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G = C27⋊C18  order 486 = 2·35

The semidirect product of C27 and C18 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C27⋊C18, D27⋊C9, C27⋊C9⋊C2, C9.(S3×C9), C27⋊C3.C6, C27⋊C6.C3, C9⋊C9.1S3, C3.2(C9⋊C18), C32.5(C9⋊C6), (C3×C9).1(C3×S3), Aut(D27), Hol(C27), SmallGroup(486,31)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C27 — C27⋊C18
Chief series
C1 — C3 — C9 — C27 — C27⋊C3 — C27⋊C9 — C27⋊C18
Lower central
C27 — C27⋊C18
Upper central
C1

Generators and relations for C27⋊C18
 G = < a,b | a27=b18=1, bab-1=a11 >

C1
27C2
C3
3C3
9S3
27C6
C9
C32
2C9
9C9
3D9
9C3×S3
27C18
C27
C3×C9
2C27
3C3×C9
6C27
D27
3C3×D9
9S3×C9
C9⋊C9
C27⋊C3
2C27⋊C3
C27⋊C6
3C9⋊C18
C27⋊C9
C27⋊C18

Permutation representations of C27⋊C18
►On 27 points - transitive group 27T176
Generators in S27
(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)

(2 6 26 18 5 21 20 15 17 27 23 3 11 24 8 9 14 12)(4 16 22 25 13 7)(10 19)
 
G:=sub<Sym(27)| (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), (2,6,26,18,5,21,20,15,17,27,23,3,11,24,8,9,14,12)(4,16,22,25,13,7)(10,19)>;
 
G:=Group( (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), (2,6,26,18,5,21,20,15,17,27,23,3,11,24,8,9,14,12)(4,16,22,25,13,7)(10,19) );
 
G=PermutationGroup([[(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)], [(2,6,26,18,5,21,20,15,17,27,23,3,11,24,8,9,14,12),(4,16,22,25,13,7),(10,19)]])
 
G:=TransitiveGroup(27,176);
 

31 conjugacy classes

class 1  2 3A3B3C6A6B9A9B9C9D···9I18A···18F27A···27I
order12333669999···918···1827···27
size12723327276669···927···2718···18

31 irreducible representations

dim1111111822266
type+++++
imageC1C2C3C6C9C18C27⋊C18S3C3×S3S3×C9C9⋊C6C9⋊C18
kernelC27⋊C18C27⋊C9C27⋊C6C27⋊C3D27C27C1C9⋊C9C3×C9C9C32C3
# reps112266112612

Matrix representation of C27⋊C18 ►in GL18(ℤ)

000000001000000000
000000000100000000
000000000010000000
000000000001000000
000000010000000000
000000-1-10000000000
000000000000001000
000000000000000100
000000000000000010
000000000000000001
000000000000010000
000000000000-1-10000
100000000000000000
010000000000000000
001000000000000000
000100000000000000
000010000000000000
000001000000000000
,
100000000000000000
-1-10000000000000000
000010000000000000
0000-1-1000000000000
000100000000000000
001000000000000000
000000000000-1-10000
000000000000010000
0000000000000000-1-1
000000000000000001
000000000000001000
00000000000000-1-100
000000000010000000
0000000000-1-1000000
000000000100000000
000000001000000000
000000-1-10000000000
000000010000000000

G:=sub<GL(18,Integers())| [0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0] >;
 

C27⋊C18 in GAP, Magma, Sage, TeX 

C_{27}\rtimes C_{18}
 
% in TeX
 
G:=Group("C27:C18");
 
// GroupNames label
 
G:=SmallGroup(486,31);
 
// by ID
 
G=gap.SmallGroup(486,31);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,2163,873,735,597,12964,5680,118,11669]);
 
// Polycyclic
 
G:=Group<a,b|a^27=b^18=1,b*a*b^-1=a^11>;
 
// generators/relations
 

Export

Subgroup lattice of C27⋊C18 in TeX

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