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G = C32⋊2D9  order 162 = 2·34

2nd semidirect product of C32 and D9 acting via D9/C3=S3

non-abelian, supersoluble, monomial

Aliases: C32⋊2D9, C33.2S3, (C3×C9)⋊4S3, C32⋊C9⋊3C2, C3.3(C9⋊S3), C32.8(C3⋊S3), C3.1(He3⋊C2), SmallGroup(162,17)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C32⋊C9 — C32⋊2D9
Chief series
C1 — C3 — C32 — C33 — C32⋊C9 — C32⋊2D9
Lower central
C32⋊C9 — C32⋊2D9
Upper central
C1 — C3

Generators and relations for C32⋊2D9
 G = < a,b,c,d | a3=b3=c9=d2=1, ab=ba, cac-1=ab-1, dad=a-1b, bc=cb, bd=db, dcd=c-1 >

C1
27C2
C3
C3
2C3
3C3
3C3
3C3
9S3
9S3
9S3
9S3
27C6
C32
C32
C32
C32
3C9
3C9
3C9
3C32
6C32
3D9
3D9
3D9
3C3⋊S3
9C3×S3
9C3×S3
9C3×S3
9C3×S3
C3×C9
C33
C3×C9
C3×C9
3C3×D9
3C3×C3⋊S3
3C3×D9
3C3×D9
C32⋊C9
C32⋊2D9

Character table of C32⋊2D9

 class 123A3B3C3D3E3F3G3H6A6B9A9B9C9D9E9F9G9H9I
 size 127112226662727666666666
ρ1111111111111111111111    trivial
ρ21-111111111-1-1111111111    linear of order 2
ρ3202222222200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ42022222-1-1-100-1-1-1222-1-1-1    orthogonal lifted from S3
ρ52022222-1-1-100222-1-1-1-1-1-1    orthogonal lifted from S3
ρ62022222-1-1-100-1-1-1-1-1-1222    orthogonal lifted from S3
ρ72022-1-1-1-1-1200ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ82022-1-1-1-12-100ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ92022-1-1-1-1-1200ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ102022-1-1-1-12-100ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ112022-1-1-1-1-1200ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ122022-1-1-12-1-100ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ132022-1-1-12-1-100ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ142022-1-1-1-12-100ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ152022-1-1-12-1-100ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ1631-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/23000ζ32ζ3000000000    complex lifted from He3⋊C2
ρ173-1-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/23000ζ65ζ6000000000    complex lifted from He3⋊C2
ρ183-1-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/23000ζ6ζ65000000000    complex lifted from He3⋊C2
ρ1931-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/23000ζ3ζ32000000000    complex lifted from He3⋊C2
ρ2060-3-3√-3-3+3√-33-3√-3/23+3√-3/2-300000000000000    complex faithful
ρ2160-3+3√-3-3-3√-33+3√-3/23-3√-3/2-300000000000000    complex faithful

Permutation representations of C32⋊2D9
►On 18 points - transitive group 18T87
Generators in S18
(2 8 5)(3 6 9)(11 14 17)(12 18 15)

(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)
 
G:=sub<Sym(18)| (2,8,5)(3,6,9)(11,14,17)(12,18,15), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)>;
 
G:=Group( (2,8,5)(3,6,9)(11,14,17)(12,18,15), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10) );
 
G=PermutationGroup([[(2,8,5),(3,6,9),(11,14,17),(12,18,15)], [(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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]])
 
G:=TransitiveGroup(18,87);
 

►On 27 points - transitive group 27T65
Generators in S27
(2 12 19)(3 20 13)(5 15 22)(6 23 16)(8 18 25)(9 26 10)

(1 27 11)(2 19 12)(3 20 13)(4 21 14)(5 22 15)(6 23 16)(7 24 17)(8 25 18)(9 26 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 9)(2 8)(3 7)(4 6)(10 11)(12 18)(13 17)(14 16)(19 25)(20 24)(21 23)(26 27)
 
G:=sub<Sym(27)| (2,12,19)(3,20,13)(5,15,22)(6,23,16)(8,18,25)(9,26,10), (1,27,11)(2,19,12)(3,20,13)(4,21,14)(5,22,15)(6,23,16)(7,24,17)(8,25,18)(9,26,10), (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), (1,9)(2,8)(3,7)(4,6)(10,11)(12,18)(13,17)(14,16)(19,25)(20,24)(21,23)(26,27)>;
 
G:=Group( (2,12,19)(3,20,13)(5,15,22)(6,23,16)(8,18,25)(9,26,10), (1,27,11)(2,19,12)(3,20,13)(4,21,14)(5,22,15)(6,23,16)(7,24,17)(8,25,18)(9,26,10), (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), (1,9)(2,8)(3,7)(4,6)(10,11)(12,18)(13,17)(14,16)(19,25)(20,24)(21,23)(26,27) );
 
G=PermutationGroup([[(2,12,19),(3,20,13),(5,15,22),(6,23,16),(8,18,25),(9,26,10)], [(1,27,11),(2,19,12),(3,20,13),(4,21,14),(5,22,15),(6,23,16),(7,24,17),(8,25,18),(9,26,10)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,11),(12,18),(13,17),(14,16),(19,25),(20,24),(21,23),(26,27)]])
 
G:=TransitiveGroup(27,65);
 

C32⋊2D9 is a maximal subgroup of
 C32⋊D18  C32⋊C9.S3  C32⋊C9⋊C6  C3.3C3≀S3  C32⋊C9.C6  C33.(C3×S3)  C32⋊2D9.C3  C33⋊1D9  (C3×C9)⋊D9  (C3×C9)⋊3D9  C92⋊4S3  C34.7S3  (C32×C9)⋊S3  C9⋊C9⋊2S3  C92⋊6S3  C92⋊5S3  C33⋊6D9  He3⋊4D9
C32⋊2D9 is a maximal quotient of
 C32⋊2Dic9  C3.2(C9⋊D9)  C32⋊2D27  C33⋊2D9  (C3×C9)⋊5D9  (C3×C9)⋊6D9  C33.D9  He3⋊2D9  3- 1+2⋊D9  He3.3D9  He3.4D9  C33⋊6D9

Matrix representation of C32⋊2D9 ►in GL5(𝔽19)

1611000
82000
00001
00100
00010
,
10000
01000
00700
00070
00007
,
018000
13000
00100
00070
000011
,
82000
1611000
001800
00008
000120

G:=sub<GL(5,GF(19))| [16,8,0,0,0,11,2,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[0,1,0,0,0,18,3,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[8,16,0,0,0,2,11,0,0,0,0,0,18,0,0,0,0,0,0,12,0,0,0,8,0] >;
 

C32⋊2D9 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_2D_9
 
% in TeX
 
G:=Group("C3^2:2D9");
 
// GroupNames label
 
G:=SmallGroup(162,17);
 
// by ID
 
G=gap.SmallGroup(162,17);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-3,-3,221,186,182,457,723]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^9=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊2D9 in TeX
Character table of C32⋊2D9 in TeX

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