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G = C32:D9  order 162 = 2·34

1st semidirect product of C32 and D9 acting via D9/C3=S3

metabelian, supersoluble, monomial

Aliases: C32:1D9, C33.1S3, C9:S3:1C3, C3.(C9:C6), (C3xC9):1C6, C32:C9:2C2, C3.1(C3xD9), C3.1(C32:C6), C32.13(C3xS3), SmallGroup(162,5)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC9 — C32:D9
Chief series
C1 — C3 — C32 — C3xC9 — C32:C9 — C32:D9
Lower central
C3xC9 — C32:D9
Upper central
C1

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

Subgroups: 184 in 34 conjugacy classes, 12 normal (10 characteristic)
Quotients: C1, C2, C3, S3, C6, D9, C3xS3, C3xD9, C32:C6, C9:C6, C32:D9
C1
27C2
C3
C3
C3
C3
3C3
6C3
9S3
9S3
9S3
9S3
27C6
C32
C32
2C32
3C32
3C32
3C9
3C32
6C9
3C3:S3
9C3xS3
9D9
9C3xS3
9C3xS3
9C3xS3
C3xC9
C33
2C3xC9
C9:S3
3C3xC3:S3
C32:C9
C32:D9

Character table of C32:D9

 class 123A3B3C3D3E3F3G3H6A6B9A9B9C9D9E9F9G9H9I
 size 127222233662727666666666
ρ1111111111111111111111    trivial
ρ21-111111111-1-1111111111    linear of order 2
ρ31-11111ζ32ζ3ζ32ζ3ζ6ζ65ζ311ζ32ζ32ζ321ζ3ζ3    linear of order 6
ρ4111111ζ32ζ3ζ32ζ3ζ32ζ3ζ311ζ32ζ32ζ321ζ3ζ3    linear of order 3
ρ51-11111ζ3ζ32ζ3ζ32ζ65ζ6ζ3211ζ3ζ3ζ31ζ32ζ32    linear of order 6
ρ6111111ζ3ζ32ζ3ζ32ζ3ζ32ζ3211ζ3ζ3ζ31ζ32ζ32    linear of order 3
ρ7202222222200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ8202-1-1-122-1-100ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ9202-1-1-122-1-100ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ10202-1-1-122-1-100ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ11202222-1+√-3-1-√-3-1+√-3-1-√-300ζ6-1-1ζ65ζ65ζ65-1ζ6ζ6    complex lifted from C3xS3
ρ12202-1-1-1-1-√-3-1+√-3ζ6ζ6500ζ95+ζ9ζ97+ζ92ζ95+ζ94ζ92+ζ9ζ97+ζ95ζ98+ζ94ζ98+ζ9ζ98+ζ97ζ94+ζ92    complex lifted from C3xD9
ρ13202222-1-√-3-1+√-3-1-√-3-1+√-300ζ65-1-1ζ6ζ6ζ6-1ζ65ζ65    complex lifted from C3xS3
ρ14202-1-1-1-1+√-3-1-√-3ζ65ζ600ζ98+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ97ζ94+ζ92ζ95+ζ9ζ98+ζ9ζ92+ζ9ζ97+ζ95    complex lifted from C3xD9
ρ15202-1-1-1-1+√-3-1-√-3ζ65ζ600ζ97+ζ95ζ98+ζ9ζ97+ζ92ζ95+ζ9ζ98+ζ97ζ94+ζ92ζ95+ζ94ζ98+ζ94ζ92+ζ9    complex lifted from C3xD9
ρ16202-1-1-1-1-√-3-1+√-3ζ6ζ6500ζ98+ζ97ζ95+ζ94ζ98+ζ9ζ97+ζ95ζ98+ζ94ζ92+ζ9ζ97+ζ92ζ94+ζ92ζ95+ζ9    complex lifted from C3xD9
ρ17202-1-1-1-1-√-3-1+√-3ζ6ζ6500ζ94+ζ92ζ98+ζ9ζ97+ζ92ζ98+ζ94ζ92+ζ9ζ97+ζ95ζ95+ζ94ζ95+ζ9ζ98+ζ97    complex lifted from C3xD9
ρ18202-1-1-1-1+√-3-1-√-3ζ65ζ600ζ92+ζ9ζ95+ζ94ζ98+ζ9ζ94+ζ92ζ95+ζ9ζ98+ζ97ζ97+ζ92ζ97+ζ95ζ98+ζ94    complex lifted from C3xD9
ρ1960-36-3-3000000000000000    orthogonal lifted from C9:C6
ρ2060-3-3-36000000000000000    orthogonal lifted from C32:C6
ρ2160-3-36-3000000000000000    orthogonal lifted from C9:C6

