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G = S3×C21order 126 = 2·32·7

Direct product of C21 and S3

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

Aliases: S3×C21, C3⋊C42, C217C6, C321C14, (C3×C21)⋊4C2, SmallGroup(126,12)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C21
C1C3C21C3×C21 — S3×C21
C3 — S3×C21
C1C21

Generators and relations for S3×C21
 G = < a,b,c | a21=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
2C3
3C6
3C14
2C21
3C42

Smallest permutation representation of S3×C21
On 42 points
Generators in S42
(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)
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)

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

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

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

63 conjugacy classes

class 1  2 3A3B3C3D3E6A6B7A···7F14A···14F21A···21L21M···21AD42A···42L
order1233333667···714···1421···2121···2142···42
size1311222331···13···31···12···23···3

63 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C7C14C21C42S3C3×S3S3×C7S3×C21
kernelS3×C21C3×C21S3×C7C21C3×S3C32S3C3C21C7C3C1
# reps112266121212612

Matrix representation of S3×C21 in GL2(𝔽43) generated by

400
040
,
360
06
,
011
40
G:=sub<GL(2,GF(43))| [40,0,0,40],[36,0,0,6],[0,4,11,0] >;

S3×C21 in GAP, Magma, Sage, TeX

S_3\times C_{21}
% in TeX

G:=Group("S3xC21");
// GroupNames label

G:=SmallGroup(126,12);
// by ID

G=gap.SmallGroup(126,12);
# by ID

G:=PCGroup([4,-2,-3,-7,-3,1347]);
// Polycyclic

G:=Group<a,b,c|a^21=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×C21 in TeX

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