direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C41, C3⋊C82, C123⋊3C2, SmallGroup(246,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C41 |
Generators and relations for S3×C41
G = < a,b,c | a41=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C41
►On 123 points
Generators in S123
(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)
(1 81 90)(2 82 91)(3 42 92)(4 43 93)(5 44 94)(6 45 95)(7 46 96)(8 47 97)(9 48 98)(10 49 99)(11 50 100)(12 51 101)(13 52 102)(14 53 103)(15 54 104)(16 55 105)(17 56 106)(18 57 107)(19 58 108)(20 59 109)(21 60 110)(22 61 111)(23 62 112)(24 63 113)(25 64 114)(26 65 115)(27 66 116)(28 67 117)(29 68 118)(30 69 119)(31 70 120)(32 71 121)(33 72 122)(34 73 123)(35 74 83)(36 75 84)(37 76 85)(38 77 86)(39 78 87)(40 79 88)(41 80 89)
(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)(73 123)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)
G:=sub<Sym(123)| (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), (1,81,90)(2,82,91)(3,42,92)(4,43,93)(5,44,94)(6,45,95)(7,46,96)(8,47,97)(9,48,98)(10,49,99)(11,50,100)(12,51,101)(13,52,102)(14,53,103)(15,54,104)(16,55,105)(17,56,106)(18,57,107)(19,58,108)(20,59,109)(21,60,110)(22,61,111)(23,62,112)(24,63,113)(25,64,114)(26,65,115)(27,66,116)(28,67,117)(29,68,118)(30,69,119)(31,70,120)(32,71,121)(33,72,122)(34,73,123)(35,74,83)(36,75,84)(37,76,85)(38,77,86)(39,78,87)(40,79,88)(41,80,89), (42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(73,123)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)>;
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), (1,81,90)(2,82,91)(3,42,92)(4,43,93)(5,44,94)(6,45,95)(7,46,96)(8,47,97)(9,48,98)(10,49,99)(11,50,100)(12,51,101)(13,52,102)(14,53,103)(15,54,104)(16,55,105)(17,56,106)(18,57,107)(19,58,108)(20,59,109)(21,60,110)(22,61,111)(23,62,112)(24,63,113)(25,64,114)(26,65,115)(27,66,116)(28,67,117)(29,68,118)(30,69,119)(31,70,120)(32,71,121)(33,72,122)(34,73,123)(35,74,83)(36,75,84)(37,76,85)(38,77,86)(39,78,87)(40,79,88)(41,80,89), (42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(73,123)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91) );
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)], [(1,81,90),(2,82,91),(3,42,92),(4,43,93),(5,44,94),(6,45,95),(7,46,96),(8,47,97),(9,48,98),(10,49,99),(11,50,100),(12,51,101),(13,52,102),(14,53,103),(15,54,104),(16,55,105),(17,56,106),(18,57,107),(19,58,108),(20,59,109),(21,60,110),(22,61,111),(23,62,112),(24,63,113),(25,64,114),(26,65,115),(27,66,116),(28,67,117),(29,68,118),(30,69,119),(31,70,120),(32,71,121),(33,72,122),(34,73,123),(35,74,83),(36,75,84),(37,76,85),(38,77,86),(39,78,87),(40,79,88),(41,80,89)], [(42,92),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,121),(72,122),(73,123),(74,83),(75,84),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91)]])
123 conjugacy classes
| class | 1 | 2 | 3 | 41A | ··· | 41AN | 82A | ··· | 82AN | 123A | ··· | 123AN |
| order | 1 | 2 | 3 | 41 | ··· | 41 | 82 | ··· | 82 | 123 | ··· | 123 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C41 | C82 | S3 | S3×C41 |
| kernel | S3×C41 | C123 | S3 | C3 | C41 | C1 |
| # reps | 1 | 1 | 40 | 40 | 1 | 40 |
Matrix representation of S3×C41 ►in GL3(𝔽739) generated by
| 631 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 738 | 738 |
| 0 | 1 | 0 |
| 738 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 738 | 738 |
G:=sub<GL(3,GF(739))| [631,0,0,0,1,0,0,0,1],[1,0,0,0,738,1,0,738,0],[738,0,0,0,1,738,0,0,738] >;
S3×C41 in GAP, Magma, Sage, TeX
S_3\times C_{41} % in TeX
G:=Group("S3xC41"); // GroupNames label
G:=SmallGroup(246,1);
// by ID
G=gap.SmallGroup(246,1);
# by ID
G:=PCGroup([3,-2,-41,-3,1478]);
// Polycyclic
G:=Group<a,b,c|a^41=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export