direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D43, D86.C2, C172⋊2C2, C2.1D86, Dic43⋊2C2, C86.2C22, C43⋊1(C2×C4), SmallGroup(344,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C4×D43 |
Generators and relations for C4×D43
G = < a,b,c | a4=b43=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D43
►On 172 points
Generators in S172
(1 168 77 115)(2 169 78 116)(3 170 79 117)(4 171 80 118)(5 172 81 119)(6 130 82 120)(7 131 83 121)(8 132 84 122)(9 133 85 123)(10 134 86 124)(11 135 44 125)(12 136 45 126)(13 137 46 127)(14 138 47 128)(15 139 48 129)(16 140 49 87)(17 141 50 88)(18 142 51 89)(19 143 52 90)(20 144 53 91)(21 145 54 92)(22 146 55 93)(23 147 56 94)(24 148 57 95)(25 149 58 96)(26 150 59 97)(27 151 60 98)(28 152 61 99)(29 153 62 100)(30 154 63 101)(31 155 64 102)(32 156 65 103)(33 157 66 104)(34 158 67 105)(35 159 68 106)(36 160 69 107)(37 161 70 108)(38 162 71 109)(39 163 72 110)(40 164 73 111)(41 165 74 112)(42 166 75 113)(43 167 76 114)
(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)
(1 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(100 129)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(130 162)(131 161)(132 160)(133 159)(134 158)(135 157)(136 156)(137 155)(138 154)(139 153)(140 152)(141 151)(142 150)(143 149)(144 148)(145 147)(163 172)(164 171)(165 170)(166 169)(167 168)
G:=sub<Sym(172)| (1,168,77,115)(2,169,78,116)(3,170,79,117)(4,171,80,118)(5,172,81,119)(6,130,82,120)(7,131,83,121)(8,132,84,122)(9,133,85,123)(10,134,86,124)(11,135,44,125)(12,136,45,126)(13,137,46,127)(14,138,47,128)(15,139,48,129)(16,140,49,87)(17,141,50,88)(18,142,51,89)(19,143,52,90)(20,144,53,91)(21,145,54,92)(22,146,55,93)(23,147,56,94)(24,148,57,95)(25,149,58,96)(26,150,59,97)(27,151,60,98)(28,152,61,99)(29,153,62,100)(30,154,63,101)(31,155,64,102)(32,156,65,103)(33,157,66,104)(34,158,67,105)(35,159,68,106)(36,160,69,107)(37,161,70,108)(38,162,71,109)(39,163,72,110)(40,164,73,111)(41,165,74,112)(42,166,75,113)(43,167,76,114), (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(145,147)(163,172)(164,171)(165,170)(166,169)(167,168)>;
G:=Group( (1,168,77,115)(2,169,78,116)(3,170,79,117)(4,171,80,118)(5,172,81,119)(6,130,82,120)(7,131,83,121)(8,132,84,122)(9,133,85,123)(10,134,86,124)(11,135,44,125)(12,136,45,126)(13,137,46,127)(14,138,47,128)(15,139,48,129)(16,140,49,87)(17,141,50,88)(18,142,51,89)(19,143,52,90)(20,144,53,91)(21,145,54,92)(22,146,55,93)(23,147,56,94)(24,148,57,95)(25,149,58,96)(26,150,59,97)(27,151,60,98)(28,152,61,99)(29,153,62,100)(30,154,63,101)(31,155,64,102)(32,156,65,103)(33,157,66,104)(34,158,67,105)(35,159,68,106)(36,160,69,107)(37,161,70,108)(38,162,71,109)(39,163,72,110)(40,164,73,111)(41,165,74,112)(42,166,75,113)(43,167,76,114), (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(145,147)(163,172)(164,171)(165,170)(166,169)(167,168) );
G=PermutationGroup([[(1,168,77,115),(2,169,78,116),(3,170,79,117),(4,171,80,118),(5,172,81,119),(6,130,82,120),(7,131,83,121),(8,132,84,122),(9,133,85,123),(10,134,86,124),(11,135,44,125),(12,136,45,126),(13,137,46,127),(14,138,47,128),(15,139,48,129),(16,140,49,87),(17,141,50,88),(18,142,51,89),(19,143,52,90),(20,144,53,91),(21,145,54,92),(22,146,55,93),(23,147,56,94),(24,148,57,95),(25,149,58,96),(26,150,59,97),(27,151,60,98),(28,152,61,99),(29,153,62,100),(30,154,63,101),(31,155,64,102),(32,156,65,103),(33,157,66,104),(34,158,67,105),(35,159,68,106),(36,160,69,107),(37,161,70,108),(38,162,71,109),(39,163,72,110),(40,164,73,111),(41,165,74,112),(42,166,75,113),(43,167,76,114)], [(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)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(44,66),(45,65),(46,64),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(100,129),(101,128),(102,127),(103,126),(104,125),(105,124),(106,123),(107,122),(108,121),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(130,162),(131,161),(132,160),(133,159),(134,158),(135,157),(136,156),(137,155),(138,154),(139,153),(140,152),(141,151),(142,150),(143,149),(144,148),(145,147),(163,172),(164,171),(165,170),(166,169),(167,168)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 43A | ··· | 43U | 86A | ··· | 86U | 172A | ··· | 172AP |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 43 | ··· | 43 | 86 | ··· | 86 | 172 | ··· | 172 |
| size | 1 | 1 | 43 | 43 | 1 | 1 | 43 | 43 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D43 | D86 | C4×D43 |
| kernel | C4×D43 | Dic43 | C172 | D86 | D43 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 21 | 21 | 42 |
Matrix representation of C4×D43 ►in GL3(𝔽173) generated by
| 93 | 0 | 0 |
| 0 | 172 | 0 |
| 0 | 0 | 172 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 172 | 11 |
| 172 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(173))| [93,0,0,0,172,0,0,0,172],[1,0,0,0,0,172,0,1,11],[172,0,0,0,0,1,0,1,0] >;
C4×D43 in GAP, Magma, Sage, TeX
C_4\times D_{43} % in TeX
G:=Group("C4xD43"); // GroupNames label
G:=SmallGroup(344,4);
// by ID
G=gap.SmallGroup(344,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-43,21,5379]);
// Polycyclic
G:=Group<a,b,c|a^4=b^43=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export