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G = S3×C32order 54 = 2·33

Direct product of C32 and S3

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

Aliases: S3×C32, C331C2, C323C6, C3⋊(C3×C6), SmallGroup(54,12)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C32
C1C3C32C33 — S3×C32
C3 — S3×C32
C1C32

Generators and relations for S3×C32
 G = < a,b,c,d | a3=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
2C3
2C3
2C3
2C3
3C6
3C6
3C6
3C6
2C32
2C32
2C32
2C32
3C3×C6

Character table of S3×C32

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q6A6B6C6D6E6F6G6H
 size 131111111122222222233333333
ρ1111111111111111111111111111    trivial
ρ21-111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311ζ3ζ3ζ321ζ32ζ31ζ32ζ321ζ3ζ311ζ32ζ3ζ32ζ3ζ32ζ3ζ321ζ321ζ3    linear of order 3
ρ41-1ζ32ζ32ζ31ζ3ζ321ζ3ζ31ζ32ζ3211ζ3ζ32ζ3ζ6ζ65ζ6ζ65-1ζ65-1ζ6    linear of order 6
ρ51-1ζ31ζ32ζ31ζ32ζ32ζ31ζ3ζ3211ζ32ζ32ζ3ζ3ζ6ζ65-1ζ6ζ65-1ζ6ζ65    linear of order 6
ρ611ζ32ζ3ζ3ζ3ζ321ζ321ζ32ζ31ζ31ζ32ζ3ζ32111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ71-11ζ321ζ3ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ321ζ3211ζ32ζ65ζ6ζ6-1ζ65ζ65ζ6-1    linear of order 6
ρ81-1ζ321ζ3ζ321ζ3ζ3ζ321ζ32ζ311ζ3ζ3ζ32ζ32ζ65ζ6-1ζ65ζ6-1ζ65ζ6    linear of order 6
ρ911ζ321ζ3ζ321ζ3ζ3ζ321ζ32ζ311ζ3ζ3ζ32ζ32ζ3ζ321ζ3ζ321ζ3ζ32    linear of order 3
ρ1011ζ32ζ32ζ31ζ3ζ321ζ3ζ31ζ32ζ3211ζ3ζ32ζ3ζ32ζ3ζ32ζ31ζ31ζ32    linear of order 3
ρ111-11ζ31ζ32ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ31ζ311ζ3ζ6ζ65ζ65-1ζ6ζ6ζ65-1    linear of order 6
ρ1211ζ3ζ32ζ32ζ32ζ31ζ31ζ3ζ321ζ321ζ3ζ32ζ3111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ131-1ζ3ζ3ζ321ζ32ζ31ζ32ζ321ζ3ζ311ζ32ζ3ζ32ζ65ζ6ζ65ζ6-1ζ6-1ζ65    linear of order 6
ρ14111ζ31ζ32ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ31ζ311ζ3ζ32ζ3ζ31ζ32ζ32ζ31    linear of order 3
ρ1511ζ31ζ32ζ31ζ32ζ32ζ31ζ3ζ3211ζ32ζ32ζ3ζ3ζ32ζ31ζ32ζ31ζ32ζ3    linear of order 3
ρ161-1ζ32ζ3ζ3ζ3ζ321ζ321ζ32ζ31ζ31ζ32ζ3ζ321-1-1ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ17111ζ321ζ3ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ321ζ3211ζ32ζ3ζ32ζ321ζ3ζ3ζ321    linear of order 3
ρ181-1ζ3ζ32ζ32ζ32ζ31ζ31ζ3ζ321ζ321ζ3ζ32ζ31-1-1ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ192022222222-1-1-1-1-1-1-1-1-100000000    orthogonal lifted from S3
ρ20202-1+-32-1--3-1--3-1--3-1+-3-1+-3ζ6ζ6ζ6ζ65-1ζ65-1-1ζ6500000000    complex lifted from C3×S3
ρ2120-1--3-1--3-1+-32-1+-3-1--32-1+-3ζ65-1ζ6ζ6-1-1ζ65ζ6ζ6500000000    complex lifted from C3×S3
ρ2220-1+-3-1+-3-1--32-1--3-1+-32-1--3ζ6-1ζ65ζ65-1-1ζ6ζ65ζ600000000    complex lifted from C3×S3
ρ2320-1--32-1+-3-1--32-1+-3-1+-3-1--3-1ζ6ζ65-1-1ζ65ζ65ζ6ζ600000000    complex lifted from C3×S3
ρ24202-1--32-1+-3-1+-3-1+-3-1--3-1--3ζ65ζ65ζ65ζ6-1ζ6-1-1ζ600000000    complex lifted from C3×S3
ρ2520-1--3-1+-3-1+-3-1+-3-1--32-1--32ζ6ζ65-1ζ65-1ζ6ζ65ζ6-100000000    complex lifted from C3×S3
ρ2620-1+-3-1--3-1--3-1--3-1+-32-1+-32ζ65ζ6-1ζ6-1ζ65ζ6ζ65-100000000    complex lifted from C3×S3
ρ2720-1+-32-1--3-1+-32-1--3-1--3-1+-3-1ζ65ζ6-1-1ζ6ζ6ζ65ζ6500000000    complex lifted from C3×S3

Permutation representations of S3×C32
On 18 points - transitive group 18T17
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 10 14)(2 11 15)(3 12 13)(4 8 18)(5 9 16)(6 7 17)
(1 10 14)(2 11 15)(3 12 13)(4 18 8)(5 16 9)(6 17 7)
(1 16)(2 17)(3 18)(4 12)(5 10)(6 11)(7 15)(8 13)(9 14)

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

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

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

G:=TransitiveGroup(18,17);

On 27 points - transitive group 27T15
Generators in S27
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)
(1 8 26)(2 9 27)(3 7 25)(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)
(7 25)(8 26)(9 27)(10 18)(11 16)(12 17)(19 23)(20 24)(21 22)

G:=sub<Sym(27)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,27,19)(11,25,20)(12,26,21), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22)>;

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)(25,26,27), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,27,19)(11,25,20)(12,26,21), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22) );

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),(25,26,27)], [(1,6,14),(2,4,15),(3,5,13),(7,24,16),(8,22,17),(9,23,18),(10,27,19),(11,25,20),(12,26,21)], [(1,8,26),(2,9,27),(3,7,25),(4,23,19),(5,24,20),(6,22,21),(10,15,18),(11,13,16),(12,14,17)], [(7,25),(8,26),(9,27),(10,18),(11,16),(12,17),(19,23),(20,24),(21,22)])

G:=TransitiveGroup(27,15);

Matrix representation of S3×C32 in GL3(𝔽7) generated by

200
020
002
,
400
010
001
,
100
020
004
,
100
001
010
G:=sub<GL(3,GF(7))| [2,0,0,0,2,0,0,0,2],[4,0,0,0,1,0,0,0,1],[1,0,0,0,2,0,0,0,4],[1,0,0,0,0,1,0,1,0] >;

S3×C32 in GAP, Magma, Sage, TeX

S_3\times C_3^2
% in TeX

G:=Group("S3xC3^2");
// GroupNames label

G:=SmallGroup(54,12);
// by ID

G=gap.SmallGroup(54,12);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,579]);
// Polycyclic

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

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