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G = C3×Dic6order 72 = 23·32

Direct product of C3 and Dic6

direct product, metacyclic, supersoluble, monomial

Aliases: C3×Dic6, C12.1C6, C323Q8, C12.7S3, C6.17D6, Dic3.C6, C3⋊(C3×Q8), C4.(C3×S3), C2.3(S3×C6), C6.1(C2×C6), (C3×C12).2C2, (C3×C6).6C22, (C3×Dic3).2C2, SmallGroup(72,26)

Series: Derived Chief Lower central Upper central

C1C6 — C3×Dic6
C1C3C6C3×C6C3×Dic3 — C3×Dic6
C3C6 — C3×Dic6
C1C6C12

Generators and relations for C3×Dic6
 G = < a,b,c | a3=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

2C3
3C4
3C4
2C6
3Q8
2C12
3C12
3C12
3C3×Q8

Character table of C3×Dic6

 class 123A3B3C3D3E4A4B4C6A6B6C6D6E12A12B12C12D12E12F12G12H12I12J12K12L
 size 111122226611222222222226666
ρ1111111111111111111111111111    trivial
ρ211111111-1-11111111111111-1-1-1-1    linear of order 2
ρ31111111-11-111111-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ41111111-1-1111111-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ511ζ3ζ32ζ31ζ321-1-1ζ32ζ3ζ31ζ321ζ3ζ3ζ32ζ32ζ321ζ3ζ6ζ65ζ65ζ6    linear of order 6
ρ611ζ3ζ32ζ31ζ32-1-11ζ32ζ3ζ31ζ32-1ζ65ζ65ζ6ζ6ζ6-1ζ65ζ32ζ3ζ65ζ6    linear of order 6
ρ711ζ32ζ3ζ321ζ3-1-11ζ3ζ32ζ321ζ3-1ζ6ζ6ζ65ζ65ζ65-1ζ6ζ3ζ32ζ6ζ65    linear of order 6
ρ811ζ32ζ3ζ321ζ31-1-1ζ3ζ32ζ321ζ31ζ32ζ32ζ3ζ3ζ31ζ32ζ65ζ6ζ6ζ65    linear of order 6
ρ911ζ3ζ32ζ31ζ32-11-1ζ32ζ3ζ31ζ32-1ζ65ζ65ζ6ζ6ζ6-1ζ65ζ6ζ65ζ3ζ32    linear of order 6
ρ1011ζ32ζ3ζ321ζ3111ζ3ζ32ζ321ζ31ζ32ζ32ζ3ζ3ζ31ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ1111ζ3ζ32ζ31ζ32111ζ32ζ3ζ31ζ321ζ3ζ3ζ32ζ32ζ321ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ1211ζ32ζ3ζ321ζ3-11-1ζ3ζ32ζ321ζ3-1ζ6ζ6ζ65ζ65ζ65-1ζ6ζ65ζ6ζ32ζ3    linear of order 6
ρ132222-1-1-1-20022-1-1-1111-2111-20000    orthogonal lifted from D6
ρ142222-1-1-120022-1-1-1-1-1-12-1-1-120000    orthogonal lifted from S3
ρ152-222222000-2-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ162-222-1-1-1000-2-211133-303-3-300000    symplectic lifted from Dic6, Schur index 2
ρ172-222-1-1-1000-2-2111-3-330-33300000    symplectic lifted from Dic6, Schur index 2
ρ182-2-1+-3-1--3-1+-32-1--30001+-31--31--3-21+-3000000000000    complex lifted from C3×Q8
ρ1922-1+-3-1--3ζ65-1ζ6200-1--3-1+-3ζ65-1ζ6-1ζ65ζ65-1--3ζ6ζ6-1-1+-30000    complex lifted from C3×S3
ρ2022-1+-3-1--3ζ65-1ζ6-200-1--3-1+-3ζ65-1ζ61ζ3ζ31+-3ζ32ζ3211--30000    complex lifted from S3×C6
ρ212-2-1--3-1+-3-1--32-1+-30001--31+-31+-3-21--3000000000000    complex lifted from C3×Q8
ρ222-2-1--3-1+-3ζ6-1ζ650001--31+-3ζ321ζ3-3ζ4ζ32+2ζ4ζ43ζ32+2ζ430ζ43ζ3+2ζ43ζ4ζ3+2ζ4300000    complex faithful
ρ2322-1--3-1+-3ζ6-1ζ65200-1+-3-1--3ζ6-1ζ65-1ζ6ζ6-1+-3ζ65ζ65-1-1--30000    complex lifted from C3×S3
ρ242-2-1--3-1+-3ζ6-1ζ650001--31+-3ζ321ζ33ζ43ζ32+2ζ43ζ4ζ32+2ζ40ζ4ζ3+2ζ4ζ43ζ3+2ζ43-300000    complex faithful
ρ2522-1--3-1+-3ζ6-1ζ65-200-1+-3-1--3ζ6-1ζ651ζ32ζ321--3ζ3ζ311+-30000    complex lifted from S3×C6
ρ262-2-1+-3-1--3ζ65-1ζ60001+-31--3ζ31ζ323ζ4ζ3+2ζ4ζ43ζ3+2ζ430ζ43ζ32+2ζ43ζ4ζ32+2ζ4-300000    complex faithful
ρ272-2-1+-3-1--3ζ65-1ζ60001+-31--3ζ31ζ32-3ζ43ζ3+2ζ43ζ4ζ3+2ζ40ζ4ζ32+2ζ4ζ43ζ32+2ζ43300000    complex faithful

Permutation representations of C3×Dic6
On 24 points - transitive group 24T64
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 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 22 7 16)(2 21 8 15)(3 20 9 14)(4 19 10 13)(5 18 11 24)(6 17 12 23)

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

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

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

G:=TransitiveGroup(24,64);

C3×Dic6 is a maximal subgroup of
Dic6⋊S3  C325SD16  C322Q16  C323Q16  D12⋊S3  Dic3.D6  D6.6D6  C3×S3×Q8  He33Q8  C36.C6  He34Q8  Dic6.A4
C3×Dic6 is a maximal quotient of
He33Q8  C36.C6

Matrix representation of C3×Dic6 in GL2(𝔽13) generated by

30
03
,
70
02
,
01
120
G:=sub<GL(2,GF(13))| [3,0,0,3],[7,0,0,2],[0,12,1,0] >;

C3×Dic6 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_6
% in TeX

G:=Group("C3xDic6");
// GroupNames label

G:=SmallGroup(72,26);
// by ID

G=gap.SmallGroup(72,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-3,60,141,66,1204]);
// Polycyclic

G:=Group<a,b,c|a^3=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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Subgroup lattice of C3×Dic6 in TeX
Character table of C3×Dic6 in TeX

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