direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D57, C57⋊5C6, C57⋊2S3, C32⋊1D19, C3⋊(C3×D19), C19⋊3(C3×S3), (C3×C57)⋊2C2, SmallGroup(342,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — C3×D57 |
Generators and relations for C3×D57
G = < a,b,c | a3=b57=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D57
►On 114 points
Generators in S114
(1 39 20)(2 40 21)(3 41 22)(4 42 23)(5 43 24)(6 44 25)(7 45 26)(8 46 27)(9 47 28)(10 48 29)(11 49 30)(12 50 31)(13 51 32)(14 52 33)(15 53 34)(16 54 35)(17 55 36)(18 56 37)(19 57 38)(58 77 96)(59 78 97)(60 79 98)(61 80 99)(62 81 100)(63 82 101)(64 83 102)(65 84 103)(66 85 104)(67 86 105)(68 87 106)(69 88 107)(70 89 108)(71 90 109)(72 91 110)(73 92 111)(74 93 112)(75 94 113)(76 95 114)
(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)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 100)(12 99)(13 98)(14 97)(15 96)(16 95)(17 94)(18 93)(19 92)(20 91)(21 90)(22 89)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 114)(55 113)(56 112)(57 111)
G:=sub<Sym(114)| (1,39,20)(2,40,21)(3,41,22)(4,42,23)(5,43,24)(6,44,25)(7,45,26)(8,46,27)(9,47,28)(10,48,29)(11,49,30)(12,50,31)(13,51,32)(14,52,33)(15,53,34)(16,54,35)(17,55,36)(18,56,37)(19,57,38)(58,77,96)(59,78,97)(60,79,98)(61,80,99)(62,81,100)(63,82,101)(64,83,102)(65,84,103)(66,85,104)(67,86,105)(68,87,106)(69,88,107)(70,89,108)(71,90,109)(72,91,110)(73,92,111)(74,93,112)(75,94,113)(76,95,114), (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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,114)(55,113)(56,112)(57,111)>;
G:=Group( (1,39,20)(2,40,21)(3,41,22)(4,42,23)(5,43,24)(6,44,25)(7,45,26)(8,46,27)(9,47,28)(10,48,29)(11,49,30)(12,50,31)(13,51,32)(14,52,33)(15,53,34)(16,54,35)(17,55,36)(18,56,37)(19,57,38)(58,77,96)(59,78,97)(60,79,98)(61,80,99)(62,81,100)(63,82,101)(64,83,102)(65,84,103)(66,85,104)(67,86,105)(68,87,106)(69,88,107)(70,89,108)(71,90,109)(72,91,110)(73,92,111)(74,93,112)(75,94,113)(76,95,114), (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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,114)(55,113)(56,112)(57,111) );
G=PermutationGroup([[(1,39,20),(2,40,21),(3,41,22),(4,42,23),(5,43,24),(6,44,25),(7,45,26),(8,46,27),(9,47,28),(10,48,29),(11,49,30),(12,50,31),(13,51,32),(14,52,33),(15,53,34),(16,54,35),(17,55,36),(18,56,37),(19,57,38),(58,77,96),(59,78,97),(60,79,98),(61,80,99),(62,81,100),(63,82,101),(64,83,102),(65,84,103),(66,85,104),(67,86,105),(68,87,106),(69,88,107),(70,89,108),(71,90,109),(72,91,110),(73,92,111),(74,93,112),(75,94,113),(76,95,114)], [(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)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,100),(12,99),(13,98),(14,97),(15,96),(16,95),(17,94),(18,93),(19,92),(20,91),(21,90),(22,89),(23,88),(24,87),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,114),(55,113),(56,112),(57,111)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 19A | ··· | 19I | 57A | ··· | 57BT |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 19 | ··· | 19 | 57 | ··· | 57 |
| size | 1 | 57 | 1 | 1 | 2 | 2 | 2 | 57 | 57 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | D19 | C3×D19 | D57 | C3×D57 |
| kernel | C3×D57 | C3×C57 | D57 | C57 | C57 | C19 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 9 | 18 | 18 | 36 |
Matrix representation of C3×D57 ►in GL2(𝔽229) generated by
| 94 | 0 |
| 0 | 94 |
| 184 | 0 |
| 0 | 173 |
| 0 | 173 |
| 184 | 0 |
G:=sub<GL(2,GF(229))| [94,0,0,94],[184,0,0,173],[0,184,173,0] >;
C3×D57 in GAP, Magma, Sage, TeX
C_3\times D_{57} % in TeX
G:=Group("C3xD57"); // GroupNames label
G:=SmallGroup(342,15);
// by ID
G=gap.SmallGroup(342,15);
# by ID
G:=PCGroup([4,-2,-3,-3,-19,146,5187]);
// Polycyclic
G:=Group<a,b,c|a^3=b^57=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export