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G = S3×C38order 228 = 22·3·19

Direct product of C38 and S3

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C38, C6⋊C38, C1143C2, C574C22, C3⋊(C2×C38), SmallGroup(228,13)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C38
C1C3C57S3×C19 — S3×C38
C3 — S3×C38
C1C38

Generators and relations for S3×C38
 G = < a,b,c | a38=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C38
3C38
3C2×C38

Smallest permutation representation of S3×C38
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 65 104)(2 66 105)(3 67 106)(4 68 107)(5 69 108)(6 70 109)(7 71 110)(8 72 111)(9 73 112)(10 74 113)(11 75 114)(12 76 77)(13 39 78)(14 40 79)(15 41 80)(16 42 81)(17 43 82)(18 44 83)(19 45 84)(20 46 85)(21 47 86)(22 48 87)(23 49 88)(24 50 89)(25 51 90)(26 52 91)(27 53 92)(28 54 93)(29 55 94)(30 56 95)(31 57 96)(32 58 97)(33 59 98)(34 60 99)(35 61 100)(36 62 101)(37 63 102)(38 64 103)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 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), (1,65,104)(2,66,105)(3,67,106)(4,68,107)(5,69,108)(6,70,109)(7,71,110)(8,72,111)(9,73,112)(10,74,113)(11,75,114)(12,76,77)(13,39,78)(14,40,79)(15,41,80)(16,42,81)(17,43,82)(18,44,83)(19,45,84)(20,46,85)(21,47,86)(22,48,87)(23,49,88)(24,50,89)(25,51,90)(26,52,91)(27,53,92)(28,54,93)(29,55,94)(30,56,95)(31,57,96)(32,58,97)(33,59,98)(34,60,99)(35,61,100)(36,62,101)(37,63,102)(38,64,103), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)>;

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

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

S3×C38 is a maximal subgroup of   C57⋊D4  C19⋊D12

114 conjugacy classes

class 1 2A2B2C 3  6 19A···19R38A···38R38S···38BB57A···57R114A···114R
order12223619···1938···3838···3857···57114···114
size1133221···11···13···32···22···2

114 irreducible representations

dim1111112222
type+++++
imageC1C2C2C19C38C38S3D6S3×C19S3×C38
kernelS3×C38S3×C19C114D6S3C6C38C19C2C1
# reps121183618111818

Matrix representation of S3×C38 in GL2(𝔽229) generated by

40
04
,
0228
1228
,
2281
01
G:=sub<GL(2,GF(229))| [4,0,0,4],[0,1,228,228],[228,0,1,1] >;

S3×C38 in GAP, Magma, Sage, TeX

S_3\times C_{38}
% in TeX

G:=Group("S3xC38");
// GroupNames label

G:=SmallGroup(228,13);
// by ID

G=gap.SmallGroup(228,13);
# by ID

G:=PCGroup([4,-2,-2,-19,-3,2435]);
// Polycyclic

G:=Group<a,b,c|a^38=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×C38 in TeX

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