direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C69, C3⋊C138, C69⋊3C6, C32⋊1C46, (C3×C69)⋊4C2, SmallGroup(414,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C69 |
Generators and relations for S3×C69
G = < a,b,c | a69=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C69
►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 47 24)(2 48 25)(3 49 26)(4 50 27)(5 51 28)(6 52 29)(7 53 30)(8 54 31)(9 55 32)(10 56 33)(11 57 34)(12 58 35)(13 59 36)(14 60 37)(15 61 38)(16 62 39)(17 63 40)(18 64 41)(19 65 42)(20 66 43)(21 67 44)(22 68 45)(23 69 46)(70 93 116)(71 94 117)(72 95 118)(73 96 119)(74 97 120)(75 98 121)(76 99 122)(77 100 123)(78 101 124)(79 102 125)(80 103 126)(81 104 127)(82 105 128)(83 106 129)(84 107 130)(85 108 131)(86 109 132)(87 110 133)(88 111 134)(89 112 135)(90 113 136)(91 114 137)(92 115 138)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)
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), (1,47,24)(2,48,25)(3,49,26)(4,50,27)(5,51,28)(6,52,29)(7,53,30)(8,54,31)(9,55,32)(10,56,33)(11,57,34)(12,58,35)(13,59,36)(14,60,37)(15,61,38)(16,62,39)(17,63,40)(18,64,41)(19,65,42)(20,66,43)(21,67,44)(22,68,45)(23,69,46)(70,93,116)(71,94,117)(72,95,118)(73,96,119)(74,97,120)(75,98,121)(76,99,122)(77,100,123)(78,101,124)(79,102,125)(80,103,126)(81,104,127)(82,105,128)(83,106,129)(84,107,130)(85,108,131)(86,109,132)(87,110,133)(88,111,134)(89,112,135)(90,113,136)(91,114,137)(92,115,138), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)>;
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), (1,47,24)(2,48,25)(3,49,26)(4,50,27)(5,51,28)(6,52,29)(7,53,30)(8,54,31)(9,55,32)(10,56,33)(11,57,34)(12,58,35)(13,59,36)(14,60,37)(15,61,38)(16,62,39)(17,63,40)(18,64,41)(19,65,42)(20,66,43)(21,67,44)(22,68,45)(23,69,46)(70,93,116)(71,94,117)(72,95,118)(73,96,119)(74,97,120)(75,98,121)(76,99,122)(77,100,123)(78,101,124)(79,102,125)(80,103,126)(81,104,127)(82,105,128)(83,106,129)(84,107,130)(85,108,131)(86,109,132)(87,110,133)(88,111,134)(89,112,135)(90,113,136)(91,114,137)(92,115,138), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84) );
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)], [(1,47,24),(2,48,25),(3,49,26),(4,50,27),(5,51,28),(6,52,29),(7,53,30),(8,54,31),(9,55,32),(10,56,33),(11,57,34),(12,58,35),(13,59,36),(14,60,37),(15,61,38),(16,62,39),(17,63,40),(18,64,41),(19,65,42),(20,66,43),(21,67,44),(22,68,45),(23,69,46),(70,93,116),(71,94,117),(72,95,118),(73,96,119),(74,97,120),(75,98,121),(76,99,122),(77,100,123),(78,101,124),(79,102,125),(80,103,126),(81,104,127),(82,105,128),(83,106,129),(84,107,130),(85,108,131),(86,109,132),(87,110,133),(88,111,134),(89,112,135),(90,113,136),(91,114,137),(92,115,138)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,121),(38,122),(39,123),(40,124),(41,125),(42,126),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84)]])
207 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 23A | ··· | 23V | 46A | ··· | 46V | 69A | ··· | 69AR | 69AS | ··· | 69DF | 138A | ··· | 138AR |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 69 | ··· | 69 | 138 | ··· | 138 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
207 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C23 | C46 | C69 | C138 | S3 | C3×S3 | S3×C23 | S3×C69 |
| kernel | S3×C69 | C3×C69 | S3×C23 | C69 | C3×S3 | C32 | S3 | C3 | C69 | C23 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 22 | 22 | 44 | 44 | 1 | 2 | 22 | 44 |
Matrix representation of S3×C69 ►in GL2(𝔽139) generated by
| 54 | 0 |
| 0 | 54 |
| 96 | 59 |
| 0 | 42 |
| 115 | 111 |
| 95 | 24 |
G:=sub<GL(2,GF(139))| [54,0,0,54],[96,0,59,42],[115,95,111,24] >;
S3×C69 in GAP, Magma, Sage, TeX
S_3\times C_{69} % in TeX
G:=Group("S3xC69"); // GroupNames label
G:=SmallGroup(414,6);
// by ID
G=gap.SmallGroup(414,6);
# by ID
G:=PCGroup([4,-2,-3,-23,-3,4419]);
// Polycyclic
G:=Group<a,b,c|a^69=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export