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G = C2×C9⋊C6  order 108 = 22·33

Direct product of C2 and C9⋊C6

direct product, metacyclic, supersoluble, monomial

Aliases: C2×C9⋊C6, C18⋊C6, D9⋊C6, D18⋊C3, C32.D6, 3- 1+2⋊C22, C9⋊(C2×C6), C6.6(C3×S3), C3.3(S3×C6), (C3×C6).4S3, (C2×3- 1+2)⋊C2, Aut(D18), Hol(C18), SmallGroup(108,26)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C2×C9⋊C6
Chief series
C1 — C3 — C9 — 3- 1+2 — C9⋊C6 — C2×C9⋊C6
Lower central
C9 — C2×C9⋊C6
Upper central
C1 — C2

Generators and relations for C2×C9⋊C6
 G = < a,b,c | a2=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

C1
C2
9C2
9C2
C3
3C3
9C22
C6
3S3
3S3
3C6
9C6
9C6
C32
C9
2C9
3D6
9C2×C6
D9
C18
D9
C3×C6
2C18
3C3×S3
3C3×S3
3- 1+2
D18
3S3×C6
C9⋊C6
C9⋊C6
C2×3- 1+2
C2×C9⋊C6

Character table of C2×C9⋊C6

 class 12A2B2C3A3B3C6A6B6C6D6E6F6G9A9B9C18A18B18C
 size 11992332339999666666
ρ111111111111111111111    trivial
ρ211-1-1111111-1-1-1-1111111    linear of order 2
ρ31-11-1111-1-1-11-1-11111-1-1-1    linear of order 2
ρ41-1-11111-1-1-1-111-1111-1-1-1    linear of order 2
ρ511-1-11ζ32ζ31ζ32ζ3ζ65ζ6ζ65ζ61ζ32ζ31ζ3ζ32    linear of order 6
ρ61-11-11ζ32ζ3-1ζ6ζ65ζ3ζ6ζ65ζ321ζ32ζ3-1ζ65ζ6    linear of order 6
ρ711111ζ32ζ31ζ32ζ3ζ3ζ32ζ3ζ321ζ32ζ31ζ3ζ32    linear of order 3
ρ811-1-11ζ3ζ321ζ3ζ32ζ6ζ65ζ6ζ651ζ3ζ321ζ32ζ3    linear of order 6
ρ91-1-111ζ32ζ3-1ζ6ζ65ζ65ζ32ζ3ζ61ζ32ζ3-1ζ65ζ6    linear of order 6
ρ1011111ζ3ζ321ζ3ζ32ζ32ζ3ζ32ζ31ζ3ζ321ζ32ζ3    linear of order 3
ρ111-11-11ζ3ζ32-1ζ65ζ6ζ32ζ65ζ6ζ31ζ3ζ32-1ζ6ζ65    linear of order 6
ρ121-1-111ζ3ζ32-1ζ65ζ6ζ6ζ3ζ32ζ651ζ3ζ32-1ζ6ζ65    linear of order 6
ρ132-200222-2-2-20000-1-1-1111    orthogonal lifted from D6
ρ1422002222220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ1522002-1-√-3-1+√-32-1-√-3-1+√-30000-1ζ6ζ65-1ζ65ζ6    complex lifted from C3×S3
ρ1622002-1+√-3-1-√-32-1+√-3-1-√-30000-1ζ65ζ6-1ζ6ζ65    complex lifted from C3×S3
ρ172-2002-1+√-3-1-√-3-21-√-31+√-30000-1ζ65ζ61ζ32ζ3    complex lifted from S3×C6
ρ182-2002-1-√-3-1+√-3-21+√-31-√-30000-1ζ6ζ651ζ3ζ32    complex lifted from S3×C6
ρ196-600-3003000000000000    orthogonal faithful
ρ206600-300-3000000000000    orthogonal lifted from C9⋊C6

Permutation representations of C2×C9⋊C6
►On 18 points - transitive group 18T45
Generators in S18
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)

(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

(2 6 8 9 5 3)(4 7)(10 15 13 12 16 18)(14 17)
 
G:=sub<Sym(18)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,6,8,9,5,3)(4,7)(10,15,13,12,16,18)(14,17)>;
 
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,6,8,9,5,3)(4,7)(10,15,13,12,16,18)(14,17) );
 
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,10)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(2,6,8,9,5,3),(4,7),(10,15,13,12,16,18),(14,17)]])
 
G:=TransitiveGroup(18,45);
 

C2×C9⋊C6 is a maximal subgroup of   D36⋊C3  Dic9⋊C6  C32.GL2(𝔽3)  D18.A4
C2×C9⋊C6 is a maximal quotient of   C36.C6  D36⋊C3  Dic9⋊C6

Matrix representation of C2×C9⋊C6 ►in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
000100
00-1-100
000001
0000-1-1
100000
010000
,
-100000
110000
0000-10
000011
001100
000-100

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

C2×C9⋊C6 in GAP, Magma, Sage, TeX 

C_2\times C_9\rtimes C_6
 
% in TeX
 
G:=Group("C2xC9:C6");
 
// GroupNames label
 
G:=SmallGroup(108,26);
 
// by ID
 
G=gap.SmallGroup(108,26);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-3,1203,253,138,1804]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=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 C2×C9⋊C6 in TeX
Character table of C2×C9⋊C6 in TeX

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