 83 167)(61 143 84 166)(62 142 85 165)(63 141 86 164)(64 140 87 163)(65 139 88 162)(66 184 89 161)(67 183 90 160)(68 182 91 159)(69 181 92 158)
G:=sub<Sym(184)| (1,88)(2,89)(3,90)(4,91)(5,92)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)(116,162)(117,163)(118,164)(119,165)(120,166)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,183)(138,184), (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), (1,116,24,93)(2,115,25,138)(3,114,26,137)(4,113,27,136)(5,112,28,135)(6,111,29,134)(7,110,30,133)(8,109,31,132)(9,108,32,131)(10,107,33,130)(11,106,34,129)(12,105,35,128)(13,104,36,127)(14,103,37,126)(15,102,38,125)(16,101,39,124)(17,100,40,123)(18,99,41,122)(19,98,42,121)(20,97,43,120)(21,96,44,119)(22,95,45,118)(23,94,46,117)(47,157,70,180)(48,156,71,179)(49,155,72,178)(50,154,73,177)(51,153,74,176)(52,152,75,175)(53,151,76,174)(54,150,77,173)(55,149,78,172)(56,148,79,171)(57,147,80,170)(58,146,81,169)(59,145,82,168)(60,144,83,167)(61,143,84,166)(62,142,85,165)(63,141,86,164)(64,140,87,163)(65,139,88,162)(66,184,89,161)(67,183,90,160)(68,182,91,159)(69,181,92,158)>;
G:=Group( (1,88)(2,89)(3,90)(4,91)(5,92)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)(116,162)(117,163)(118,164)(119,165)(120,166)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,183)(138,184), (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), (1,116,24,93)(2,115,25,138)(3,114,26,137)(4,113,27,136)(5,112,28,135)(6,111,29,134)(7,110,30,133)(8,109,31,132)(9,108,32,131)(10,107,33,130)(11,106,34,129)(12,105,35,128)(13,104,36,127)(14,103,37,126)(15,102,38,125)(16,101,39,124)(17,100,40,123)(18,99,41,122)(19,98,42,121)(20,97,43,120)(21,96,44,119)(22,95,45,118)(23,94,46,117)(47,157,70,180)(48,156,71,179)(49,155,72,178)(50,154,73,177)(51,153,74,176)(52,152,75,175)(53,151,76,174)(54,150,77,173)(55,149,78,172)(56,148,79,171)(57,147,80,170)(58,146,81,169)(59,145,82,168)(60,144,83,167)(61,143,84,166)(62,142,85,165)(63,141,86,164)(64,140,87,163)(65,139,88,162)(66,184,89,161)(67,183,90,160)(68,182,91,159)(69,181,92,158) );
G=PermutationGroup([[(1,88),(2,89),(3,90),(4,91),(5,92),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,161),(116,162),(117,163),(118,164),(119,165),(120,166),(121,167),(122,168),(123,169),(124,170),(125,171),(126,172),(127,173),(128,174),(129,175),(130,176),(131,177),(132,178),(133,179),(134,180),(135,181),(136,182),(137,183),(138,184)], [(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)], [(1,116,24,93),(2,115,25,138),(3,114,26,137),(4,113,27,136),(5,112,28,135),(6,111,29,134),(7,110,30,133),(8,109,31,132),(9,108,32,131),(10,107,33,130),(11,106,34,129),(12,105,35,128),(13,104,36,127),(14,103,37,126),(15,102,38,125),(16,101,39,124),(17,100,40,123),(18,99,41,122),(19,98,42,121),(20,97,43,120),(21,96,44,119),(22,95,45,118),(23,94,46,117),(47,157,70,180),(48,156,71,179),(49,155,72,178),(50,154,73,177),(51,153,74,176),(52,152,75,175),(53,151,76,174),(54,150,77,173),(55,149,78,172),(56,148,79,171),(57,147,80,170),(58,146,81,169),(59,145,82,168),(60,144,83,167),(61,143,84,166),(62,142,85,165),(63,141,86,164),(64,140,87,163),(65,139,88,162),(66,184,89,161),(67,183,90,160),(68,182,91,159),(69,181,92,158)]])
C2×Dic23 is a maximal subgroup of
Dic23⋊C4 C92⋊C4 D46⋊C4 C23.D23 C2×C4×D23 D4⋊2D23
C2×Dic23 is a maximal quotient of C92.C4 C92⋊C4 C23.D23
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 23A | ··· | 23K | 46A | ··· | 46AG |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 |
| size | 1 | 1 | 1 | 1 | 23 | 23 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D23 | Dic23 | D46 |
| kernel | C2×Dic23 | Dic23 | C2×C46 | C46 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 11 | 22 | 11 |
Matrix representation of C2×Dic23 ►in GL4(𝔽277) generated by
| 1 | 0 | 0 | 0 |
| 0 | 276 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 276 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 236 |
| 217 | 0 | 0 | 0 |
| 0 | 276 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(277))| [1,0,0,0,0,276,0,0,0,0,1,0,0,0,0,1],[276,0,0,0,0,1,0,0,0,0,27,0,0,0,0,236],[217,0,0,0,0,276,0,0,0,0,0,1,0,0,1,0] >;
C2×Dic23 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{23} % in TeX
G:=Group("C2xDic23"); // GroupNames label
G:=SmallGroup(184,6);
// by ID
G=gap.SmallGroup(184,6);
# by ID
G:=PCGroup([4,-2,-2,-2,-23,16,2819]);
// Polycyclic
G:=Group<a,b,c|a^2=b^46=1,c^2=b^23,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export