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G = C3×D57order 342 = 2·32·19

Direct product of C3 and D57

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×D57, C575C6, C572S3, C321D19, C3⋊(C3×D19), C193(C3×S3), (C3×C57)⋊2C2, SmallGroup(342,15)

Series: Derived Chief Lower central Upper central

C1C57 — C3×D57
C1C19C57C3×C57 — C3×D57
C57 — C3×D57
C1C3

Generators and relations for C3×D57
 G = < a,b,c | a3=b57=c2=1, ab=ba, ac=ca, cbc=b-1 >

57C2
2C3
19S3
57C6
3D19
2C57
19C3×S3
3C3×D19

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

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

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

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

90 conjugacy classes

class 1  2 3A3B3C3D3E6A6B19A···19I57A···57BT
order12333336619···1957···57
size1571122257572···22···2

90 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6S3C3×S3D19C3×D19D57C3×D57
kernelC3×D57C3×C57D57C57C57C19C32C3C3C1
# reps1122129181836

Matrix representation of C3×D57 in GL2(𝔽229) generated by

940
094
,
1840
0173
,
0173
1840
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

Subgroup lattice of C3×D57 in TeX

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