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G = C92⋊C3order 243 = 35

1st semidirect product of C92 and C3 acting faithfully

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

Aliases: C921C3, C32.1He3, C3.He31C3, (C3×C9).16C32, He3⋊C3.1C3, C3.6(He3⋊C3), SmallGroup(243,25)

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=c3=1, ab=ba, cac-1=a7b2, cbc-1=a3b7 >

3C3
27C3
3C9
3C9
3C9
3C9
9C32
9C9
9C9
33- 1+2
3He3
33- 1+2
3C3×C9

Permutation representations of C92⋊C3
On 27 points - transitive group 27T108
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 8 4 3 7 6 2 9 5)(10 14 18 13 17 12 16 11 15)(19 21 23 25 27 20 22 24 26)
(1 21 10)(2 24 16)(3 27 13)(4 19 18)(5 22 15)(6 25 12)(7 26 17)(8 20 14)(9 23 11)

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

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

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

G:=TransitiveGroup(27,108);

C92⋊C3 is a maximal subgroup of   C92⋊C6

35 conjugacy classes

class 1 3A3B3C3D3E3F9A···9X9Y9Z9AA9AB
order13333339···99999
size1113327273···327272727

35 irreducible representations

dim1111333
type+
imageC1C3C3C3He3He3⋊C3C92⋊C3
kernelC92⋊C3C92He3⋊C3C3.He3C32C3C1
# reps12242618

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

700
1340
1505
,
600
060
609
,
1100
0181
0180
G:=sub<GL(3,GF(19))| [7,13,15,0,4,0,0,0,5],[6,0,6,0,6,0,0,0,9],[1,0,0,10,18,18,0,1,0] >;

C92⋊C3 in GAP, Magma, Sage, TeX

C_9^2\rtimes C_3
% in TeX

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

G:=SmallGroup(243,25);
// by ID

G=gap.SmallGroup(243,25);
# by ID

G:=PCGroup([5,-3,3,-3,-3,-3,121,186,542,282,2163]);
// Polycyclic

G:=Group<a,b,c|a^9=b^9=c^3=1,a*b=b*a,c*a*c^-1=a^7*b^2,c*b*c^-1=a^3*b^7>;
// generators/relations

Export

Subgroup lattice of C92⋊C3 in TeX

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