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G = C34.C3  order 243 = 35

3rd non-split extension by C34 of C3 acting faithfully

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C34.3C3, C32.7He3, C32.21C33, C33.27C32, C32⋊23- 1+2, C32⋊C9⋊6C3, (C3×C9)⋊1C32, C3.4(C3×He3), (C3×3- 1+2)⋊2C3, C3.4(C3×3- 1+2), SmallGroup(243,38)

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

Derived series
C1 — C32 — C34.C3
Chief series
C1 — C3 — C32 — C33 — C34 — C34.C3
Lower central
C1 — C32 — C34.C3
Upper central
C1 — C32 — C34.C3
Jennings
C1 — C3 — C32 — C34.C3

Generators and relations for C34.C3
 G = < a,b,c,d,e | a3=b3=c3=d3=1, e3=c, ab=ba, ac=ca, ad=da, eae-1=ac-1, bc=cb, bd=db, ebe-1=bd-1, cd=dc, ce=ec, de=ed >

Subgroups: 288 in 126 conjugacy classes, 45 normal (6 characteristic)
C1, C3, C3, C3, C9, C32, C32, C32, C3×C9, 3- 1+2, C33, C33, C33, C32⋊C9, C3×3- 1+2, C34, C34.C3
Quotients: C1, C3, C32, He3, 3- 1+2, C33, C3×He3, C3×3- 1+2, C34.C3

Permutation representations of C34.C3
►On 27 points - transitive group 27T110
Generators in S27
(1 27 11)(2 25 15)(3 23 10)(4 21 14)(5 19 18)(6 26 13)(7 24 17)(8 22 12)(9 20 16)

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

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

(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)
 
G:=sub<Sym(27)| (1,27,11)(2,25,15)(3,23,10)(4,21,14)(5,19,18)(6,26,13)(7,24,17)(8,22,12)(9,20,16), (1,4,7)(2,18,22)(3,20,13)(5,12,25)(6,23,16)(8,15,19)(9,26,10)(11,14,17)(21,24,27), (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), (1,24,14)(2,25,15)(3,26,16)(4,27,17)(5,19,18)(6,20,10)(7,21,11)(8,22,12)(9,23,13), (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)>;
 
G:=Group( (1,27,11)(2,25,15)(3,23,10)(4,21,14)(5,19,18)(6,26,13)(7,24,17)(8,22,12)(9,20,16), (1,4,7)(2,18,22)(3,20,13)(5,12,25)(6,23,16)(8,15,19)(9,26,10)(11,14,17)(21,24,27), (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), (1,24,14)(2,25,15)(3,26,16)(4,27,17)(5,19,18)(6,20,10)(7,21,11)(8,22,12)(9,23,13), (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) );
 
G=PermutationGroup([[(1,27,11),(2,25,15),(3,23,10),(4,21,14),(5,19,18),(6,26,13),(7,24,17),(8,22,12),(9,20,16)], [(1,4,7),(2,18,22),(3,20,13),(5,12,25),(6,23,16),(8,15,19),(9,26,10),(11,14,17),(21,24,27)], [(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)], [(1,24,14),(2,25,15),(3,26,16),(4,27,17),(5,19,18),(6,20,10),(7,21,11),(8,22,12),(9,23,13)], [(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)]])
 
G:=TransitiveGroup(27,110);
 

C34.C3 is a maximal subgroup of   C34.C6  C34.S3  C34.7S3

51 conjugacy classes

class 1 3A···3H3I···3AF9A···9R
order13···33···39···9
size11···13···39···9

51 irreducible representations

dim111133
type+
imageC1C3C3C3He33- 1+2
kernelC34.C3C32⋊C9C3×3- 1+2C34C32C32
# reps11862618

Matrix representation of C34.C3 ►in GL6(𝔽19)

1100000
010000
1127000
000700
000070
000007
,
700000
070000
007000
000100
000070
0000011
,
700000
070000
007000
000100
000010
000001
,
100000
010000
001000
000700
000070
000007
,
010000
8126000
107000
000010
000001
000100

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

C34.C3 in GAP, Magma, Sage, TeX 

C_3^4.C_3
 
% in TeX
 
G:=Group("C3^4.C3");
 
// GroupNames label
 
G:=SmallGroup(243,38);
 
// by ID
 
G=gap.SmallGroup(243,38);
 
# by ID
 
G:=PCGroup([5,-3,3,3,-3,3,135,301,1352]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=1,e^3=c,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*c^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b*d^-1,c*d=d*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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