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G = C3⋊S3×C3×C9  order 486 = 2·35

Direct product of C3×C9 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3⋊S3×C3×C9, C33⋊8C18, C34.13C6, (C33×C9)⋊2C2, C32⋊5(S3×C9), (C32×C9)⋊27S3, C32⋊6(C3×C18), (C32×C9)⋊38C6, C33.53(C3×C6), C33.83(C3×S3), C32.46(S3×C32), C3⋊(S3×C3×C9), (C3×C9)⋊29(C3×S3), C3.7(C32×C3⋊S3), (C32×C3⋊S3).4C3, (C3×C3⋊S3).6C32, C32.56(C3×C3⋊S3), SmallGroup(486,228)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊S3×C3×C9
Chief series
C1 — C3 — C32 — C33 — C32×C9 — C33×C9 — C3⋊S3×C3×C9
Lower central
C32 — C3⋊S3×C3×C9
Upper central
C1 — C3×C9

Generators and relations for C3⋊S3×C3×C9
 G = < a,b,c,d,e | a3=b9=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 616 in 288 conjugacy classes, 70 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, C33, C33, S3×C9, C3×C18, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C32×C9, C32×C9, C34, S3×C3×C9, C9×C3⋊S3, C32×C3⋊S3, C33×C9, C3⋊S3×C3×C9
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3⋊S3, C3×C6, C3×C9, S3×C9, C3×C18, S3×C32, C3×C3⋊S3, S3×C3×C9, C9×C3⋊S3, C32×C3⋊S3, C3⋊S3×C3×C9

Smallest permutation representation of C3⋊S3×C3×C9
►On 54 points
Generators in S54
(1 36 42)(2 28 43)(3 29 44)(4 30 45)(5 31 37)(6 32 38)(7 33 39)(8 34 40)(9 35 41)(10 25 49)(11 26 50)(12 27 51)(13 19 52)(14 20 53)(15 21 54)(16 22 46)(17 23 47)(18 24 48)

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

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

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

162 conjugacy classes

class 1  2 3A···3H3I···3AR6A···6H9A···9R9S···9CL18A···18R
order123···33···36···69···99···918···18
size191···12···29···91···12···29···9

162 irreducible representations

dim111111112222
type+++
imageC1C2C3C3C6C6C9C18S3C3×S3C3×S3S3×C9
kernelC3⋊S3×C3×C9C33×C9C9×C3⋊S3C32×C3⋊S3C32×C9C34C3×C3⋊S3C33C32×C9C3×C9C33C32
# reps1162621818424872

Matrix representation of C3⋊S3×C3×C9 ►in GL4(𝔽19) generated by

11000
01100
0070
0007
,
6000
0600
0040
0004
,
11000
0700
0070
00011
,
11000
0700
0010
0001
,
0100
1000
0001
0010
G:=sub<GL(4,GF(19))| [11,0,0,0,0,11,0,0,0,0,7,0,0,0,0,7],[6,0,0,0,0,6,0,0,0,0,4,0,0,0,0,4],[11,0,0,0,0,7,0,0,0,0,7,0,0,0,0,11],[11,0,0,0,0,7,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
 

C3⋊S3×C3×C9 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3\times C_3\times C_9
 
% in TeX
 
G:=Group("C3:S3xC3xC9");
 
// GroupNames label
 
G:=SmallGroup(486,228);
 
// by ID
 
G=gap.SmallGroup(486,228);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^9=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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