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G = S3×C9order 54 = 2·33

Direct product of C9 and S3

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

Aliases: S3×C9, C3⋊C18, C32.2C6, (C3×C9)⋊1C2, (C3×S3).C3, C3.4(C3×S3), SmallGroup(54,4)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C9
C1C3C32C3×C9 — S3×C9
C3 — S3×C9
C1C9

Generators and relations for S3×C9
 G = < a,b,c | a9=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
2C3
3C6
2C9
3C18

Character table of S3×C9

 class 123A3B3C3D3E6A6B9A9B9C9D9E9F9G9H9I9J9K9L18A18B18C18D18E18F
 size 131122233111111222222333333
ρ1111111111111111111111111111    trivial
ρ21-111111-1-1111111111111-1-1-1-1-1-1    linear of order 2
ρ31-111111-1-1ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ41-111111-1-1ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ5111111111ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ6111111111ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ711ζ3ζ32ζ3ζ321ζ32ζ3ζ95ζ97ζ98ζ92ζ9ζ94ζ95ζ9ζ92ζ97ζ94ζ98ζ97ζ9ζ94ζ95ζ98ζ92    linear of order 9
ρ811ζ3ζ32ζ3ζ321ζ32ζ3ζ98ζ94ζ92ζ95ζ97ζ9ζ98ζ97ζ95ζ94ζ9ζ92ζ94ζ97ζ9ζ98ζ92ζ95    linear of order 9
ρ911ζ32ζ3ζ32ζ31ζ3ζ32ζ9ζ95ζ97ζ94ζ92ζ98ζ9ζ92ζ94ζ95ζ98ζ97ζ95ζ92ζ98ζ9ζ97ζ94    linear of order 9
ρ1011ζ3ζ32ζ3ζ321ζ32ζ3ζ92ζ9ζ95ζ98ζ94ζ97ζ92ζ94ζ98ζ9ζ97ζ95ζ9ζ94ζ97ζ92ζ95ζ98    linear of order 9
ρ111-1ζ32ζ3ζ32ζ31ζ65ζ6ζ94ζ92ζ9ζ97ζ98ζ95ζ94ζ98ζ97ζ92ζ95ζ992989594997    linear of order 18
ρ121-1ζ32ζ3ζ32ζ31ζ65ζ6ζ97ζ98ζ94ζ9ζ95ζ92ζ97ζ95ζ9ζ98ζ92ζ9498959297949    linear of order 18
ρ131-1ζ3ζ32ζ3ζ321ζ6ζ65ζ92ζ9ζ95ζ98ζ94ζ97ζ92ζ94ζ98ζ9ζ97ζ9599497929598    linear of order 18
ρ1411ζ32ζ3ζ32ζ31ζ3ζ32ζ97ζ98ζ94ζ9ζ95ζ92ζ97ζ95ζ9ζ98ζ92ζ94ζ98ζ95ζ92ζ97ζ94ζ9    linear of order 9
ρ151-1ζ32ζ3ζ32ζ31ζ65ζ6ζ9ζ95ζ97ζ94ζ92ζ98ζ9ζ92ζ94ζ95ζ98ζ9795929899794    linear of order 18
ρ161-1ζ3ζ32ζ3ζ321ζ6ζ65ζ98ζ94ζ92ζ95ζ97ζ9ζ98ζ97ζ95ζ94ζ9ζ9294979989295    linear of order 18
ρ1711ζ32ζ3ζ32ζ31ζ3ζ32ζ94ζ92ζ9ζ97ζ98ζ95ζ94ζ98ζ97ζ92ζ95ζ9ζ92ζ98ζ95ζ94ζ9ζ97    linear of order 9
ρ181-1ζ3ζ32ζ3ζ321ζ6ζ65ζ95ζ97ζ98ζ92ζ9ζ94ζ95ζ9ζ92ζ97ζ94ζ9897994959892    linear of order 18
ρ192022-1-1-100222222-1-1-1-1-1-1000000    orthogonal lifted from S3
ρ202022-1-1-100-1+-3-1--3-1+-3-1+-3-1--3-1--3ζ65ζ6ζ65ζ6ζ6ζ65000000    complex lifted from C3×S3
ρ212022-1-1-100-1--3-1+-3-1--3-1--3-1+-3-1+-3ζ6ζ65ζ6ζ65ζ65ζ6000000    complex lifted from C3×S3
ρ2220-1+-3-1--3ζ65ζ6-1009597989299495992979498000000    complex faithful
ρ2320-1+-3-1--3ζ65ζ6-1009299598949792949899795000000    complex faithful
ρ2420-1--3-1+-3ζ6ζ65-1009492997989594989792959000000    complex faithful
ρ2520-1+-3-1--3ζ65ζ6-1009894929597998979594992000000    complex faithful
ρ2620-1--3-1+-3ζ6ζ65-1009959794929899294959897000000    complex faithful
ρ2720-1--3-1+-3ζ6ζ65-1009798949959297959989294000000    complex faithful

Permutation representations of S3×C9
On 18 points - transitive group 18T16
Generators in S18
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)

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

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

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

G:=TransitiveGroup(18,16);

On 27 points - transitive group 27T12
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 14 24)(2 15 25)(3 16 26)(4 17 27)(5 18 19)(6 10 20)(7 11 21)(8 12 22)(9 13 23)
(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)

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

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

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

G:=TransitiveGroup(27,12);

Matrix representation of S3×C9 in GL2(𝔽19) generated by

60
06
,
70
011
,
01
10
G:=sub<GL(2,GF(19))| [6,0,0,6],[7,0,0,11],[0,1,1,0] >;

S3×C9 in GAP, Magma, Sage, TeX

S_3\times C_9
% in TeX

G:=Group("S3xC9");
// GroupNames label

G:=SmallGroup(54,4);
// by ID

G=gap.SmallGroup(54,4);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,29,579]);
// Polycyclic

G:=Group<a,b,c|a^9=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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