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G = C32:C18  order 162 = 2·34

The semidirect product of C32 and C18 acting via C18/C3=C6

metabelian, supersoluble, monomial

Aliases: C32:C18, C33.1C6, C3:S3:C9, (C3xC9):1S3, C3.2(S3xC9), C32:C9:1C2, C3.5(C32:C6), C32.12(C3xS3), (C3xC3:S3).C3, SmallGroup(162,4)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C32:C18
Chief series
C1 — C3 — C32 — C33 — C32:C9 — C32:C18
Lower central
C32 — C32:C18
Upper central
C1 — C3

Generators and relations for C32:C18
 G = < a,b,c | a3=b3=c18=1, ab=ba, cac-1=a-1b-1, cbc-1=b-1 >

Subgroups: 98 in 28 conjugacy classes, 11 normal (all characteristic)
Quotients: C1, C2, C3, S3, C6, C9, C18, C3xS3, S3xC9, C32:C6, C32:C18
C1
9C2
C3
C3
2C3
3C3
6C3
3S3
9C6
9S3
C32
C32
2C32
3C32
3C9
6C9
6C32
C3:S3
3C3xS3
9C3xS3
9C18
C3xC9
C33
2C3xC9
C3xC3:S3
3S3xC9
C32:C9
C32:C18

