metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D46⋊C4, C2.2D92, C46.6D4, C22.6D46, (C2×C92)⋊1C2, (C2×C4)⋊1D23, C46.5(C2×C4), C2.5(C4×D23), C23⋊1(C22⋊C4), (C22×D23).C2, (C2×Dic23)⋊1C2, C2.2(C23⋊D4), (C2×C46).6C22, SmallGroup(368,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D46⋊C4
G = < a,b,c | a46=b2=c4=1, bab=a-1, ac=ca, cbc-1=a23b >
Smallest permutation representation of D46⋊C4
►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)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 92)(11 91)(12 90)(13 89)(14 88)(15 87)(16 86)(17 85)(18 84)(19 83)(20 82)(21 81)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 60)(43 59)(44 58)(45 57)(46 56)(93 177)(94 176)(95 175)(96 174)(97 173)(98 172)(99 171)(100 170)(101 169)(102 168)(103 167)(104 166)(105 165)(106 164)(107 163)(108 162)(109 161)(110 160)(111 159)(112 158)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 184)(133 183)(134 182)(135 181)(136 180)(137 179)(138 178)
(1 141 56 107)(2 142 57 108)(3 143 58 109)(4 144 59 110)(5 145 60 111)(6 146 61 112)(7 147 62 113)(8 148 63 114)(9 149 64 115)(10 150 65 116)(11 151 66 117)(12 152 67 118)(13 153 68 119)(14 154 69 120)(15 155 70 121)(16 156 71 122)(17 157 72 123)(18 158 73 124)(19 159 74 125)(20 160 75 126)(21 161 76 127)(22 162 77 128)(23 163 78 129)(24 164 79 130)(25 165 80 131)(26 166 81 132)(27 167 82 133)(28 168 83 134)(29 169 84 135)(30 170 85 136)(31 171 86 137)(32 172 87 138)(33 173 88 93)(34 174 89 94)(35 175 90 95)(36 176 91 96)(37 177 92 97)(38 178 47 98)(39 179 48 99)(40 180 49 100)(41 181 50 101)(42 182 51 102)(43 183 52 103)(44 184 53 104)(45 139 54 105)(46 140 55 106)
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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,56)(93,177)(94,176)(95,175)(96,174)(97,173)(98,172)(99,171)(100,170)(101,169)(102,168)(103,167)(104,166)(105,165)(106,164)(107,163)(108,162)(109,161)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,140)(131,139)(132,184)(133,183)(134,182)(135,181)(136,180)(137,179)(138,178), (1,141,56,107)(2,142,57,108)(3,143,58,109)(4,144,59,110)(5,145,60,111)(6,146,61,112)(7,147,62,113)(8,148,63,114)(9,149,64,115)(10,150,65,116)(11,151,66,117)(12,152,67,118)(13,153,68,119)(14,154,69,120)(15,155,70,121)(16,156,71,122)(17,157,72,123)(18,158,73,124)(19,159,74,125)(20,160,75,126)(21,161,76,127)(22,162,77,128)(23,163,78,129)(24,164,79,130)(25,165,80,131)(26,166,81,132)(27,167,82,133)(28,168,83,134)(29,169,84,135)(30,170,85,136)(31,171,86,137)(32,172,87,138)(33,173,88,93)(34,174,89,94)(35,175,90,95)(36,176,91,96)(37,177,92,97)(38,178,47,98)(39,179,48,99)(40,180,49,100)(41,181,50,101)(42,182,51,102)(43,183,52,103)(44,184,53,104)(45,139,54,105)(46,140,55,106)>;
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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,56)(93,177)(94,176)(95,175)(96,174)(97,173)(98,172)(99,171)(100,170)(101,169)(102,168)(103,167)(104,166)(105,165)(106,164)(107,163)(108,162)(109,161)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,140)(131,139)(132,184)(133,183)(134,182)(135,181)(136,180)(137,179)(138,178), (1,141,56,107)(2,142,57,108)(3,143,58,109)(4,144,59,110)(5,145,60,111)(6,146,61,112)(7,147,62,113)(8,148