Copied to
clipboard

G = S3×C6order 36 = 22·32

Direct product of C6 and S3

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

Aliases: S3×C6, C6⋊C6, C322C22, C3⋊(C2×C6), (C3×C6)⋊1C2, SmallGroup(36,12)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C6
C1C3C32C3×S3 — S3×C6
C3 — S3×C6
C1C6

Generators and relations for S3×C6
 G = < a,b,c | a6=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
2C3
3C22
2C6
3C6
3C6
3C2×C6

Character table of S3×C6

 class 12A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I
 size 113311222112223333
ρ1111111111111111111    trivial
ρ21-11-111111-1-1-1-1-1-1-111    linear of order 2
ρ311-1-11111111111-1-1-1-1    linear of order 2
ρ41-1-1111111-1-1-1-1-111-1-1    linear of order 2
ρ51-11-1ζ32ζ3ζ3ζ321ζ6ζ65ζ65-1ζ6ζ6ζ65ζ3ζ32    linear of order 6
ρ611-1-1ζ3ζ32ζ32ζ31ζ3ζ32ζ321ζ3ζ65ζ6ζ6ζ65    linear of order 6
ρ71111ζ3ζ32ζ32ζ31ζ3ζ32ζ321ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ81-1-11ζ32ζ3ζ3ζ321ζ6ζ65ζ65-1ζ6ζ32ζ3ζ65ζ6    linear of order 6
ρ911-1-1ζ32ζ3ζ3ζ321ζ32ζ3ζ31ζ32ζ6ζ65ζ65ζ6    linear of order 6
ρ101111ζ32ζ3ζ3ζ321ζ32ζ3ζ31ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ111-1-11ζ3ζ32ζ32ζ31ζ65ζ6ζ6-1ζ65ζ3ζ32ζ6ζ65    linear of order 6
ρ121-11-1ζ3ζ32ζ32ζ31ζ65ζ6ζ6-1ζ65ζ65ζ6ζ32ζ3    linear of order 6
ρ13220022-1-1-122-1-1-10000    orthogonal lifted from S3
ρ142-20022-1-1-1-2-21110000    orthogonal lifted from D6
ρ152-200-1--3-1+-3ζ65ζ6-11+-31--3ζ31ζ320000    complex faithful
ρ162200-1+-3-1--3ζ6ζ65-1-1+-3-1--3ζ6-1ζ650000    complex lifted from C3×S3
ρ172-200-1+-3-1--3ζ6ζ65-11--31+-3ζ321ζ30000    complex faithful
ρ182200-1--3-1+-3ζ65ζ6-1-1--3-1+-3ζ65-1ζ60000    complex lifted from C3×S3

Permutation representations of S3×C6
On 12 points - transitive group 12T18
Generators in S12
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)

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

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

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

G:=TransitiveGroup(12,18);

On 18 points - transitive group 18T6
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 11 14)(2 12 15)(3 7 16)(4 8 17)(5 9 18)(6 10 13)
(1 4)(2 5)(3 6)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)

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

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

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

G:=TransitiveGroup(18,6);

Polynomial with Galois group S3×C6 over ℚ
actionf(x)Disc(f)
12T18x12-2x11+6x9+x8-8x7+3x6+2x4-6x3+7x2-4x+1212·38·56·5092

Matrix representation of S3×C6 in GL2(𝔽7) generated by

50
05
,
40
22
,
61
01
G:=sub<GL(2,GF(7))| [5,0,0,5],[4,2,0,2],[6,0,1,1] >;

S3×C6 in GAP, Magma, Sage, TeX

S_3\times C_6
% in TeX

G:=Group("S3xC6");
// GroupNames label

G:=SmallGroup(36,12);
// by ID

G=gap.SmallGroup(36,12);
# by ID

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

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

׿
×
𝔽