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G = C9⋊C18order 162 = 2·34

The semidirect product of C9 and C18 acting via C18/C3=C6

metacyclic, supersoluble, monomial

Aliases: C9⋊C18, D9⋊C9, C9⋊C9⋊C2, (C3×C9).C6, (C3×D9).C3, C3.3(S3×C9), (C3×C9).1S3, C3.2(C9⋊C6), C32.14(C3×S3), SmallGroup(162,6)

Series: Derived Chief Lower central Upper central

C1C9 — C9⋊C18
C1C3C9C3×C9C9⋊C9 — C9⋊C18
C9 — C9⋊C18
C1C3

Generators and relations for C9⋊C18
 G = < a,b | a9=b18=1, bab-1=a2 >

9C2
2C3
3S3
9C6
2C9
3C9
6C9
3C3×S3
9C18
2C3×C9
3S3×C9

Character table of C9⋊C18

 class 123A3B3C3D3E6A6B9A9B9C9D9E9F9G9H9I9J9K9L9M9N9O18A18B18C18D18E18F
 size 191122299333333666666666999999
ρ1111111111111111111111111111111    trivial
ρ21-111111-1-1111111111111111-1-1-1-1-1-1    linear of order 2
ρ3111111111ζ3ζ3ζ3ζ32ζ32ζ321ζ3ζ32ζ31ζ3ζ321ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ4111111111ζ32ζ32ζ32ζ3ζ3ζ31ζ32ζ3ζ321ζ32ζ31ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ51-111111-1-1ζ32ζ32ζ32ζ3ζ3ζ31ζ32ζ3ζ321ζ32ζ31ζ3ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ61-111111-1-1ζ3ζ3ζ3ζ32ζ32ζ321ζ3ζ32ζ31ζ3ζ321ζ32ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ711ζ32ζ3ζ32ζ31ζ3ζ32ζ94ζ9ζ97ζ98ζ95ζ921ζ94ζ92ζ9ζ32ζ97ζ98ζ3ζ95ζ92ζ98ζ95ζ94ζ9ζ97    linear of order 9
ρ81-1ζ32ζ3ζ32ζ31ζ65ζ6ζ94ζ9ζ97ζ98ζ95ζ921ζ94ζ92ζ9ζ32ζ97ζ98ζ3ζ9592989594997    linear of order 18
ρ911ζ32ζ3ζ32ζ31ζ3ζ32ζ9ζ97ζ94ζ92ζ98ζ951ζ9ζ95ζ97ζ32ζ94ζ92ζ3ζ98ζ95ζ92ζ98ζ9ζ97ζ94    linear of order 9
ρ1011ζ3ζ32ζ3ζ321ζ32ζ3ζ92ζ95ζ98ζ94ζ97ζ91ζ92ζ9ζ95ζ3ζ98ζ94ζ32ζ97ζ9ζ94ζ97ζ92ζ95ζ98    linear of order 9
ρ111-1ζ3ζ32ζ3ζ321ζ6ζ65ζ92ζ95ζ98ζ94ζ97ζ91ζ92ζ9ζ95ζ3ζ98ζ94ζ32ζ9799497929598    linear of order 18
ρ121-1ζ3ζ32ζ3ζ321ζ6ζ65ζ95ζ98ζ92ζ9ζ94ζ971ζ95ζ97ζ98ζ3ζ92ζ9ζ32ζ9497994959892    linear of order 18
ρ131-1ζ32ζ3ζ32ζ31ζ65ζ6ζ9ζ97ζ94ζ92ζ98ζ951ζ9ζ95ζ97ζ32ζ94ζ92ζ3ζ9895929899794    linear of order 18
ρ141-1ζ3ζ32ζ3ζ321ζ6ζ65ζ98ζ92ζ95ζ97ζ9ζ941ζ98ζ94ζ92ζ3ζ95ζ97ζ32ζ994979989295    linear of order 18
ρ151-1ζ32ζ3ζ32ζ31ζ65ζ6ζ97ζ94ζ9ζ95ζ92ζ981ζ97ζ98ζ94ζ32ζ9ζ95ζ3ζ9298959297949    linear of order 18
ρ1611ζ32ζ3ζ32ζ31ζ3ζ32ζ97ζ94ζ9ζ95ζ92ζ981ζ97ζ98ζ94ζ32ζ9ζ95ζ3ζ92ζ98ζ95ζ92ζ97ζ94ζ9    linear of order 9
