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G = C32×C32⋊C4  order 324 = 22·34

Direct product of C32 and C32⋊C4

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

Aliases: C32×C32⋊C4, C34⋊1C4, C33⋊5C12, C32⋊2(C3×C12), C3⋊S3.(C3×C6), (C3×C3⋊S3).4C6, (C32×C3⋊S3).1C2, SmallGroup(324,161)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C32×C32⋊C4
Chief series
C1 — C32 — C3⋊S3 — C3×C3⋊S3 — C32×C3⋊S3 — C32×C32⋊C4
Lower central
C32 — C32×C32⋊C4
Upper central
C1 — C32

Generators and relations for C32×C32⋊C4
 G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 404 in 104 conjugacy classes, 24 normal (8 characteristic)
C1, C2, C3, C3, C4, S3, C6, C32, C32, C12, C3×S3, C3⋊S3, C3×C6, C33, C33, C3×C12, C32⋊C4, S3×C32, C3×C3⋊S3, C34, C3×C32⋊C4, C32×C3⋊S3, C32×C32⋊C4
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, C3×C12, C32⋊C4, C3×C32⋊C4, C32×C32⋊C4

Smallest permutation representation of C32×C32⋊C4
►On 36 points
Generators in S36
(1 9 27)(2 10 28)(3 11 25)(4 12 26)(5 20 23)(6 17 24)(7 18 21)(8 19 22)(13 30 34)(14 31 35)(15 32 36)(16 29 33)

(1 7 16)(2 8 13)(3 5 14)(4 6 15)(9 18 29)(10 19 30)(11 20 31)(12 17 32)(21 33 27)(22 34 28)(23 35 25)(24 36 26)

(1 7 16)(2 19 34)(3 14 5)(4 36 17)(6 26 32)(8 30 28)(9 18 29)(10 22 13)(11 31 20)(12 15 24)(21 33 27)(23 25 35)

(1 29 21)(2 10 28)(3 23 31)(4 26 12)(5 35 11)(6 24 17)(7 9 33)(8 19 22)(13 30 34)(14 25 20)(15 36 32)(16 18 27)

(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)
 
G:=sub<Sym(36)| (1,9,27)(2,10,28)(3,11,25)(4,12,26)(5,20,23)(6,17,24)(7,18,21)(8,19,22)(13,30,34)(14,31,35)(15,32,36)(16,29,33), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (1,7,16)(2,19,34)(3,14,5)(4,36,17)(6,26,32)(8,30,28)(9,18,29)(10,22,13)(11,31,20)(12,15,24)(21,33,27)(23,25,35), (1,29,21)(2,10,28)(3,23,31)(4,26,12)(5,35,11)(6,24,17)(7,9,33)(8,19,22)(13,30,34)(14,25,20)(15,36,32)(16,18,27), (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)>;
 
G:=Group( (1,9,27)(2,10,28)(3,11,25)(4,12,26)(5,20,23)(6,17,24)(7,18,21)(8,19,22)(13,30,34)(14,31,35)(15,32,36)(16,29,33), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (1,7,16)(2,19,34)(3,14,5)(4,36,17)(6,26,32)(8,30,28)(9,18,29)(10,22,13)(11,31,20)(12,15,24)(21,33,27)(23,25,35), (1,29,21)(2,10,28)(3,23,31)(4,26,12)(5,35,11)(6,24,17)(7,9,33)(8,19,22)(13,30,34)(14,25,20)(15,36,32)(16,18,27), (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) );
 
G=PermutationGroup([[(1,9,27),(2,10,28),(3,11,25),(4,12,26),(5,20,23),(6,17,24),(7,18,21),(8,19,22),(13,30,34),(14,31,35),(15,32,36),(16,29,33)], [(1,7,16),(2,8,13),(3,5,14),(4,6,15),(9,18,29),(10,19,30),(11,20,31),(12,17,32),(21,33,27),(22,34,28),(23,35,25),(24,36,26)], [(1,7,16),(2,19,34),(3,14,5),(4,36,17),(6,26,32),(8,30,28),(9,18,29),(10,22,13),(11,31,20),(12,15,24),(21,33,27),(23,25,35)], [(1,29,21),(2,10,28),(3,23,31),(4,26,12),(5,35,11),(6,24,17),(7,9,33),(8,19,22),(13,30,34),(14,25,20),(15,36,32),(16,18,27)], [(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)]])
 

54 conjugacy classes

class 1  2 3A···3H3I···3Z4A4B6A···6H12A···12P
order123···33···3446···612···12
size191···14···4999···99···9

54 irreducible representations

dim11111144
type+++
imageC1C2C3C4C6C12C32⋊C4C3×C32⋊C4
kernelC32×C32⋊C4C32×C3⋊S3C3×C32⋊C4C34C3×C3⋊S3C33C32C3
# reps1182816216

Matrix representation of C32×C32⋊C4 ►in GL5(𝔽13)

10000
09000
00900
00090
00009
,
30000
09000
00900
00090
00009
,
10000
09406
00300
00090
00003
,
10000
01025
00100
00030
00009
,
50000
05000
00001
00100
08158

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,9,0,0,0,0,4,3,0,0,0,0,0,9,0,0,6,0,0,3],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,2,0,3,0,0,5,0,0,9],[5,0,0,0,0,0,5,0,0,8,0,0,0,1,1,0,0,0,0,5,0,0,1,0,8] >;
 

C32×C32⋊C4 in GAP, Magma, Sage, TeX 

C_3^2\times C_3^2\rtimes C_4
 
% in TeX
 
G:=Group("C3^2xC3^2:C4");
 
// GroupNames label
 
G:=SmallGroup(324,161);
 
// by ID
 
G=gap.SmallGroup(324,161);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-2,-3,3,108,7564,142,10373,455]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^4=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,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
 
// generators/relations
 

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