Character table of C32:C18

 class 123A3B3C3D3E3F3G3H6A6B9A9B9C9D9E9F9G9H9I9J9K9L18A18B18C18D18E18F
 size 191122266699333333666666999999
ρ1111111111111111111111111111111    trivial
ρ21-111111111-1-1111111111111-1-1-1-1-1-1    linear of order 2
ρ3111111111111ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ32ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ41-111111111-1-1ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ3ζ32ζ3ζ32ζ65ζ6ζ6ζ65ζ65ζ6    linear of order 6
ρ51-111111111-1-1ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ32ζ3ζ32ζ3ζ6ζ65ζ65ζ6ζ6ζ65    linear of order 6
ρ6111111111111ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ3ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ71-1ζ32ζ31ζ3ζ32ζ3ζ321ζ6ζ65ζ95ζ98ζ92ζ9ζ94ζ97ζ98ζ94ζ92ζ9ζ95ζ97-ζ92-ζ9-ζ94-ζ95-ζ98-ζ97    linear of order 18
ρ811ζ3ζ321ζ32ζ3ζ32ζ31ζ3ζ32ζ97ζ94ζ9ζ95ζ92ζ98ζ94ζ92ζ9ζ95ζ97ζ98ζ9ζ95ζ92ζ97ζ94ζ98    linear of order 9
ρ91-1ζ3ζ321ζ32ζ3ζ32ζ31ζ65ζ6ζ97ζ94ζ9ζ95ζ92ζ98ζ94ζ92ζ9ζ95ζ97ζ98-ζ9-ζ95-ζ92-ζ97-ζ94-ζ98    linear of order 18
ρ101-1ζ32ζ31ζ3ζ32ζ3ζ321ζ6ζ65ζ92ζ95ζ98ζ94ζ97ζ9ζ95ζ97ζ98ζ94ζ92ζ9-ζ98-ζ94-ζ97-ζ92-ζ95-ζ9    linear of order 18
ρ111-1ζ3ζ321ζ32ζ3ζ32ζ31ζ65ζ6ζ94ζ9ζ97ζ98ζ95ζ92ζ9ζ95ζ97ζ98ζ94ζ92-ζ97-ζ98-ζ95-ζ94-ζ9-ζ92    linear of order 18
ρ1211ζ32ζ31ζ3ζ32ζ3ζ321ζ32ζ3ζ95ζ98ζ92ζ9ζ94ζ97ζ98ζ94ζ92ζ9ζ95ζ97ζ92ζ9ζ94ζ95ζ98ζ97    linear of order 9
ρ1311ζ3ζ321ζ32ζ3ζ32ζ31ζ3ζ32ζ94ζ9ζ97ζ98ζ95ζ92ζ9ζ95ζ97ζ98ζ94ζ92ζ97ζ98ζ95ζ94ζ9ζ92    linear of order 9
ρ141-1ζ3ζ321ζ32ζ3ζ32ζ31ζ65ζ6ζ9ζ97ζ94ζ92ζ98ζ95ζ97ζ98ζ94ζ92ζ9ζ95-ζ94-ζ92-ζ98-ζ9-ζ97-ζ95    linear of order 18
ρ1511ζ32ζ31ζ3ζ32ζ3ζ321ζ32ζ3ζ98ζ92ζ95ζ97ζ9ζ94ζ92ζ9ζ95ζ97ζ98ζ94ζ95ζ97ζ9ζ98ζ92ζ94    linear of order 9
ρ161-1ζ32ζ31ζ3ζ32ζ3ζ321ζ6ζ65ζ98ζ92ζ95ζ97ζ9ζ94ζ92ζ9ζ95ζ97ζ98ζ94-ζ95-ζ97-ζ9-ζ98-ζ92-ζ94    linear of order 18
ρ1711ζ32ζ31ζ3ζ32ζ3ζ321ζ32ζ3ζ92ζ95ζ98ζ94ζ97ζ9ζ95ζ97ζ98ζ94ζ92ζ9ζ98ζ94ζ97ζ92ζ95ζ9    linear of order 9
ρ1811ζ3ζ321ζ32ζ3ζ32ζ31ζ3ζ32ζ9ζ97ζ94ζ92ζ98ζ95ζ97ζ98ζ94ζ92ζ9ζ95ζ94ζ92ζ98ζ9ζ97ζ95    linear of order 9
ρ192022222-1-1-100222222-1-1-1-1-1-1000000    orthogonal lifted from S3
ρ202022222-1-1-100-1-√-3-1-√-3-1-√-3-1+√-3-1+√-3-1+√-3ζ6ζ65ζ6ζ65ζ6ζ65000000    complex lifted from C3xS3
ρ212022222-1-1-100-1+√-3-1+√-3-1+√-3-1-√-3-1-√-3-1-√-3ζ65ζ6ζ65ζ6ζ65ζ6000000    complex lifted from C3xS3
ρ2220-1+√-3-1-√-32-1-√-3-1+√-3ζ6ζ65-1002ζ92ζ972ζ942ζ922ζ982ζ95-ζ97-ζ98-ζ94-ζ92-ζ9-ζ95000000    complex lifted from S3xC9
ρ2320-1-√-3-1+√-32-1+√-3-1-√-3ζ65ζ6-1002ζ982ζ922ζ952ζ972ζ92ζ94-ζ92-ζ9-ζ95-ζ97-ζ98-ζ94000000    complex lifted from S3xC9
ρ2420-1+√-3-1-√-32-1-√-3-1+√-3ζ6ζ65-1002ζ972ζ942ζ92ζ952ζ922ζ98-ζ94-ζ92-ζ9-ζ95-ζ97-ζ98000000    complex lifted from S3xC9
ρ2520-1+√-3-1-√-32-1-√-3-1+√-3ζ6ζ65-1002ζ942ζ92ζ972ζ982ζ952ζ92-ζ9-ζ95-ζ97-ζ98-ζ94-ζ92000000    complex lifted from S3xC9
ρ2620-1-√-3-1+√-32-1+√-3-1-√-3ζ65ζ6-1002ζ952ζ982ζ922ζ92ζ942ζ97-ζ98-ζ94-ζ92-ζ9-ζ95-ζ97000000    complex lifted from S3xC9
ρ2720-1-√-3-1+√-32-1+√-3-1-√-3ζ65ζ6-1002ζ922ζ952ζ982ζ942ζ972ζ9-ζ95-ζ97-ζ98-ζ94-ζ92-ζ9000000    complex lifted from S3xC9
ρ286066-3-3-300000000000000000000000    orthogonal lifted from C32:C6
ρ2960-3-3√-3-3+3√-3-33-3√-3/23+3√-3/200000000000000000000000    complex faithful
ρ3060-3+3√-3-3-3√-3-33+3√-3/23-3√-3/200000000000000000000000    complex faithful

