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G = C32×C32⋊C6  order 486 = 2·35

Direct product of C32 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C32×C32⋊C6, C34⋊8S3, C34⋊6C6, C3⋊S3⋊C33, He3⋊5(C3×C6), C33⋊8(C3×C6), C33⋊9(C3×S3), C32⋊(C32×C6), (C3×He3)⋊20C6, C3.2(S3×C33), (C32×He3)⋊1C2, C32⋊1(S3×C32), (C3×C3⋊S3)⋊C32, (C32×C3⋊S3)⋊2C3, SmallGroup(486,222)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C32×C32⋊C6
Chief series
C1 — C3 — C32 — C33 — C34 — C32×He3 — C32×C32⋊C6
Lower central
C32 — C32×C32⋊C6
Upper central
C1 — C32

Generators and relations for C32×C32⋊C6
 G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 1300 in 360 conjugacy classes, 90 normal (11 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, C3×C6, He3, He3, C33, C33, C33, C32⋊C6, S3×C32, C3×C3⋊S3, C32×C6, C3×He3, C3×He3, C34, C34, C3×C32⋊C6, S3×C33, C32×C3⋊S3, C32×He3, C32×C32⋊C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C33, C32⋊C6, S3×C32, C32×C6, C3×C32⋊C6, S3×C33, C32×C32⋊C6

Smallest permutation representation of C32×C32⋊C6
►On 54 points
Generators in S54
(1 24 25)(2 19 26)(3 20 27)(4 21 28)(5 22 29)(6 23 30)(7 15 46)(8 16 47)(9 17 48)(10 18 43)(11 13 44)(12 14 45)(31 50 37)(32 51 38)(33 52 39)(34 53 40)(35 54 41)(36 49 42)

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

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

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

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

90 conjugacy classes

class 1  2 3A···3H3I···3Q3R···3AI3AJ···3BJ6A···6Z
order123···33···33···33···36···6
size191···12···23···36···69···9

90 irreducible representations

dim1111112266
type++++
imageC1C2C3C3C6C6S3C3×S3C32⋊C6C3×C32⋊C6
kernelC32×C32⋊C6C32×He3C3×C32⋊C6C32×C3⋊S3C3×He3C34C34C33C32C3
# reps1124224212618

Matrix representation of C32×C32⋊C6 ►in GL8(𝔽7)

20000000
02000000
00200000
00020000
00002000
00000200
00000020
00000002
,
20000000
02000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
02000000
00450000
00034000
00030000
00000206
00000005
00000025
,
10000000
01000000
00200000
00020000
00002000
00000400
00000040
00000004
,
04000000
40000000
00000405
00000603
00000423
00405000
00603000
00423000

G:=sub<GL(8,GF(7))| [2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,3,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,5,5],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,6,4,0,0,0,0,0,0,0,2,0,0,0,0,0,5,3,3,0,0,4,6,4,0,0,0,0,0,0,0,2,0,0,0,0,0,5,3,3,0,0,0] >;
 

C32×C32⋊C6 in GAP, Magma, Sage, TeX 

C_3^2\times C_3^2\rtimes C_6
 
% in TeX
 
G:=Group("C3^2xC3^2:C6");
 
// GroupNames label
 
G:=SmallGroup(486,222);
 
// by ID
 
G=gap.SmallGroup(486,222);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3244,3250,11669]);
 
// 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,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1*d^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

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