ρ1711ζ3ζ32ζ3ζ321ζ32ζ3ζ95ζ98ζ92ζ9ζ94ζ971ζ95ζ97ζ98ζ3ζ92ζ9ζ32ζ94ζ97ζ9ζ94ζ95ζ98ζ92    linear of order 9
ρ1811ζ3ζ32ζ3ζ321ζ32ζ3ζ98ζ92ζ95ζ97ζ9ζ941ζ98ζ94ζ92ζ3ζ95ζ97ζ32ζ9ζ94ζ97ζ9ζ98ζ92ζ95    linear of order 9
ρ19202222200222222-1-1-1-1-1-1-1-1-1000000    orthogonal lifted from S3
ρ20202222200-1+-3-1+-3-1+-3-1--3-1--3-1--3-1ζ65ζ6ζ65-1ζ65ζ6-1ζ6000000    complex lifted from C3×S3
ρ21202222200-1--3-1--3-1--3-1+-3-1+-3-1+-3-1ζ6ζ65ζ6-1ζ6ζ65-1ζ65000000    complex lifted from C3×S3
ρ2220-1+-3-1--3-1+-3-1--320095989299497-1959798ζ65929ζ694000000    complex lifted from S3×C9
ρ2320-1--3-1+-3-1--3-1+-320097949959298-1979894ζ6995ζ6592000000    complex lifted from S3×C9
ρ2420-1+-3-1--3-1+-3-1--320098929597994-1989492ζ659597ζ69000000    complex lifted from S3×C9
ρ2520-1+-3-1--3-1+-3-1--320092959894979-192995ζ659894ζ697000000    complex lifted from S3×C9
ρ2620-1--3-1+-3-1--3-1+-320094997989592-194929ζ69798ζ6595000000    complex lifted from S3×C9
ρ2720-1--3-1+-3-1--3-1+-320099794929895-199597ζ69492ζ6598000000    complex lifted from S3×C9
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 C9⋊C18
On 18 points - transitive group 18T80
Generators in S18
(1 5 15 13 17 9 7 11 3)(2 16 18 8 4 6 14 10 12)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,80);

C9⋊C18 is a maximal subgroup of
C27⋊C18  C9⋊C9.S3  C9⋊C9.3S3  C9⋊C9⋊S3  C9×C9⋊C6  D9⋊3- 1+2  C927C6  C928C6  C9⋊(S3×C9)  C923S3  C9⋊C92S3  C926S3  C925S3
C9⋊C18 is a maximal quotient of
C9⋊C36  C9⋊S3⋊C9  C9⋊C54  C27⋊C18  C9⋊(S3×C9)

Matrix representation of C9⋊C18 in GL6(𝔽19)

0110000
0011000
100000
000001
000700
000070
,
000010
0000011
0001100
010000
0011000
1100000

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

C9⋊C18 in GAP, Magma, Sage, TeX

C_9\rtimes C_{18}
% in TeX

G:=Group("C9:C18");
// GroupNames label

G:=SmallGroup(162,6);
// by ID

G=gap.SmallGroup(162,6);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,36,1803,728,138,2704]);
// Polycyclic

G:=Group<a,b|a^9=b^18=1,b*a*b^-1=a^2>;
// generators/relations

Export

Subgroup lattice of C9⋊C18 in TeX
Character table of C9⋊C18 in TeX

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