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G = C34.C3order 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, C3223- 1+2, C32⋊C96C3, (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

C1C32 — C34.C3
C1C3C32C33C34 — C34.C3
C1C32 — C34.C3
C1C32 — C34.C3
C1C3C32 — 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 [×3], C3 [×12], C9 [×9], C32, C32 [×12], C32 [×39], C3×C9 [×9], 3- 1+2 [×9], C33, C33 [×3], C33 [×12], C32⋊C9 [×9], C3×3- 1+2 [×3], C34, C34.C3
Quotients: C1, C3 [×13], C32 [×13], He3 [×3], 3- 1+2 [×9], C33, C3×He3, C3×3- 1+2 [×3], C34.C3

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