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G = C92.C3order 243 = 35

2nd non-split extension by C92 of C3 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C92.2C3, C32.3He3, (C3×C9).18C32, C3.He3.1C3, C3.8(He3⋊C3), SmallGroup(243,27)

Series: Derived Chief Lower central Upper central Jennings

C1C3×C9 — C92.C3
C1C3C32C3×C9C92 — C92.C3
C1C3C32C3×C9 — C92.C3
C1C3C32C3×C9 — C92.C3
C1C3C3C3C32C3×C9 — C92.C3

Generators and relations for C92.C3
 G = < a,b,c | a9=b9=1, c3=b6, ab=ba, cac-1=a7b-1, cbc-1=a3b >

3C3
3C9
3C9
3C9
3C9
9C9
9C9
9C9
33- 1+2
33- 1+2
3C3×C9
33- 1+2

Permutation representations of C92.C3
On 27 points - transitive group 27T112
Generators in S27
(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 7 4 3 9 6 2 8 5)(10 11 12 13 14 15 16 17 18)(19 21 23 25 27 20 22 24 26)
(1 27 16 2 21 13 3 24 10)(4 22 15 5 25 12 6 19 18)(7 20 11 8 23 17 9 26 14)

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

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

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

G:=TransitiveGroup(27,112);

C92.C3 is a maximal subgroup of   C92.S3

35 conjugacy classes

class 1 3A3B3C3D9A···9X9Y···9AD
order133339···99···9
size111333···327···27

35 irreducible representations

dim111333
type+
imageC1C3C3He3He3⋊C3C92.C3
kernelC92.C3C92C3.He3C32C3C1
# reps1262618

Matrix representation of C92.C3 in GL3(𝔽19) generated by

700
040
0017
,
600
090
006
,
010
001
1100
G:=sub<GL(3,GF(19))| [7,0,0,0,4,0,0,0,17],[6,0,0,0,9,0,0,0,6],[0,0,11,1,0,0,0,1,0] >;

C92.C3 in GAP, Magma, Sage, TeX

C_9^2.C_3
% in TeX

G:=Group("C9^2.C3");
// GroupNames label

G:=SmallGroup(243,27);
// by ID

G=gap.SmallGroup(243,27);
# by ID

G:=PCGroup([5,-3,3,-3,-3,-3,405,121,456,542,282,2163]);
// Polycyclic

G:=Group<a,b,c|a^9=b^9=1,c^3=b^6,a*b=b*a,c*a*c^-1=a^7*b^-1,c*b*c^-1=a^3*b>;
// generators/relations

Export

Subgroup lattice of C92.C3 in TeX

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