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## G = C6×Dic3order 72 = 23·32

### Direct product of C6 and Dic3

Aliases: C6×Dic3, C6⋊C12, C6.20D6, C62.1C2, (C3×C6)⋊2C4, C32(C2×C12), C2.2(S3×C6), C6.4(C2×C6), (C2×C6).6S3, (C2×C6).3C6, C326(C2×C4), C22.(C3×S3), (C3×C6).9C22, SmallGroup(72,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C6×Dic3
 Chief series C1 — C3 — C6 — C3×C6 — C3×Dic3 — C6×Dic3
 Lower central C3 — C6×Dic3
 Upper central C1 — C2×C6

Generators and relations for C6×Dic3
G = < a,b,c | a6=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Permutation representations of C6×Dic3
On 24 points - transitive group 24T66
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 14 3 16 5 18)(2 15 4 17 6 13)(7 21 11 19 9 23)(8 22 12 20 10 24)
(1 11 16 23)(2 12 17 24)(3 7 18 19)(4 8 13 20)(5 9 14 21)(6 10 15 22)

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

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

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

G:=TransitiveGroup(24,66);

C6×Dic3 is a maximal subgroup of   D6⋊Dic3  C6.D12  Dic3⋊Dic3  C62.C22  D6.3D6  S3×C2×C12

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 12A ··· 12H order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 1 1 2 2 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 kernel C6×Dic3 C3×Dic3 C62 C2×Dic3 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 2 1 2 4 2

Matrix representation of C6×Dic3 in GL4(𝔽13) generated by

 4 0 0 0 0 4 0 0 0 0 3 0 0 0 0 3
,
 12 0 0 0 0 1 0 0 0 0 3 0 0 0 0 9
,
 8 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[8,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C6×Dic3 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_3
% in TeX

G:=Group("C6xDic3");
// GroupNames label

G:=SmallGroup(72,29);
// by ID

G=gap.SmallGroup(72,29);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-3,60,1204]);
// Polycyclic

G:=Group<a,b,c|a^6=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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