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G = C322D9order 162 = 2·34

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

non-abelian, supersoluble, monomial

Aliases: C322D9, C33.2S3, (C3×C9)⋊4S3, C32⋊C93C2, C3.3(C9⋊S3), C32.8(C3⋊S3), C3.1(He3⋊C2), SmallGroup(162,17)

Series: Derived Chief Lower central Upper central

C1C3C32⋊C9 — C322D9
C1C3C32C33C32⋊C9 — C322D9
C32⋊C9 — C322D9
C1C3

Generators and relations for C322D9
 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 >

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

Character table of C322D9

 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
ρ72022-1-1-1-1-1200ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ82022-1-1-1-12-100ζ989ζ9792ζ9594ζ9792ζ9594ζ989ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ92022-1-1-1-1-1200ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ102022-1-1-1-12-100ζ9792ζ9594ζ989ζ9594ζ989ζ9792ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ112022-1-1-1-1-1200ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ122022-1-1-12-1-100ζ989ζ9792ζ9594ζ9594ζ989ζ9792ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ132022-1-1-12-1-100ζ9594ζ989ζ9792ζ9792ζ9594ζ989ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ142022-1-1-1-12-100ζ9594ζ989ζ9792ζ989ζ9792ζ9594ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ152022-1-1-12-1-100ζ9792ζ9594ζ989ζ989ζ9792ζ9594ζ9594ζ989ζ9792    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 C322D9
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);

C322D9 is a maximal subgroup of
C32⋊D18  C32⋊C9.S3  C32⋊C9⋊C6  C3.3C3≀S3  C32⋊C9.C6  C33.(C3×S3)  C322D9.C3  C331D9  (C3×C9)⋊D9  (C3×C9)⋊3D9  C924S3  C34.7S3  (C32×C9)⋊S3  C9⋊C92S3  C926S3  C925S3  C336D9  He34D9
C322D9 is a maximal quotient of
C322Dic9  C3.2(C9⋊D9)  C322D27  C332D9  (C3×C9)⋊5D9  (C3×C9)⋊6D9  C33.D9  He32D9  3- 1+2⋊D9  He3.3D9  He3.4D9  C336D9

Matrix representation of C322D9 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] >;

C322D9 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 C322D9 in TeX
Character table of C322D9 in TeX

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