,63,114)(9,149,64,115)(10,150,65,116)(11,151,66,117)(12,152,67,118)(13,153,68,119)(14,154,69,120)(15,155,70,121)(16,156,71,122)(17,157,72,123)(18,158,73,124)(19,159,74,125)(20,160,75,126)(21,161,76,127)(22,162,77,128)(23,163,78,129)(24,164,79,130)(25,165,80,131)(26,166,81,132)(27,167,82,133)(28,168,83,134)(29,169,84,135)(30,170,85,136)(31,171,86,137)(32,172,87,138)(33,173,88,93)(34,174,89,94)(35,175,90,95)(36,176,91,96)(37,177,92,97)(38,178,47,98)(39,179,48,99)(40,180,49,100)(41,181,50,101)(42,182,51,102)(43,183,52,103)(44,184,53,104)(45,139,54,105)(46,140,55,106) );
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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,92),(11,91),(12,90),(13,89),(14,88),(15,87),(16,86),(17,85),(18,84),(19,83),(20,82),(21,81),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,74),(29,73),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,61),(42,60),(43,59),(44,58),(45,57),(46,56),(93,177),(94,176),(95,175),(96,174),(97,173),(98,172),(99,171),(100,170),(101,169),(102,168),(103,167),(104,166),(105,165),(106,164),(107,163),(108,162),(109,161),(110,160),(111,159),(112,158),(113,157),(114,156),(115,155),(116,154),(117,153),(118,152),(119,151),(120,150),(121,149),(122,148),(123,147),(124,146),(125,145),(126,144),(127,143),(128,142),(129,141),(130,140),(131,139),(132,184),(133,183),(134,182),(135,181),(136,180),(137,179),(138,178)], [(1,141,56,107),(2,142,57,108),(3,143,58,109),(4,144,59,110),(5,145,60,111),(6,146,61,112),(7,147,62,113),(8,148,63,114),(9,149,64,115),(10,150,65,116),(11,151,66,117),(12,152,67,118),(13,153,68,119),(14,154,69,120),(15,155,70,121),(16,156,71,122),(17,157,72,123),(18,158,73,124),(19,159,74,125),(20,160,75,126),(21,161,76,127),(22,162,77,128),(23,163,78,129),(24,164,79,130),(25,165,80,131),(26,166,81,132),(27,167,82,133),(28,168,83,134),(29,169,84,135),(30,170,85,136),(31,171,86,137),(32,172,87,138),(33,173,88,93),(34,174,89,94),(35,175,90,95),(36,176,91,96),(37,177,92,97),(38,178,47,98),(39,179,48,99),(40,180,49,100),(41,181,50,101),(42,182,51,102),(43,183,52,103),(44,184,53,104),(45,139,54,105),(46,140,55,106)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 23A | ··· | 23K | 46A | ··· | 46AG | 92A | ··· | 92AR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 1 | 1 | 46 | 46 | 2 | 2 | 46 | 46 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D23 | D46 | C4×D23 | D92 | C23⋊D4 |
| kernel | D46⋊C4 | C2×Dic23 | C2×C92 | C22×D23 | D46 | C46 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 11 | 11 | 22 | 22 | 22 |
Matrix representation of D46⋊C4 ►in GL3(𝔽277) generated by
| 1 | 0 | 0 |
| 0 | 98 | 98 |
| 0 | 179 | 114 |
| 1 | 0 | 0 |
| 0 | 98 | 98 |
| 0 | 114 | 179 |
| 60 | 0 | 0 |
| 0 | 125 | 200 |
| 0 | 77 | 152 |
G:=sub<GL(3,GF(277))| [1,0,0,0,98,179,0,98,114],[1,0,0,0,98,114,0,98,179],[60,0,0,0,125,77,0,200,152] >;
D46⋊C4 in GAP, Magma, Sage, TeX
D_{46}\rtimes C_4 % in TeX
G:=Group("D46:C4"); // GroupNames label
G:=SmallGroup(368,13);
// by ID
G=gap.SmallGroup(368,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,101,26,8804]);
// Polycyclic
G:=Group<a,b,c|a^46=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^23*b>;
// generators/relations
Export