direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D43, C43⋊3C6, C129⋊2C2, SmallGroup(258,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C3×D43 |
Generators and relations for C3×D43
G = < a,b,c | a3=b43=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D43
►On 129 points
Generators in S129
(1 121 62)(2 122 63)(3 123 64)(4 124 65)(5 125 66)(6 126 67)(7 127 68)(8 128 69)(9 129 70)(10 87 71)(11 88 72)(12 89 73)(13 90 74)(14 91 75)(15 92 76)(16 93 77)(17 94 78)(18 95 79)(19 96 80)(20 97 81)(21 98 82)(22 99 83)(23 100 84)(24 101 85)(25 102 86)(26 103 44)(27 104 45)(28 105 46)(29 106 47)(30 107 48)(31 108 49)(32 109 50)(33 110 51)(34 111 52)(35 112 53)(36 113 54)(37 114 55)(38 115 56)(39 116 57)(40 117 58)(41 118 59)(42 119 60)(43 120 61)
(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)
(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 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(80 86)(81 85)(82 84)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(112 129)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)
G:=sub<Sym(129)| (1,121,62)(2,122,63)(3,123,64)(4,124,65)(5,125,66)(6,126,67)(7,127,68)(8,128,69)(9,129,70)(10,87,71)(11,88,72)(12,89,73)(13,90,74)(14,91,75)(15,92,76)(16,93,77)(17,94,78)(18,95,79)(19,96,80)(20,97,81)(21,98,82)(22,99,83)(23,100,84)(24,101,85)(25,102,86)(26,103,44)(27,104,45)(28,105,46)(29,106,47)(30,107,48)(31,108,49)(32,109,50)(33,110,51)(34,111,52)(35,112,53)(36,113,54)(37,114,55)(38,115,56)(39,116,57)(40,117,58)(41,118,59)(42,119,60)(43,120,61), (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), (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)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(80,86)(81,85)(82,84)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)>;
G:=Group( (1,121,62)(2,122,63)(3,123,64)(4,124,65)(5,125,66)(6,126,67)(7,127,68)(8,128,69)(9,129,70)(10,87,71)(11,88,72)(12,89,73)(13,90,74)(14,91,75)(15,92,76)(16,93,77)(17,94,78)(18,95,79)(19,96,80)(20,97,81)(21,98,82)(22,99,83)(23,100,84)(24,101,85)(25,102,86)(26,103,44)(27,104,45)(28,105,46)(29,106,47)(30,107,48)(31,108,49)(32,109,50)(33,110,51)(34,111,52)(35,112,53)(36,113,54)(37,114,55)(38,115,56)(39,116,57)(40,117,58)(41,118,59)(42,119,60)(43,120,61), (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), (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)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(80,86)(81,85)(82,84)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121) );
G=PermutationGroup([[(1,121,62),(2,122,63),(3,123,64),(4,124,65),(5,125,66),(6,126,67),(7,127,68),(8,128,69),(9,129,70),(10,87,71),(11,88,72),(12,89,73),(13,90,74),(14,91,75),(15,92,76),(16,93,77),(17,94,78),(18,95,79),(19,96,80),(20,97,81),(21,98,82),(22,99,83),(23,100,84),(24,101,85),(25,102,86),(26,103,44),(27,104,45),(28,105,46),(29,106,47),(30,107,48),(31,108,49),(32,109,50),(33,110,51),(34,111,52),(35,112,53),(36,113,54),(37,114,55),(38,115,56),(39,116,57),(40,117,58),(41,118,59),(42,119,60),(43,120,61)], [(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)], [(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),(44,79),(45,78),(46,77),(47,76),(48,75),(49,74),(50,73),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(80,86),(81,85),(82,84),(87,111),(88,110),(89,109),(90,108),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(112,129),(113,128),(114,127),(115,126),(116,125),(117,124),(118,123),(119,122),(120,121)]])
69 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 43A | ··· | 43U | 129A | ··· | 129AP |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 43 | ··· | 43 | 129 | ··· | 129 |
| size | 1 | 43 | 1 | 1 | 43 | 43 | 2 | ··· | 2 | 2 | ··· | 2 |
69 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D43 | C3×D43 |
| kernel | C3×D43 | C129 | D43 | C43 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 21 | 42 |
Matrix representation of C3×D43 ►in GL2(𝔽1033) generated by
| 195 | 0 |
| 0 | 195 |
| 912 | 1 |
| 759 | 173 |
| 173 | 1032 |
| 1004 | 860 |
G:=sub<GL(2,GF(1033))| [195,0,0,195],[912,759,1,173],[173,1004,1032,860] >;
C3×D43 in GAP, Magma, Sage, TeX
C_3\times D_{43} % in TeX
G:=Group("C3xD43"); // GroupNames label
G:=SmallGroup(258,4);
// by ID
G=gap.SmallGroup(258,4);
# by ID
G:=PCGroup([3,-2,-3,-43,2270]);
// Polycyclic
G:=Group<a,b,c|a^3=b^43=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export