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G = C3×Dic3order 36 = 22·32

Direct product of C3 and Dic3

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

Aliases: C3×Dic3, C3⋊C12, C6.C6, C6.4S3, C322C4, C2.(C3×S3), (C3×C6).1C2, SmallGroup(36,6)

Series: Derived Chief Lower central Upper central

C1C3 — C3×Dic3
C1C3C6C3×C6 — C3×Dic3
C3 — C3×Dic3
C1C6

Generators and relations for C3×Dic3
 G = < a,b,c | a3=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

2C3
3C4
2C6
3C12

Character table of C3×Dic3

 class 123A3B3C3D3E4A4B6A6B6C6D6E12A12B12C12D
 size 111122233112223333
ρ1111111111111111111    trivial
ρ21111111-1-111111-1-1-1-1    linear of order 2
ρ311ζ3ζ32ζ321ζ3-1-1ζ32ζ3ζ31ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ411ζ32ζ3ζ31ζ32-1-1ζ3ζ32ζ321ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ511ζ3ζ32ζ321ζ311ζ32ζ3ζ31ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ611ζ32ζ3ζ31ζ3211ζ3ζ32ζ321ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ71-111111i-i-1-1-1-1-1-ii-ii    linear of order 4
ρ81-111111-ii-1-1-1-1-1i-ii-i    linear of order 4
ρ91-1ζ62ζ32ζ321ζ62ζ2ζ2ζ6ζ65ζ65-1ζ6ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ101-1ζ62ζ32ζ321ζ62ζ2ζ2ζ6ζ65ζ65-1ζ6ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ111-1ζ32ζ62ζ621ζ32ζ2ζ2ζ65ζ6ζ6-1ζ65ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ121-1ζ32ζ62ζ621ζ32ζ2ζ2ζ65ζ6ζ6-1ζ65ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ132222-1-1-10022-1-1-10000    orthogonal lifted from S3
ρ142-222-1-1-100-2-21110000    symplectic lifted from Dic3, Schur index 2
ρ152-2-1+-3-1--3ζ6-1ζ65001+-31--3ζ31ζ320000    complex faithful
ρ162-2-1--3-1+-3ζ65-1ζ6001--31+-3ζ321ζ30000    complex faithful
ρ1722-1+-3-1--3ζ6-1ζ6500-1--3-1+-3ζ65-1ζ60000    complex lifted from C3×S3
ρ1822-1--3-1+-3ζ65-1ζ600-1+-3-1--3ζ6-1ζ650000    complex lifted from C3×S3

Permutation representations of C3×Dic3
On 12 points - transitive group 12T19
Generators in S12
(1 3 5)(2 4 6)(7 11 9)(8 12 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 10 4 7)(2 9 5 12)(3 8 6 11)

G:=sub<Sym(12)| (1,3,5)(2,4,6)(7,11,9)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10,4,7)(2,9,5,12)(3,8,6,11)>;

G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10,4,7)(2,9,5,12)(3,8,6,11) );

G=PermutationGroup([(1,3,5),(2,4,6),(7,11,9),(8,12,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,10,4,7),(2,9,5,12),(3,8,6,11)])

G:=TransitiveGroup(12,19);

Polynomial with Galois group C3×Dic3 over ℚ
actionf(x)Disc(f)
12T19x12-4x11-15x10+68x9+62x8-362x7-107x6+798x5+112x4-654x3-63x2+84x+1216·59·194·5412·1019992

Matrix representation of C3×Dic3 in GL2(𝔽7) generated by

20
02
,
50
03
,
06
10
G:=sub<GL(2,GF(7))| [2,0,0,2],[5,0,0,3],[0,1,6,0] >;

C3×Dic3 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3
% in TeX

G:=Group("C3xDic3");
// GroupNames label

G:=SmallGroup(36,6);
// by ID

G=gap.SmallGroup(36,6);
# by ID

G:=PCGroup([4,-2,-3,-2,-3,24,387]);
// Polycyclic

G:=Group<a,b,c|a^3=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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