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)
(1 130 62)(2 131 63)(3 132 64)(4 133 65)(5 134 66)(6 135 67)(7 136 68)(8 137 69)(9 138 70)(10 93 71)(11 94 72)(12 95 73)(13 96 74)(14 97 75)(15 98 76)(16 99 77)(17 100 78)(18 101 79)(19 102 80)(20 103 81)(21 104 82)(22 105 83)(23 106 84)(24 107 85)(25 108 86)(26 109 87)(27 110 88)(28 111 89)(29 112 90)(30 113 91)(31 114 92)(32 115 47)(33 116 48)(34 117 49)(35 118 50)(36 119 51)(37 120 52)(38 121 53)(39 122 54)(40 123 55)(41 124 56)(42 125 57)(43 126 58)(44 127 59)(45 128 60)(46 129 61)
(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)(47 138)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)
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), (1,130,62)(2,131,63)(3,132,64)(4,133,65)(5,134,66)(6,135,67)(7,136,68)(8,137,69)(9,138,70)(10,93,71)(11,94,72)(12,95,73)(13,96,74)(14,97,75)(15,98,76)(16,99,77)(17,100,78)(18,101,79)(19,102,80)(20,103,81)(21,104,82)(22,105,83)(23,106,84)(24,107,85)(25,108,86)(26,109,87)(27,110,88)(28,111,89)(29,112,90)(30,113,91)(31,114,92)(32,115,47)(33,116,48)(34,117,49)(35,118,50)(36,119,51)(37,120,52)(38,121,53)(39,122,54)(40,123,55)(41,124,56)(42,125,57)(43,126,58)(44,127,59)(45,128,60)(46,129,61), (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)(47,138)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)>;
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), (1,130,62)(2,131,63)(3,132,64)(4,133,65)(5,134,66)(6,135,67)(7,136,68)(8,137,69)(9,138,70)(10,93,71)(11,94,72)(12,95,73)(13,96,74)(14,97,75)(15,98,76)(16,99,77)(17,100,78)(18,101,79)(19,102,80)(20,103,81)(21,104,82)(22,105,83)(23,106,84)(24,107,85)(25,108,86)(26,109,87)(27,110,88)(28,111,89)(29,112,90)(30,113,91)(31,114,92)(32,115,47)(33,116,48)(34,117,49)(35,118,50)(36,119,51)(37,120,52)(38,121,53)(39,122,54)(40,123,55)(41,124,56)(42,125,57)(43,126,58)(44,127,59)(45,128,60)(46,129,61), (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)(47,138)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137) );
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)], [(1,130,62),(2,131,63),(3,132,64),(4,133,65),(5,134,66),(6,135,67),(7,136,68),(8,137,69),(9,138,70),(10,93,71),(11,94,72),(12,95,73),(13,96,74),(14,97,75),(15,98,76),(16,99,77),(17,100,78),(18,101,79),(19,102,80),(20,103,81),(21,104,82),(22,105,83),(23,106,84),(24,107,85),(25,108,86),(26,109,87),(27,110,88),(28,111,89),(29,112,90),(30,113,91),(31,114,92),(32,115,47),(33,116,48),(34,117,49),(35,118,50),(36,119,51),(37,120,52),(38,121,53),(39,122,54),(40,123,55),(41,124,56),(42,125,57),(43,126,58),(44,127,59),(45,128,60),(46,129,61)], [(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),(47,138),(48,93),(49,94),(50,95),(51,96),(52,97),(53,98),(54,99),(55,100),(56,101),(57,102),(58,103),(59,104),(60,105),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,127),(83,128),(84,129),(85,130),(86,131),(87,132),(88,133),(89,134),(90,135),(91,136),(92,137)]])
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