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G = C27⋊C9order 243 = 35

The semidirect product of C27 and C9 acting faithfully

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

Aliases: C27⋊C9, C9.53- 1+2, C32.73- 1+2, C27⋊C3.C3, C9⋊C9.1C3, C9.2(C3×C9), C3.3(C9⋊C9), (C3×C9).1C32, SmallGroup(243,22)

Series: Derived Chief Lower central Upper central Jennings

C1C9 — C27⋊C9
C1C3C32C3×C9C27⋊C3 — C27⋊C9
C1C3C9 — C27⋊C9
C1C3C3×C9 — C27⋊C9
C1C3C3C3C3C3C3C3×C9C3×C9 — C27⋊C9

Generators and relations for C27⋊C9
 G = < a,b | a27=b9=1, bab-1=a7 >

3C3
9C9
3C27
3C3×C9
3C27

Permutation representations of C27⋊C9
On 27 points - transitive group 27T107
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)
(2 5 17 11 14 26 20 23 8)(3 9 6 21 27 24 12 18 15)(4 13 22)(7 25 16)

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

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

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

G:=TransitiveGroup(27,107);

C27⋊C9 is a maximal subgroup of   C27⋊C18

35 conjugacy classes

class 1 3A3B3C3D9A···9F9G···9L27A···27R
order133339···99···927···27
size111333···39···99···9

35 irreducible representations

dim1111339
type+
imageC1C3C3C93- 1+23- 1+2C27⋊C9
kernelC27⋊C9C9⋊C9C27⋊C3C27C9C32C1
# reps12618422

Matrix representation of C27⋊C9 in GL9(𝔽109)

000100000
000010000
000001000
0000006300
0000000630
0000000063
0450000000
0045000000
100000000
,
100000000
0450000000
0063000000
000001000
000100000
0000450000
0000000630
000000001
000000100

G:=sub<GL(9,GF(109))| [0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,45,0,0,0,0,0,0,0,0,0,45,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,63,0,0,0],[1,0,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,1,0] >;

C27⋊C9 in GAP, Magma, Sage, TeX

C_{27}\rtimes C_9
% in TeX

G:=Group("C27:C9");
// GroupNames label

G:=SmallGroup(243,22);
// by ID

G=gap.SmallGroup(243,22);
# by ID

G:=PCGroup([5,-3,3,-3,3,-3,135,121,36,1352,147,1268]);
// Polycyclic

G:=Group<a,b|a^27=b^9=1,b*a*b^-1=a^7>;
// generators/relations

Export

Subgroup lattice of C27⋊C9 in TeX

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