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G = C6×Dic3order 72 = 23·32

Direct product of C6 and Dic3

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

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

C1C3 — C6×Dic3
C1C3C6C3×C6C3×Dic3 — C6×Dic3
C3 — C6×Dic3
C1C2×C6

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

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

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 2A2B2C3A3B3C3D3E4A4B4C4D6A···6F6G···6O12A···12H
order12223333344446···66···612···12
size11111122233331···12···23···3

36 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6C3×S3C3×Dic3S3×C6
kernelC6×Dic3C3×Dic3C62C2×Dic3C3×C6Dic3C2×C6C6C2×C6C6C6C22C2C2
# reps12124428121242

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

4000
0400
0030
0003
,
12000
0100
0030
0009
,
8000
0100
0001
0010
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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Subgroup lattice of C6×Dic3 in TeX

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