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G = C3×C9⋊C6order 162 = 2·34

Direct product of C3 and C9⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C9⋊C6, D9⋊C32, C33.3S3, 3- 1+23C6, C9⋊(C3×C6), (C3×D9)⋊C3, (C3×C9)⋊4C6, C3.3(S3×C32), C32.4(C3×S3), (C3×3- 1+2)⋊1C2, SmallGroup(162,36)

Series: Derived Chief Lower central Upper central

C1C9 — C3×C9⋊C6
C1C3C9C3×C9C3×3- 1+2 — C3×C9⋊C6
C9 — C3×C9⋊C6
C1C3

Generators and relations for C3×C9⋊C6
 G = < a,b,c | a3=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

9C2
2C3
3C3
3C3
3C3
3S3
9C6
9C6
9C6
9C6
2C9
2C9
2C9
2C9
3C32
6C32
3C3×S3
3C3×S3
3C3×S3
3C3×S3
9C3×C6
23- 1+2
23- 1+2
23- 1+2
2C3×C9
3S3×C32

Character table of C3×C9⋊C6

 class 123A3B3C3D3E3F3G3H3I3J3K6A6B6C6D6E6F6G6H9A9B9C9D9E9F9G9H9I
 size 191122233333399999999666666666
ρ1111111111111111111111111111111    trivial
ρ21-111111111111-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ31-1ζ32ζ3ζ32ζ311ζ31ζ32ζ3ζ32ζ65ζ6ζ65-1ζ6ζ65-1ζ6ζ32ζ3111ζ32ζ32ζ3ζ3    linear of order 6
ρ41111111ζ3ζ32ζ32ζ3ζ3ζ32ζ311ζ32ζ32ζ32ζ3ζ3ζ32ζ321ζ3ζ32ζ31ζ31    linear of order 3
ρ51-1ζ3ζ32ζ3ζ3211ζ321ζ3ζ32ζ3ζ6ζ65ζ6-1ζ65ζ6-1ζ65ζ3ζ32111ζ3ζ3ζ32ζ32    linear of order 6
ρ61-111111ζ32ζ3ζ3ζ32ζ32ζ3ζ6-1-1ζ65ζ65ζ65ζ6ζ6ζ3ζ31ζ32ζ3ζ321ζ321    linear of order 6
ρ711ζ3ζ32ζ3ζ321ζ321ζ31ζ3ζ32ζ3ζ3ζ32ζ3ζ321ζ321ζ3211ζ32ζ31ζ3ζ3ζ32    linear of order 3
ρ811ζ3ζ32ζ3ζ3211ζ321ζ3ζ32ζ3ζ32ζ3ζ321ζ3ζ321ζ3ζ3ζ32111ζ3ζ3ζ32ζ32    linear of order 3
ρ91-1ζ32ζ3ζ32ζ31ζ31ζ321ζ32ζ3ζ6ζ6ζ65ζ6ζ65-1ζ65-1ζ311ζ3ζ321ζ32ζ32ζ3    linear of order 6
ρ101-111111ζ3ζ32ζ32ζ3ζ3ζ32ζ65-1-1ζ6ζ6ζ6ζ65ζ65ζ32ζ321ζ3ζ32ζ31ζ31    linear of order 6
ρ111111111ζ32ζ3ζ3ζ32ζ32ζ3ζ3211ζ3ζ3ζ3ζ32ζ32ζ3ζ31ζ32ζ3ζ321ζ321    linear of order 3
ρ1211ζ32ζ3ζ32ζ31ζ31ζ321ζ32ζ3ζ32ζ32ζ3ζ32ζ31ζ31ζ311ζ3ζ321ζ32ζ32ζ3    linear of order 3
ρ131-1ζ32ζ3ζ32ζ31ζ32ζ32ζ3ζ311-1ζ6ζ65ζ65-1ζ6ζ6ζ651ζ321ζ32ζ3ζ3ζ321ζ3    linear of order 6
ρ1411ζ3ζ32ζ3ζ321ζ3ζ3ζ32ζ32111ζ3ζ32ζ321ζ3ζ3ζ321ζ31ζ3ζ32ζ32ζ31ζ32    linear of order 3
ρ151-1ζ3ζ32ζ3ζ321ζ321ζ31ζ3ζ32ζ65ζ65ζ6ζ65ζ6-1ζ6-1ζ3211ζ32ζ31ζ3ζ3ζ32    linear of order 6
ρ1611ζ32ζ3ζ32ζ311ζ31ζ32ζ3ζ32ζ3ζ32ζ31ζ32ζ31ζ32ζ32ζ3111ζ32ζ32ζ3ζ3    linear of order 3
ρ1711ζ32ζ3ζ32ζ31ζ32ζ32ζ3ζ3111ζ32ζ3ζ31ζ32ζ32ζ31ζ321ζ32ζ3ζ3ζ321ζ3    linear of order 3
ρ181-1ζ3ζ32ζ3ζ321ζ3ζ3ζ32ζ3211-1ζ65ζ6ζ6-1ζ65ζ65ζ61ζ31ζ3ζ32ζ32ζ31ζ32    linear of order 6
ρ19202222222222200000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ202022222-1+-3-1--3-1--3-1+-3-1+-3-1--300000000ζ6ζ6-1ζ65ζ6ζ65-1ζ65-1    complex lifted from C3×S3
ρ2120-1--3-1+-3-1--3-1+-322-1+-32-1--3-1+-3-1--300000000ζ6ζ65-1-1-1ζ6ζ6ζ65ζ65    complex lifted from C3×S3
ρ2220-1+-3-1--3-1+-3-1--32-1--32-1+-32-1+-3-1--300000000ζ6-1-1ζ6ζ65-1ζ65ζ65ζ6    complex lifted from C3×S3
ρ2320-1+-3-1--3-1+-3-1--32-1+-3-1+-3-1--3-1--32200000000-1ζ65-1ζ65ζ6ζ6ζ65-1ζ6    complex lifted from C3×S3
ρ242022222-1--3-1+-3-1+-3-1--3-1--3-1+-300000000ζ65ζ65-1ζ6ζ65ζ6-1ζ6-1    complex lifted from C3×S3
ρ2520-1--3-1+-3-1--3-1+-32-1+-32-1--32-1--3-1+-300000000ζ65-1-1ζ65ζ6-1ζ6ζ6ζ65    complex lifted from C3×S3
ρ2620-1--3-1+-3-1--3-1+-32-1--3-1--3-1+-3-1+-32200000000-1ζ6-1ζ6ζ65ζ65ζ6-1ζ65    complex lifted from C3×S3
ρ2720-1+-3-1--3-1+-3-1--322-1--32-1+-3-1--3-1+-300000000ζ65ζ6-1-1-1ζ65ζ65ζ6ζ6    complex lifted from C3×S3
ρ286066-3-3-300000000000000000000000    orthogonal lifted from C9⋊C6
ρ2960-3+3-3-3-3-33-3-3/23+3-3/2-300000000000000000000000    complex faithful
ρ3060-3-3-3-3+3-33+3-3/23-3-3/2-300000000000000000000000    complex faithful

