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G = S3×C57order 342 = 2·32·19

Direct product of C57 and S3

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

Aliases: S3×C57, C3⋊C114, C577C6, C321C38, (C3×C57)⋊4C2, SmallGroup(342,14)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C57
C1C3C57C3×C57 — S3×C57
C3 — S3×C57
C1C57

Generators and relations for S3×C57
 G = < a,b,c | a57=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
2C3
3C6
3C38
2C57
3C114

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 3A3B3C3D3E6A6B19A···19R38A···38R57A···57AJ57AK···57CL114A···114AJ
order12333336619···1938···3857···5757···57114···114
size1311222331···13···31···12···23···3

171 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C19C38C57C114S3C3×S3S3×C19S3×C57
kernelS3×C57C3×C57S3×C19C57C3×S3C32S3C3C57C19C3C1
# reps112218183636121836

Matrix representation of S3×C57 in GL2(𝔽229) generated by

90
09
,
940
108134
,
153133
5376
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

Subgroup lattice of S3×C57 in TeX

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