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## G = C3×D57order 342 = 2·32·19

### Direct product of C3 and D57

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

 Derived series C1 — C57 — C3×D57
 Chief series C1 — C19 — C57 — C3×C57 — C3×D57
 Lower central C57 — C3×D57
 Upper central C1 — C3

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

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