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