direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C57, C3⋊C114, C57⋊7C6, C32⋊1C38, (C3×C57)⋊4C2, SmallGroup(342,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C57 |
Generators and relations for S3×C57
G = < a,b,c | a57=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C57
►On 114 points
Generators in S114
(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)
(1 20 39)(2 21 40)(3 22 41)(4 23 42)(5 24 43)(6 25 44)(7 26 45)(8 27 46)(9 28 47)(10 29 48)(11 30 49)(12 31 50)(13 32 51)(14 33 52)(15 34 53)(16 35 54)(17 36 55)(18 37 56)(19 38 57)(58 96 77)(59 97 78)(60 98 79)(61 99 80)(62 100 81)(63 101 82)(64 102 83)(65 103 84)(66 104 85)(67 105 86)(68 106 87)(69 107 88)(70 108 89)(71 109 90)(72 110 91)(73 111 92)(74 112 93)(75 113 94)(76 114 95)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(57 105)
G:=sub<Sym(114)| (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), (1,20,39)(2,21,40)(3,22,41)(4,23,42)(5,24,43)(6,25,44)(7,26,45)(8,27,46)(9,28,47)(10,29,48)(11,30,49)(12,31,50)(13,32,51)(14,33,52)(15,34,53)(16,35,54)(17,36,55)(18,37,56)(19,38,57)(58,96,77)(59,97,78)(60,98,79)(61,99,80)(62,100,81)(63,101,82)(64,102,83)(65,103,84)(66,104,85)(67,105,86)(68,106,87)(69,107,88)(70,108,89)(71,109,90)(72,110,91)(73,111,92)(74,112,93)(75,113,94)(76,114,95), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,105)>;
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), (1,20,39)(2,21,40)(3,22,41)(4,23,42)(5,24,43)(6,25,44)(7,26,45)(8,27,46)(9,28,47)(10,29,48)(11,30,49)(12,31,50)(13,32,51)(14,33,52)(15,34,53)(16,35,54)(17,36,55)(18,37,56)(19,38,57)(58,96,77)(59,97,78)(60,98,79)(61,99,80)(62,100,81)(63,101,82)(64,102,83)(65,103,84)(66,104,85)(67,105,86)(68,106,87)(69,107,88)(70,108,89)(71,109,90)(72,110,91)(73,111,92)(74,112,93)(75,113,94)(76,114,95), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,105) );
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)], [(1,20,39),(2,21,40),(3,22,41),(4,23,42),(5,24,43),(6,25,44),(7,26,45),(8,27,46),(9,28,47),(10,29,48),(11,30,49),(12,31,50),(13,32,51),(14,33,52),(15,34,53),(16,35,54),(17,36,55),(18,37,56),(19,38,57),(58,96,77),(59,97,78),(60,98,79),(61,99,80),(62,100,81),(63,101,82),(64,102,83),(65,103,84),(66,104,85),(67,105,86),(68,106,87),(69,107,88),(70,108,89),(71,109,90),(72,110,91),(73,111,92),(74,112,93),(75,113,94),(76,114,95)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104),(57,105)]])
171 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 19A | ··· | 19R | 38A | ··· | 38R | 57A | ··· | 57AJ | 57AK | ··· | 57CL | 114A | ··· | 114AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
171 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C19 | C38 | C57 | C114 | S3 | C3×S3 | S3×C19 | S3×C57 |
| kernel | S3×C57 | C3×C57 | S3×C19 | C57 | C3×S3 | C32 | S3 | C3 | C57 | C19 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 18 | 18 | 36 | 36 | 1 | 2 | 18 | 36 |
Matrix representation of S3×C57 ►in GL2(𝔽229) generated by
| 9 | 0 |
| 0 | 9 |
| 94 | 0 |
| 108 | 134 |
| 153 | 133 |
| 53 | 76 |
G:=sub<GL(2,GF(229))| [9,0,0,9],[94,108,0,134],[153,53,133,76] >;
S3×C57 in GAP, Magma, Sage, TeX
S_3\times C_{57} % in TeX
G:=Group("S3xC57"); // GroupNames label
G:=SmallGroup(342,14);
// by ID
G=gap.SmallGroup(342,14);
# by ID
G:=PCGroup([4,-2,-3,-19,-3,3651]);
// Polycyclic
G:=Group<a,b,c|a^57=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export