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G = C3.3C3≀S3  order 486 = 2·35

3rd non-split extension by C3 of C3≀S3 acting via C3≀S3/C3≀C3=C2

non-abelian, supersoluble, monomial

Aliases: C3.3C3≀S3, C32⋊C9⋊2C6, (C3×He3).2S3, C33.5(C3×S3), C32⋊2D9⋊3C3, C32.27He3⋊2C2, C32.29(C32⋊C6), C3.6(He3.2S3), SmallGroup(486,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C32⋊C9 — C3.3C3≀S3
Chief series
C1 — C3 — C32 — C33 — C32⋊C9 — C32.27He3 — C3.3C3≀S3
Lower central
C32⋊C9 — C3.3C3≀S3
Upper central
C1 — C3

Generators and relations for C3.3C3≀S3
 G = < a,b,c,d,e | a3=b3=c3=e6=1, d3=eae-1=a-1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, ebe-1=a-1b-1c, cd=dc, ce=ec, ede-1=a-1b-1d2 >

C1
27C2
C3
C3
2C3
9C3
9C3
9C3
9C3
9S3
27S3
27C6
27C6
27C6
27C6
C32
3C32
3C32
3C32
3C32
3C32
6C32
9C32
9C9
18C32
18C9
3C3⋊S3
9C3×S3
9D9
9C3×S3
9C3×S3
9C3×S3
27C3×S3
27C3×C6
C33
3He3
3He3
3C3×C9
3C33
3He3
6C3×C9
3C32⋊C6
3C32⋊C6
3C3×C3⋊S3
3C32⋊C6
9C3×D9
9S3×C32
C32⋊C9
C3×He3
2C32⋊C9
C32⋊2D9
3C3×C32⋊C6
C32.27He3
C3.3C3≀S3

Smallest permutation representation of C3.3C3≀S3
►On 54 points
Generators in S54
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)

(1 7 4)(3 6 9)(10 16 13)(12 15 18)(20 23 26)(21 27 24)(29 35 32)(30 33 36)(37 40 43)(39 45 42)(46 52 49)(47 50 53)

(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)

(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)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)

(1 46 17 31 26 37)(2 48 15 30 24 39)(3 53 16 29 19 38)(4 52 11 28 20 43)(5 54 18 36 27 45)(6 50 10 35 22 44)(7 49 14 34 23 40)(8 51 12 33 21 42)(9 47 13 32 25 41)
 
G:=sub<Sym(54)| (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51), (1,7,4)(3,6,9)(10,16,13)(12,15,18)(20,23,26)(21,27,24)(29,35,32)(30,33,36)(37,40,43)(39,45,42)(46,52,49)(47,50,53), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,46,17,31,26,37)(2,48,15,30,24,39)(3,53,16,29,19,38)(4,52,11,28,20,43)(5,54,18,36,27,45)(6,50,10,35,22,44)(7,49,14,34,23,40)(8,51,12,33,21,42)(9,47,13,32,25,41)>;
 
G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51), (1,7,4)(3,6,9)(10,16,13)(12,15,18)(20,23,26)(21,27,24)(29,35,32)(30,33,36)(37,40,43)(39,45,42)(46,52,49)(47,50,53), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,46,17,31,26,37)(2,48,15,30,24,39)(3,53,16,29,19,38)(4,52,11,28,20,43)(5,54,18,36,27,45)(6,50,10,35,22,44)(7,49,14,34,23,40)(8,51,12,33,21,42)(9,47,13,32,25,41) );
 
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51)], [(1,7,4),(3,6,9),(10,16,13),(12,15,18),(20,23,26),(21,27,24),(29,35,32),(30,33,36),(37,40,43),(39,45,42),(46,52,49),(47,50,53)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(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),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,46,17,31,26,37),(2,48,15,30,24,39),(3,53,16,29,19,38),(4,52,11,28,20,43),(5,54,18,36,27,45),(6,50,10,35,22,44),(7,49,14,34,23,40),(8,51,12,33,21,42),(9,47,13,32,25,41)]])
 

31 conjugacy classes

class 1  2 3A3B3C3D3E3F···3K3L6A···6H9A···9I
order12333333···336···69···9
size127112229···91827···2718···18

31 irreducible representations

dim1111223666
type+++++
imageC1C2C3C6S3C3×S3C3≀S3C32⋊C6He3.2S3C3.3C3≀S3
kernelC3.3C3≀S3C32.27He3C32⋊2D9C32⋊C9C3×He3C33C3C32C3C1
# reps11221212136

Matrix representation of C3.3C3≀S3 ►in GL6(𝔽19)

1100000
0110000
0011000
000700
000070
000007
,
1100000
070000
001000
000100
000070
0000011
,
1100000
0110000
0011000
0001100
0000110
0000011
,
010000
007000
100000
000001
000100
0000110
,
0000017
000500
0000170
0017000
500000
0170000

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,0,0,5,0,0,0,0,0,0,17,0,0,0,17,0,0,0,5,0,0,0,0,0,0,17,0,0,0,17,0,0,0,0,0] >;
 

C3.3C3≀S3 in GAP, Magma, Sage, TeX 

C_3._3C_3\wr S_3
 
% in TeX
 
G:=Group("C3.3C3wrS3");
 
// GroupNames label
 
G:=SmallGroup(486,8);
 
// by ID
 
G=gap.SmallGroup(486,8);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,338,4755,873,735,3244]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^6=1,d^3=e*a*e^-1=a^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=a^-1*b^-1*c,c*d=d*c,c*e=e*c,e*d*e^-1=a^-1*b^-1*d^2>;
 
// generators/relations
 

Export

Subgroup lattice of C3.3C3≀S3 in TeX

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