direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C47, C3⋊C94, C141⋊3C2, SmallGroup(282,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C47 |
Generators and relations for S3×C47
G = < a,b,c | a47=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C47
►On 141 points
Generators in S141
(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 66 128)(2 67 129)(3 68 130)(4 69 131)(5 70 132)(6 71 133)(7 72 134)(8 73 135)(9 74 136)(10 75 137)(11 76 138)(12 77 139)(13 78 140)(14 79 141)(15 80 95)(16 81 96)(17 82 97)(18 83 98)(19 84 99)(20 85 100)(21 86 101)(22 87 102)(23 88 103)(24 89 104)(25 90 105)(26 91 106)(27 92 107)(28 93 108)(29 94 109)(30 48 110)(31 49 111)(32 50 112)(33 51 113)(34 52 114)(35 53 115)(36 54 116)(37 55 117)(38 56 118)(39 57 119)(40 58 120)(41 59 121)(42 60 122)(43 61 123)(44 62 124)(45 63 125)(46 64 126)(47 65 127)
(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 141)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)
G:=sub<Sym(141)| (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), (1,66,128)(2,67,129)(3,68,130)(4,69,131)(5,70,132)(6,71,133)(7,72,134)(8,73,135)(9,74,136)(10,75,137)(11,76,138)(12,77,139)(13,78,140)(14,79,141)(15,80,95)(16,81,96)(17,82,97)(18,83,98)(19,84,99)(20,85,100)(21,86,101)(22,87,102)(23,88,103)(24,89,104)(25,90,105)(26,91,106)(27,92,107)(28,93,108)(29,94,109)(30,48,110)(31,49,111)(32,50,112)(33,51,113)(34,52,114)(35,53,115)(36,54,116)(37,55,117)(38,56,118)(39,57,119)(40,58,120)(41,59,121)(42,60,122)(43,61,123)(44,62,124)(45,63,125)(46,64,126)(47,65,127), (48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)>;
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), (1,66,128)(2,67,129)(3,68,130)(4,69,131)(5,70,132)(6,71,133)(7,72,134)(8,73,135)(9,74,136)(10,75,137)(11,76,138)(12,77,139)(13,78,140)(14,79,141)(15,80,95)(16,81,96)(17,82,97)(18,83,98)(19,84,99)(20,85,100)(21,86,101)(22,87,102)(23,88,103)(24,89,104)(25,90,105)(26,91,106)(27,92,107)(28,93,108)(29,94,109)(30,48,110)(31,49,111)(32,50,112)(33,51,113)(34,52,114)(35,53,115)(36,54,116)(37,55,117)(38,56,118)(39,57,119)(40,58,120)(41,59,121)(42,60,122)(43,61,123)(44,62,124)(45,63,125)(46,64,126)(47,65,127), (48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109) );
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)], [(1,66,128),(2,67,129),(3,68,130),(4,69,131),(5,70,132),(6,71,133),(7,72,134),(8,73,135),(9,74,136),(10,75,137),(11,76,138),(12,77,139),(13,78,140),(14,79,141),(15,80,95),(16,81,96),(17,82,97),(18,83,98),(19,84,99),(20,85,100),(21,86,101),(22,87,102),(23,88,103),(24,89,104),(25,90,105),(26,91,106),(27,92,107),(28,93,108),(29,94,109),(30,48,110),(31,49,111),(32,50,112),(33,51,113),(34,52,114),(35,53,115),(36,54,116),(37,55,117),(38,56,118),(39,57,119),(40,58,120),(41,59,121),(42,60,122),(43,61,123),(44,62,124),(45,63,125),(46,64,126),(47,65,127)], [(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,121),(60,122),(61,123),(62,124),(63,125),(64,126),(65,127),(66,128),(67,129),(68,130),(69,131),(70,132),(71,133),(72,134),(73,135),(74,136),(75,137),(76,138),(77,139),(78,140),(79,141),(80,95),(81,96),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108),(94,109)]])
141 conjugacy classes
| class | 1 | 2 | 3 | 47A | ··· | 47AT | 94A | ··· | 94AT | 141A | ··· | 141AT |
| order | 1 | 2 | 3 | 47 | ··· | 47 | 94 | ··· | 94 | 141 | ··· | 141 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
141 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C47 | C94 | S3 | S3×C47 |
| kernel | S3×C47 | C141 | S3 | C3 | C47 | C1 |
| # reps | 1 | 1 | 46 | 46 | 1 | 46 |
Matrix representation of S3×C47 ►in GL2(𝔽283) generated by
| 181 | 0 |
| 0 | 181 |
| 282 | 282 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(283))| [181,0,0,181],[282,1,282,0],[0,1,1,0] >;
S3×C47 in GAP, Magma, Sage, TeX
S_3\times C_{47} % in TeX
G:=Group("S3xC47"); // GroupNames label
G:=SmallGroup(282,1);
// by ID
G=gap.SmallGroup(282,1);
# by ID
G:=PCGroup([3,-2,-47,-3,1694]);
// Polycyclic
G:=Group<a,b,c|a^47=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export