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G = S3×C17order 102 = 2·3·17

Direct product of C17 and S3

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

Aliases: S3×C17, C3⋊C34, C513C2, SmallGroup(102,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C17
C1C3C51 — S3×C17
C3 — S3×C17
C1C17

Generators and relations for S3×C17
 G = < a,b,c | a17=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C34

Smallest permutation representation of S3×C17
On 51 points
Generators in S51
(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)
(1 23 47)(2 24 48)(3 25 49)(4 26 50)(5 27 51)(6 28 35)(7 29 36)(8 30 37)(9 31 38)(10 32 39)(11 33 40)(12 34 41)(13 18 42)(14 19 43)(15 20 44)(16 21 45)(17 22 46)
(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 35)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)

G:=sub<Sym(51)| (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), (1,23,47)(2,24,48)(3,25,49)(4,26,50)(5,27,51)(6,28,35)(7,29,36)(8,30,37)(9,31,38)(10,32,39)(11,33,40)(12,34,41)(13,18,42)(14,19,43)(15,20,44)(16,21,45)(17,22,46), (18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)>;

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), (1,23,47)(2,24,48)(3,25,49)(4,26,50)(5,27,51)(6,28,35)(7,29,36)(8,30,37)(9,31,38)(10,32,39)(11,33,40)(12,34,41)(13,18,42)(14,19,43)(15,20,44)(16,21,45)(17,22,46), (18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41) );

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)], [(1,23,47),(2,24,48),(3,25,49),(4,26,50),(5,27,51),(6,28,35),(7,29,36),(8,30,37),(9,31,38),(10,32,39),(11,33,40),(12,34,41),(13,18,42),(14,19,43),(15,20,44),(16,21,45),(17,22,46)], [(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,35),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41)])

51 conjugacy classes

class 1  2  3 17A···17P34A···34P51A···51P
order12317···1734···3451···51
size1321···13···32···2

51 irreducible representations

dim111122
type+++
imageC1C2C17C34S3S3×C17
kernelS3×C17C51S3C3C17C1
# reps111616116

Matrix representation of S3×C17 in GL2(𝔽103) generated by

340
034
,
0102
1102
,
01
10
G:=sub<GL(2,GF(103))| [34,0,0,34],[0,1,102,102],[0,1,1,0] >;

S3×C17 in GAP, Magma, Sage, TeX

S_3\times C_{17}
% in TeX

G:=Group("S3xC17");
// GroupNames label

G:=SmallGroup(102,1);
// by ID

G=gap.SmallGroup(102,1);
# by ID

G:=PCGroup([3,-2,-17,-3,614]);
// Polycyclic

G:=Group<a,b,c|a^17=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×C17 in TeX

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