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

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

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

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

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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