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G = C9⋊C6order 54 = 2·33

The semidirect product of C9 and C6 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C9⋊C6, D9⋊C3, C32.S3, 3- 1+2⋊C2, C3.3(C3×S3), Aut(D9), Hol(C9), SmallGroup(54,6)

Series: Derived Chief Lower central Upper central

C1C9 — C9⋊C6
C1C3C93- 1+2 — C9⋊C6
C9 — C9⋊C6
C1

Generators and relations for C9⋊C6
 G = < a,b | a9=b6=1, bab-1=a2 >

9C2
3C3
3S3
9C6
2C9
3C3×S3

Character table of C9⋊C6

 class 123A3B3C6A6B9A9B9C
 size 1923399666
ρ11111111111    trivial
ρ21-1111-1-1111    linear of order 2
ρ31-11ζ3ζ32ζ6ζ65ζ321ζ3    linear of order 6
ρ4111ζ32ζ3ζ3ζ32ζ31ζ32    linear of order 3
ρ5111ζ3ζ32ζ32ζ3ζ321ζ3    linear of order 3
ρ61-11ζ32ζ3ζ65ζ6ζ31ζ32    linear of order 6
ρ72022200-1-1-1    orthogonal lifted from S3
ρ8202-1+-3-1--300ζ6-1ζ65    complex lifted from C3×S3
ρ9202-1--3-1+-300ζ65-1ζ6    complex lifted from C3×S3
ρ1060-30000000    orthogonal faithful

Permutation representations of C9⋊C6
On 9 points - transitive group 9T10
Generators in S9
(1 2 3 4 5 6 7 8 9)
(2 6 8 9 5 3)(4 7)

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

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

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

G:=TransitiveGroup(9,10);

On 18 points - transitive group 18T18
Generators in S18
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 16)(2 12 8 15 5 18)(3 17 6 14 9 11)(4 13)(7 10)

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

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

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

G:=TransitiveGroup(18,18);

On 27 points - transitive group 27T14
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 17 23)(2 13 21 9 12 25)(3 18 19 8 16 27)(4 14 26 7 11 20)(5 10 24 6 15 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), (1,17,23)(2,13,21,9,12,25)(3,18,19,8,16,27)(4,14,26,7,11,20)(5,10,24,6,15,22)>;

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

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

G:=TransitiveGroup(27,14);

Polynomial with Galois group C9⋊C6 over ℚ
actionf(x)Disc(f)
9T10x9-4x8-14x7+44x6+62x5-120x4-92x3+48x2+12x-4216·194·374·42192

Matrix representation of C9⋊C6 in GL6(ℤ)

000100
00-1-100
000001
0000-1-1
100000
010000
,
100000
-1-10000
000010
0000-1-1
00-1-100
000100

G:=sub<GL(6,Integers())| [0,0,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,1,-1,0,0,0,0,0,-1,0,0] >;

C9⋊C6 in GAP, Magma, Sage, TeX

C_9\rtimes C_6
% in TeX

G:=Group("C9:C6");
// GroupNames label

G:=SmallGroup(54,6);
// by ID

G=gap.SmallGroup(54,6);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,362,150,82,579]);
// Polycyclic

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

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