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G = C3.C3≀S3  order 486 = 2·35

1st non-split extension by C3 of C3≀S3 acting via C3≀S3/C3≀C3=C2

non-abelian, supersoluble, monomial

Aliases: C3.1C3≀S3, (C3×He3)⋊1S3, (C3×He3)⋊1C6, C33.1(C3×S3), He3⋊5S3⋊1C3, C3.3(C33⋊C6), C32.24He3⋊1C2, C32.25(C32⋊C6), SmallGroup(486,4)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C3×He3 — C3.C3≀S3
Chief series
C1 — C3 — C32 — C33 — C3×He3 — C32.24He3 — C3.C3≀S3
Lower central
C3×He3 — C3.C3≀S3
Upper central
C1 — C3

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

Subgroups: 704 in 73 conjugacy classes, 10 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C3×S3, C3⋊S3, C3×C6, C3×C9, He3, C33, C33, C32⋊C6, He3⋊C2, S3×C32, C3×C3⋊S3, C32⋊C9, C3×He3, C3×C32⋊C6, He3⋊5S3, C32.24He3, C3.C3≀S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C32⋊C6, C3≀S3, C33⋊C6, C3.C3≀S3

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

(2 37 30)(3 25 38)(4 26 39)(5 40 27)(8 13 31)(9 32 14)(10 33 15)(11 16 34)(19 50 44)(20 45 51)(21 46 52)(22 53 47)

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

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

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

31 conjugacy classes

class 1  2 3A3B3C3D3E3F···3K3L3M3N3O6A···6H9A···9F
order12333333···333336···69···9
size127112229···91818181827···2718···18

31 irreducible representations

dim1111223666
type+++++
imageC1C2C3C6S3C3×S3C3≀S3C32⋊C6C33⋊C6C3.C3≀S3
kernelC3.C3≀S3C32.24He3He3⋊5S3C3×He3C3×He3C33C3C32C3C1
# reps11221212136

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

700000
070000
007000
0001100
0000110
0000011
,
100000
0110000
007000
000700
0000110
000001
,
700000
070000
007000
000700
000070
000007
,
001000
100000
010000
000010
000001
000100
,
0000017
0001700
000050
0017000
1700000
050000

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

C3.C3≀S3 in GAP, Magma, Sage, TeX 

C_3.C_3\wr S_3
 
% in TeX
 
G:=Group("C3.C3wrS3");
 
// GroupNames label
 
G:=SmallGroup(486,4);
 
// by ID
 
G=gap.SmallGroup(486,4);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,224,4755,873,735,3244]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=a*b^-1*c,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*d^-1>;
 
// generators/relations
 

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