Permutation representations of C3×C9⋊C6
On 18 points - transitive group 18T83
Generators in S18
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 13 7 16 4 10)(2 18 5 15 8 12)(3 14)(6 11)(9 17)

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

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

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

G:=TransitiveGroup(18,83);

On 27 points - transitive group 27T57
Generators in S27
(1 20 11)(2 21 12)(3 22 13)(4 23 14)(5 24 15)(6 25 16)(7 26 17)(8 27 18)(9 19 10)
(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 11 20)(2 16 27 9 15 22)(3 12 25 8 10 24)(4 17 23 7 14 26)(5 13 21 6 18 19)

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

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

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

G:=TransitiveGroup(27,57);

C3×C9⋊C6 is a maximal subgroup of
D9⋊He3  C927C6  C928C6  C34.7S3  (C32×C9)⋊S3  C33⋊(C3×S3)  He3.C32C6  He3⋊(C3×S3)  C3.He3⋊C6  3- 1+4⋊C2
C3×C9⋊C6 is a maximal quotient of
C34.S3  D9⋊He3  D9⋊3- 1+2  C927C6  C928C6

Matrix representation of C3×C9⋊C6 in GL6(𝔽19)

1100000
0110000
0011000
0001100
0000110
0000011
,
070000
007000
100000
000001
0001100
0000110
,
0001100
000010
000007
1100000
010000
007000

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0] >;

C3×C9⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes C_6
% in TeX

G:=Group("C3xC9:C6");
// GroupNames label

G:=SmallGroup(162,36);
// by ID

G=gap.SmallGroup(162,36);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,1803,728,138,2704]);
// Polycyclic

G:=Group<a,b,c|a^3=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

Export

Subgroup lattice of C3×C9⋊C6 in TeX
Character table of C3×C9⋊C6 in TeX

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