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## G = S3×C12order 72 = 23·32

### Direct product of C12 and S3

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

 Derived series C1 — C3 — S3×C12
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C12
 Lower central C3 — S3×C12
 Upper central C1 — C12

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

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 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 3 3 1 1 2 2 2 1 1 3 3 1 1 2 2 2 3 3 3 3 1 1 1 1 2 ··· 2 3 3 3 3

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D6 C3×S3 C4×S3 S3×C6 S3×C12 kernel S3×C12 C3×Dic3 C3×C12 S3×C6 C4×S3 C3×S3 Dic3 C12 D6 S3 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 4

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

 11 0 0 11
,
 3 0 0 9
,
 0 1 1 0
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

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