Permutation representations of C32:C18
►On 18 points - transitive group 18T82
Generators in S18
(2 8 14)(3 9 15)(5 17 11)(6 18 12)

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

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)
 
G:=sub<Sym(18)| (2,8,14)(3,9,15)(5,17,11)(6,18,12), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)>;
 
G:=Group( (2,8,14)(3,9,15)(5,17,11)(6,18,12), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18) );
 
G=PermutationGroup([[(2,8,14),(3,9,15),(5,17,11),(6,18,12)], [(1,13,7),(2,8,14),(3,15,9),(4,10,16),(5,17,11),(6,12,18)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)]])
 
G:=TransitiveGroup(18,82);
 

►On 27 points - transitive group 27T47
Generators in S27
(1 16 25)(3 27 18)(4 10 19)(6 21 12)(7 22 13)(9 15 24)

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

(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)
 
G:=sub<Sym(27)| (1,16,25)(3,27,18)(4,10,19)(6,21,12)(7,22,13)(9,15,24), (1,25,16)(2,17,26)(3,27,18)(4,19,10)(5,11,20)(6,21,12)(7,13,22)(8,23,14)(9,15,24), (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)>;
 
G:=Group( (1,16,25)(3,27,18)(4,10,19)(6,21,12)(7,22,13)(9,15,24), (1,25,16)(2,17,26)(3,27,18)(4,19,10)(5,11,20)(6,21,12)(7,13,22)(8,23,14)(9,15,24), (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) );
 
G=PermutationGroup([[(1,16,25),(3,27,18),(4,10,19),(6,21,12),(7,22,13),(9,15,24)], [(1,25,16),(2,17,26),(3,27,18),(4,19,10),(5,11,20),(6,21,12),(7,13,22),(8,23,14),(9,15,24)], [(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)]])
 
G:=TransitiveGroup(27,47);
 

C32:C18 is a maximal subgroup of
 C32:D18  C32:C9.S3  C32:C9:S3  (C3xHe3).C6  C32:C9.C6  C33.(C3xS3)  C32:2D9.C3  C33:1C18  (C3xC9):C18  C9:S3:3C9  C9xC32:C6  C34.C6  C9:He3:C2  C33:C18  C92:3S3  (C32xC9):8S3  C92:6S3  C92:5S3
C32:C18 is a maximal quotient of
 C32:C36  C9:S3:C9  C32:C54  C33:1C18  (C3xC9):C18  C9:S3:3C9  He3:C18  He3.C18  He3.2C18  C33:C18

Matrix representation of C32:C18 ►in GL6(F19)

100000
070000
0011000
000100
0000110
12071217
,
700000
070000
007000
0001100
0000110
818120011
,
1812818129
000700
000070
0011000
700000
1812018121

G:=sub<GL(6,GF(19))| [1,0,0,0,0,12,0,7,0,0,0,0,0,0,11,0,0,7,0,0,0,1,0,12,0,0,0,0,11,1,0,0,0,0,0,7],[7,0,0,0,0,8,0,7,0,0,0,18,0,0,7,0,0,12,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[18,0,0,0,7,18,12,0,0,0,0,12,8,0,0,11,0,0,18,7,0,0,0,18,12,0,7,0,0,12,9,0,0,0,0,1] >;
 

C32:C18 in GAP, Magma, Sage, TeX 

C_3^2\rtimes C_{18}
 
% in TeX
 
G:=Group("C3^2:C18");
 
// GroupNames label
 
G:=SmallGroup(162,4);
 
// by ID
 
G=gap.SmallGroup(162,4);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-3,-3,36,723,728,2704]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^3=c^18=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32:C18 in TeX
Character table of C32:C18 in TeX

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