Copied to
clipboard

G = S3xC38order 228 = 22·3·19

Direct product of C38 and S3

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

Aliases: S3xC38, C6:C38, C114:3C2, C57:4C22, C3:(C2xC38), SmallGroup(228,13)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC38
C1C3C57S3xC19 — S3xC38
C3 — S3xC38
C1C38

Generators and relations for S3xC38
 G = < a,b,c | a38=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 32 in 20 conjugacy classes, 14 normal (10 characteristic)
Quotients: C1, C2, C22, S3, D6, C19, C38, C2xC38, S3xC19, S3xC38
3C2
3C2
3C22
3C38
3C38
3C2xC38

Smallest permutation representation of S3xC38
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 99 50)(2 100 51)(3 101 52)(4 102 53)(5 103 54)(6 104 55)(7 105 56)(8 106 57)(9 107 58)(10 108 59)(11 109 60)(12 110 61)(13 111 62)(14 112 63)(15 113 64)(16 114 65)(17 77 66)(18 78 67)(19 79 68)(20 80 69)(21 81 70)(22 82 71)(23 83 72)(24 84 73)(25 85 74)(26 86 75)(27 87 76)(28 88 39)(29 89 40)(30 90 41)(31 91 42)(32 92 43)(33 93 44)(34 94 45)(35 95 46)(36 96 47)(37 97 48)(38 98 49)
(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)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)

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

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

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

S3xC38 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+++++
imageC1C2C2C19C38C38S3D6S3xC19S3xC38
kernelS3xC38S3xC19C114D6S3C6C38C19C2C1
# reps121183618111818

Matrix representation of S3xC38 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] >;

S3xC38 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 S3xC38 in TeX

׿
x
:
Z
F
o
wr
Q
<