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G = C3≀C3.C6  order 486 = 2·35

The non-split extension by C3≀C3 of C6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C3≀C3.C6, C3≀S3⋊2C3, C9.2He3⋊C2, C9○He3⋊3C6, He3.C3⋊4C6, He3.7(C3×C6), He3⋊C3⋊4C6, C9.2(C32⋊C6), C33.19(C3×S3), He3.2C6⋊2C3, He3.C6⋊2C3, He3.4C6⋊2C3, C32.8(S3×C32), He3⋊C2.5C32, (C3×3- 1+2)⋊17S3, (C3×C9).13(C3×S3), C3.22(C3×C32⋊C6), SmallGroup(486,132)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — He3 — C3≀C3.C6
Chief series
C1 — C3 — C32 — He3 — C9○He3 — C9.2He3 — C3≀C3.C6
Lower central
He3 — C3≀C3.C6
Upper central
C1 — C3 — C9

Generators and relations for C3≀C3.C6
 G = < a,b,c,d,e | a3=b3=c3=d3=1, e6=b-1, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1b, bc=cb, ede-1=bd=db, be=eb, dcd-1=ab-1c, ece-1=c-1 >

Subgroups: 306 in 69 conjugacy classes, 21 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, S3×C9, He3⋊C2, C2×3- 1+2, S3×C32, C3≀C3, C3≀C3, He3.C3, He3.C3, He3⋊C3, C3.He3, C3×3- 1+2, C9○He3, C9○He3, C3≀S3, He3.C6, He3.2C6, S3×3- 1+2, He3.4C6, C9.2He3, C3≀C3.C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, C3≀C3.C6

Permutation representations of C3≀C3.C6
►On 27 points - transitive group 27T201
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(10 22 16)(12 24 18)(14 26 20)

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

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

(2 5 8)(3 9 6)(10 16 22)(12 24 18)(13 19 25)(15 27 21)

(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)| (1,7,4)(2,8,5)(3,9,6)(10,22,16)(12,24,18)(14,26,20), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,10,19)(2,20,11)(3,12,21)(4,22,13)(5,14,23)(6,24,15)(7,16,25)(8,26,17)(9,18,27), (2,5,8)(3,9,6)(10,16,22)(12,24,18)(13,19,25)(15,27,21), (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( (1,7,4)(2,8,5)(3,9,6)(10,22,16)(12,24,18)(14,26,20), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,10,19)(2,20,11)(3,12,21)(4,22,13)(5,14,23)(6,24,15)(7,16,25)(8,26,17)(9,18,27), (2,5,8)(3,9,6)(10,16,22)(12,24,18)(13,19,25)(15,27,21), (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([[(1,7,4),(2,8,5),(3,9,6),(10,22,16),(12,24,18),(14,26,20)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21)], [(1,10,19),(2,20,11),(3,12,21),(4,22,13),(5,14,23),(6,24,15),(7,16,25),(8,26,17),(9,18,27)], [(2,5,8),(3,9,6),(10,16,22),(12,24,18),(13,19,25),(15,27,21)], [(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(27,201);
 

34 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H6A6B6C6D9A9B9C9D9E9F9G9H9I···9N18A···18F
order12333333336666999999999···918···18
size19116991818189927273366999918···1827···27

34 irreducible representations

dim1111111111222669
type++++
imageC1C2C3C3C3C3C6C6C6C6S3C3×S3C3×S3C32⋊C6C3×C32⋊C6C3≀C3.C6
kernelC3≀C3.C6C9.2He3C3≀S3He3.C6He3.2C6He3.4C6C3≀C3He3.C3He3⋊C3C9○He3C3×3- 1+2C3×C9C33C9C3C1
# reps1122222222162124

Matrix representation of C3≀C3.C6 ►in GL9(𝔽19)

1100000000
070000000
1121000000
0001100000
000070000
1120001000
0000001100
000000070
1120000001
,
700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
121213000000
1100000000
877000000
0012007000
0001100000
877010000
0012000007
0000001100
877000010
,
100000000
0110000000
18011000000
1200700000
0180010000
010001000
1000001100
0110000070
780000007
,
11000001500
8800000015
000000111111
1100000800
887000008
011000011011
1100700800
880007008
011007011011

G:=sub<GL(9,GF(19))| [11,0,1,0,0,1,0,0,1,0,7,12,0,0,12,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[12,11,8,0,0,8,0,0,8,12,0,7,0,0,7,0,0,7,13,0,7,12,0,7,12,0,7,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0],[1,0,18,12,0,0,1,0,7,0,11,0,0,18,1,0,11,8,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[11,8,0,11,8,0,11,8,0,0,8,0,0,8,11,0,8,11,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,15,0,11,8,0,11,8,0,11,0,0,11,0,0,0,0,0,0,0,15,11,0,8,11,0,8,11] >;
 

C3≀C3.C6 in GAP, Magma, Sage, TeX 

C_3\wr C_3.C_6
 
% in TeX
 
G:=Group("C3wrC3.C6");
 
// GroupNames label
 
G:=SmallGroup(486,132);
 
// by ID
 
G=gap.SmallGroup(486,132);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,500,867,873,8104,382]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=1,e^6=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1*b,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,d*c*d^-1=a*b^-1*c,e*c*e^-1=c^-1>;
 
// generators/relations
 

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