Copied to
clipboard

G = S3×C12order 72 = 23·32

Direct product of C12 and S3

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

Aliases: S3×C12, D6.C6, C122C6, Dic3C12, C6.18D6, Dic32C6, (C3×C12)⋊3C2, C31(C2×C12), C2.1(S3×C6), C6.2(C2×C6), C324(C2×C4), C4(C3×Dic3), (S3×C6).2C2, C12(C3×Dic3), (C3×Dic3)⋊5C2, (C3×C6).7C22, SmallGroup(72,27)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C12
C1C3C6C3×C6S3×C6 — S3×C12
C3 — S3×C12
C1C12

Generators and relations for S3×C12
 G = < a,b,c | a12=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
2C3
3C22
3C4
2C6
3C6
3C6
3C2×C4
2C12
3C12
3C2×C6
3C2×C12

Permutation representations of S3×C12
On 24 points - transitive group 24T65
Generators in S24
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)

G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)>;

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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21) );

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)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21)])

G:=TransitiveGroup(24,65);

S3×C12 is a maximal subgroup of   D6.Dic3  D125S3  D6.D6  D6.6D6

36 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I12A12B12C12D12E···12J12K12L12M12N
order12223333344446666666661212121212···1212121212
size113311222113311222333311112···23333

36 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6C3×S3C4×S3S3×C6S3×C12
kernelS3×C12C3×Dic3C3×C12S3×C6C4×S3C3×S3Dic3C12D6S3C12C6C4C3C2C1
# reps1111242228112224

Matrix representation of S3×C12 in GL2(𝔽13) generated by

110
011
,
30
09
,
01
10
G:=sub<GL(2,GF(13))| [11,0,0,11],[3,0,0,9],[0,1,1,0] >;

S3×C12 in GAP, Magma, Sage, TeX

S_3\times C_{12}
% in TeX

G:=Group("S3xC12");
// GroupNames label

G:=SmallGroup(72,27);
// by ID

G=gap.SmallGroup(72,27);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-3,66,1204]);
// Polycyclic

G:=Group<a,b,c|a^12=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×C12 in TeX

׿
×
𝔽