Permutation representations of C32:D9
►On 27 points - transitive group 27T59
Generators in S27
(2 26 10)(3 11 27)(5 20 13)(6 14 21)(8 23 16)(9 17 24)

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

(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 23)(11 22)(12 21)(13 20)(14 19)(15 27)(16 26)(17 25)(18 24)
 
G:=sub<Sym(27)| (2,26,10)(3,11,27)(5,20,13)(6,14,21)(8,23,16)(9,17,24), (1,25,18)(2,26,10)(3,27,11)(4,19,12)(5,20,13)(6,21,14)(7,22,15)(8,23,16)(9,24,17), (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)(10,23)(11,22)(12,21)(13,20)(14,19)(15,27)(16,26)(17,25)(18,24)>;
 
G:=Group( (2,26,10)(3,11,27)(5,20,13)(6,14,21)(8,23,16)(9,17,24), (1,25,18)(2,26,10)(3,27,11)(4,19,12)(5,20,13)(6,21,14)(7,22,15)(8,23,16)(9,24,17), (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)(10,23)(11,22)(12,21)(13,20)(14,19)(15,27)(16,26)(17,25)(18,24) );
 
G=PermutationGroup([[(2,26,10),(3,11,27),(5,20,13),(6,14,21),(8,23,16),(9,17,24)], [(1,25,18),(2,26,10),(3,27,11),(4,19,12),(5,20,13),(6,21,14),(7,22,15),(8,23,16),(9,24,17)], [(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),(10,23),(11,22),(12,21),(13,20),(14,19),(15,27),(16,26),(17,25),(18,24)]])
 
G:=TransitiveGroup(27,59);
 

C32:D9 is a maximal subgroup of
 C32:D18  (C3xHe3):S3  (C3xHe3).S3  C33.(C3:S3)  C32:C9:6S3  C3.(C33:S3)  C3.(He3:S3)  C32:C9.10S3  C33:2D9  (C3xC9):5D9  (C3xC9):6D9  He3:2D9  3- 1+2:D9  C34.S3  C33:D9  C92:3C6  He3:3D9  C92:9C6  C9:He3:2C2  (C32xC9):C6  C92:10C6  C92:4C6  C92:5C6  C92:11C6
C32:D9 is a maximal quotient of
 C32:Dic9  C9:S3:C9  C32:D27  C33:1D9  (C3xC9):D9  (C3xC9):3D9  He3:D9  He3.D9  He3.2D9  C33:D9

Matrix representation of C32:D9 ►in GL8(F19)

70000000
07000000
00100000
00010000
0000181800
00001000
00000001
0000001818
,
10000000
01000000
00010000
0018180000
00000100
0000181800
00000001
0000001818
,
214000000
57000000
00001000
00000100
00000010
00000001
0018180000
00100000
,
214000000
1217000000
00001000
0000181800
00100000
0018180000
00000001
00000010

G:=sub<GL(8,GF(19))| [7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[2,5,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[2,12,0,0,0,0,0,0,14,17,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C32:D9 in GAP, Magma, Sage, TeX 

C_3^2\rtimes D_9
 
% in TeX
 
G:=Group("C3^2:D9");
 
// GroupNames label
 
G:=SmallGroup(162,5);
 
// by ID
 
G=gap.SmallGroup(162,5);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-3,-3,992,187,282,723,2704]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^9=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32:D9 in TeX
Character table of C32:D9 in TeX

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