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G = C32.C33  order 243 = 35

9th non-split extension by C32 of C33 acting via C33/C32=C3

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C32.16He3, C32.9C33, C33.15C32, 3- 1+2.3C32, C3.15(C3×He3), C3.He3⋊4C3, (C3×C9).14C32, (C3×3- 1+2).9C3, SmallGroup(243,59)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C32 — C32.C33
Chief series
C1 — C3 — C32 — C33 — C3×3- 1+2 — C32.C33
Lower central
C1 — C3 — C32 — C32.C33
Upper central
C1 — C3 — C33 — C32.C33
Jennings
C1 — C3 — C32 — C32.C33

Generators and relations for C32.C33
 G = < a,b,c,d,e | a3=b3=e3=1, c3=b, d3=b-1, ab=ba, cac-1=ab-1, ad=da, ae=ea, bc=cb, ede-1=bd=db, be=eb, dcd-1=ab-1c, ce=ec >

Subgroups: 126 in 62 conjugacy classes, 33 normal (6 characteristic)
C1, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3.He3, C3×3- 1+2, C3×3- 1+2, C32.C33
Quotients: C1, C3, C32, He3, C33, C3×He3, C32.C33

Permutation representations of C32.C33
►On 27 points - transitive group 27T111
Generators in S27
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 22 25)(21 27 24)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 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)

(1 26 15 7 23 12 4 20 18)(2 21 13 8 27 10 5 24 16)(3 19 17 9 25 14 6 22 11)

(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)
 
G:=sub<Sym(27)| (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (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), (1,26,15,7,23,12,4,20,18)(2,21,13,8,27,10,5,24,16)(3,19,17,9,25,14,6,22,11), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)>;
 
G:=Group( (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (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), (1,26,15,7,23,12,4,20,18)(2,21,13,8,27,10,5,24,16)(3,19,17,9,25,14,6,22,11), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24) );
 
G=PermutationGroup([[(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,22,25),(21,27,24)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27)], [(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)], [(1,26,15,7,23,12,4,20,18),(2,21,13,8,27,10,5,24,16),(3,19,17,9,25,14,6,22,11)], [(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)]])
 
G:=TransitiveGroup(27,111);
 

C32.C33 is a maximal subgroup of   C3.He3⋊C6

35 conjugacy classes

class 1 3A3B3C···3J9A···9X
order1333···39···9
size1113···39···9

35 irreducible representations

dim11139
type+
imageC1C3C3He3C32.C33
kernelC32.C33C3.He3C3×3- 1+2C32C1
# reps118862

Matrix representation of C32.C33 ►in GL9(𝔽19)

100000000
010000000
001000000
000700000
000070000
000007000
0000001100
081112070110
118001270011
,
700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
000100000
000010000
000001000
000000100
01181801160
18100181106
700000000
18100000018
18010000118
,
010000000
001000000
1100000000
000010000
000001000
0001100000
07121207740
701121800127
01811201818120
,
100000000
070000000
0011000000
000100000
000070000
0000011000
000000100
00718012170
117001208011

G:=sub<GL(9,GF(19))| [1,0,0,0,0,0,0,0,11,0,1,0,0,0,0,0,8,8,0,0,1,0,0,0,0,11,0,0,0,0,7,0,0,0,12,0,0,0,0,0,7,0,0,0,12,0,0,0,0,0,7,0,7,7,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[0,0,0,0,0,18,7,18,18,0,0,0,0,1,1,0,1,0,0,0,0,0,18,0,0,0,1,1,0,0,0,18,0,0,0,0,0,1,0,0,0,18,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,6,0,0,0,1,0,0,0,0,0,6,0,18,18],[0,0,11,0,0,0,0,7,0,1,0,0,0,0,0,7,0,18,0,1,0,0,0,0,12,1,1,0,0,0,0,0,11,12,12,12,0,0,0,1,0,0,0,18,0,0,0,0,0,1,0,7,0,18,0,0,0,0,0,0,7,0,18,0,0,0,0,0,0,4,12,12,0,0,0,0,0,0,0,7,0],[1,0,0,0,0,0,0,0,11,0,7,0,0,0,0,0,0,7,0,0,11,0,0,0,0,7,0,0,0,0,1,0,0,0,18,0,0,0,0,0,7,0,0,0,12,0,0,0,0,0,11,0,12,0,0,0,0,0,0,0,1,1,8,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11] >;
 

C32.C33 in GAP, Magma, Sage, TeX 

C_3^2.C_3^3
 
% in TeX
 
G:=Group("C3^2.C3^3");
 
// GroupNames label
 
G:=SmallGroup(243,59);
 
// by ID
 
G=gap.SmallGroup(243,59);
 
# by ID
 
G:=PCGroup([5,-3,3,3,-3,-3,405,301,546,457,2163]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=e^3=1,c^3=b,d^3=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*e=e*a,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,d*c*d^-1=a*b^-1*c,c*e=e*c>;
 
// generators/relations
 

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