direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D47, C47⋊C6, C141⋊2C2, SmallGroup(282,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C47 — C3×D47 |
Generators and relations for C3×D47
G = < a,b,c | a3=b47=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D47
►On 141 points
Generators in S141
(1 104 77)(2 105 78)(3 106 79)(4 107 80)(5 108 81)(6 109 82)(7 110 83)(8 111 84)(9 112 85)(10 113 86)(11 114 87)(12 115 88)(13 116 89)(14 117 90)(15 118 91)(16 119 92)(17 120 93)(18 121 94)(19 122 48)(20 123 49)(21 124 50)(22 125 51)(23 126 52)(24 127 53)(25 128 54)(26 129 55)(27 130 56)(28 131 57)(29 132 58)(30 133 59)(31 134 60)(32 135 61)(33 136 62)(34 137 63)(35 138 64)(36 139 65)(37 140 66)(38 141 67)(39 95 68)(40 96 69)(41 97 70)(42 98 71)(43 99 72)(44 100 73)(45 101 74)(46 102 75)(47 103 76)
(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)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(48 58)(49 57)(50 56)(51 55)(52 54)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(113 141)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)
G:=sub<Sym(141)| (1,104,77)(2,105,78)(3,106,79)(4,107,80)(5,108,81)(6,109,82)(7,110,83)(8,111,84)(9,112,85)(10,113,86)(11,114,87)(12,115,88)(13,116,89)(14,117,90)(15,118,91)(16,119,92)(17,120,93)(18,121,94)(19,122,48)(20,123,49)(21,124,50)(22,125,51)(23,126,52)(24,127,53)(25,128,54)(26,129,55)(27,130,56)(28,131,57)(29,132,58)(30,133,59)(31,134,60)(32,135,61)(33,136,62)(34,137,63)(35,138,64)(36,139,65)(37,140,66)(38,141,67)(39,95,68)(40,96,69)(41,97,70)(42,98,71)(43,99,72)(44,100,73)(45,101,74)(46,102,75)(47,103,76), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(48,58)(49,57)(50,56)(51,55)(52,54)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)>;
G:=Group( (1,104,77)(2,105,78)(3,106,79)(4,107,80)(5,108,81)(6,109,82)(7,110,83)(8,111,84)(9,112,85)(10,113,86)(11,114,87)(12,115,88)(13,116,89)(14,117,90)(15,118,91)(16,119,92)(17,120,93)(18,121,94)(19,122,48)(20,123,49)(21,124,50)(22,125,51)(23,126,52)(24,127,53)(25,128,54)(26,129,55)(27,130,56)(28,131,57)(29,132,58)(30,133,59)(31,134,60)(32,135,61)(33,136,62)(34,137,63)(35,138,64)(36,139,65)(37,140,66)(38,141,67)(39,95,68)(40,96,69)(41,97,70)(42,98,71)(43,99,72)(44,100,73)(45,101,74)(46,102,75)(47,103,76), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(48,58)(49,57)(50,56)(51,55)(52,54)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128) );
G=PermutationGroup([[(1,104,77),(2,105,78),(3,106,79),(4,107,80),(5,108,81),(6,109,82),(7,110,83),(8,111,84),(9,112,85),(10,113,86),(11,114,87),(12,115,88),(13,116,89),(14,117,90),(15,118,91),(16,119,92),(17,120,93),(18,121,94),(19,122,48),(20,123,49),(21,124,50),(22,125,51),(23,126,52),(24,127,53),(25,128,54),(26,129,55),(27,130,56),(28,131,57),(29,132,58),(30,133,59),(31,134,60),(32,135,61),(33,136,62),(34,137,63),(35,138,64),(36,139,65),(37,140,66),(38,141,67),(39,95,68),(40,96,69),(41,97,70),(42,98,71),(43,99,72),(44,100,73),(45,101,74),(46,102,75),(47,103,76)], [(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(48,58),(49,57),(50,56),(51,55),(52,54),(59,94),(60,93),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(95,112),(96,111),(97,110),(98,109),(99,108),(100,107),(101,106),(102,105),(103,104),(113,141),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128)]])
75 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 47A | ··· | 47W | 141A | ··· | 141AT |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 47 | ··· | 47 | 141 | ··· | 141 |
| size | 1 | 47 | 1 | 1 | 47 | 47 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D47 | C3×D47 |
| kernel | C3×D47 | C141 | D47 | C47 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 23 | 46 |
Matrix representation of C3×D47 ►in GL3(𝔽283) generated by
| 238 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 257 | 1 |
| 0 | 149 | 16 |
| 282 | 0 | 0 |
| 0 | 179 | 184 |
| 0 | 195 | 104 |
G:=sub<GL(3,GF(283))| [238,0,0,0,1,0,0,0,1],[1,0,0,0,257,149,0,1,16],[282,0,0,0,179,195,0,184,104] >;
C3×D47 in GAP, Magma, Sage, TeX
C_3\times D_{47} % in TeX
G:=Group("C3xD47"); // GroupNames label
G:=SmallGroup(282,2);
// by ID
G=gap.SmallGroup(282,2);
# by ID
G:=PCGroup([3,-2,-3,-47,2486]);
// Polycyclic
G:=Group<a,b,c|a^3=b^47=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export