direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C46, C6⋊C46, C138⋊3C2, C69⋊4C22, C3⋊(C2×C46), SmallGroup(276,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C46 |
Generators and relations for S3×C46
G = < a,b,c | a46=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C46
►On 138 points
Generators in S138
(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)
(1 68 125)(2 69 126)(3 70 127)(4 71 128)(5 72 129)(6 73 130)(7 74 131)(8 75 132)(9 76 133)(10 77 134)(11 78 135)(12 79 136)(13 80 137)(14 81 138)(15 82 93)(16 83 94)(17 84 95)(18 85 96)(19 86 97)(20 87 98)(21 88 99)(22 89 100)(23 90 101)(24 91 102)(25 92 103)(26 47 104)(27 48 105)(28 49 106)(29 50 107)(30 51 108)(31 52 109)(32 53 110)(33 54 111)(34 55 112)(35 56 113)(36 57 114)(37 58 115)(38 59 116)(39 60 117)(40 61 118)(41 62 119)(42 63 120)(43 64 121)(44 65 122)(45 66 123)(46 67 124)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 121)(88 122)(89 123)(90 124)(91 125)(92 126)
G:=sub<Sym(138)| (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), (1,68,125)(2,69,126)(3,70,127)(4,71,128)(5,72,129)(6,73,130)(7,74,131)(8,75,132)(9,76,133)(10,77,134)(11,78,135)(12,79,136)(13,80,137)(14,81,138)(15,82,93)(16,83,94)(17,84,95)(18,85,96)(19,86,97)(20,87,98)(21,88,99)(22,89,100)(23,90,101)(24,91,102)(25,92,103)(26,47,104)(27,48,105)(28,49,106)(29,50,107)(30,51,108)(31,52,109)(32,53,110)(33,54,111)(34,55,112)(35,56,113)(36,57,114)(37,58,115)(38,59,116)(39,60,117)(40,61,118)(41,62,119)(42,63,120)(43,64,121)(44,65,122)(45,66,123)(46,67,124), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)>;
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), (1,68,125)(2,69,126)(3,70,127)(4,71,128)(5,72,129)(6,73,130)(7,74,131)(8,75,132)(9,76,133)(10,77,134)(11,78,135)(12,79,136)(13,80,137)(14,81,138)(15,82,93)(16,83,94)(17,84,95)(18,85,96)(19,86,97)(20,87,98)(21,88,99)(22,89,100)(23,90,101)(24,91,102)(25,92,103)(26,47,104)(27,48,105)(28,49,106)(29,50,107)(30,51,108)(31,52,109)(32,53,110)(33,54,111)(34,55,112)(35,56,113)(36,57,114)(37,58,115)(38,59,116)(39,60,117)(40,61,118)(41,62,119)(42,63,120)(43,64,121)(44,65,122)(45,66,123)(46,67,124), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126) );
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)], [(1,68,125),(2,69,126),(3,70,127),(4,71,128),(5,72,129),(6,73,130),(7,74,131),(8,75,132),(9,76,133),(10,77,134),(11,78,135),(12,79,136),(13,80,137),(14,81,138),(15,82,93),(16,83,94),(17,84,95),(18,85,96),(19,86,97),(20,87,98),(21,88,99),(22,89,100),(23,90,101),(24,91,102),(25,92,103),(26,47,104),(27,48,105),(28,49,106),(29,50,107),(30,51,108),(31,52,109),(32,53,110),(33,54,111),(34,55,112),(35,56,113),(36,57,114),(37,58,115),(38,59,116),(39,60,117),(40,61,118),(41,62,119),(42,63,120),(43,64,121),(44,65,122),(45,66,123),(46,67,124)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,93),(60,94),(61,95),(62,96),(63,97),(64,98),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,113),(80,114),(81,115),(82,116),(83,117),(84,118),(85,119),(86,120),(87,121),(88,122),(89,123),(90,124),(91,125),(92,126)])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 23A | ··· | 23V | 46A | ··· | 46V | 46W | ··· | 46BN | 69A | ··· | 69V | 138A | ··· | 138V |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 69 | ··· | 69 | 138 | ··· | 138 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
138 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C23 | C46 | C46 | S3 | D6 | S3×C23 | S3×C46 |
| kernel | S3×C46 | S3×C23 | C138 | D6 | S3 | C6 | C46 | C23 | C2 | C1 |
| # reps | 1 | 2 | 1 | 22 | 44 | 22 | 1 | 1 | 22 | 22 |
Matrix representation of S3×C46 ►in GL2(𝔽47) generated by
| 43 | 0 |
| 0 | 43 |
| 0 | 12 |
| 43 | 46 |
| 1 | 12 |
| 0 | 46 |
G:=sub<GL(2,GF(47))| [43,0,0,43],[0,43,12,46],[1,0,12,46] >;
S3×C46 in GAP, Magma, Sage, TeX
S_3\times C_{46} % in TeX
G:=Group("S3xC46"); // GroupNames label
G:=SmallGroup(276,8);
// by ID
G=gap.SmallGroup(276,8);
# by ID
G:=PCGroup([4,-2,-2,-23,-3,2947]);
// Polycyclic
G:=Group<a,b,c|a^46=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export