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G = F9⋊S3  order 432 = 24·33

The semidirect product of F9 and S3 acting via S3/C3=C2

non-abelian, soluble, monomial

Aliases: F9⋊S3, C33⋊4SD16, C3⋊1AΓL1(𝔽9), C3⋊S3.D12, (C3×F9)⋊1C2, C32⋊C4.3D6, C32⋊(C24⋊C2), C33⋊Q8⋊1C2, C32⋊2D12.1C2, (C3×C3⋊S3).4D4, (C3×C32⋊C4).1C22, SmallGroup(432,740)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C32⋊C4 — F9⋊S3
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C3×C32⋊C4 — C32⋊2D12 — F9⋊S3
Lower central
C33 — C3×C3⋊S3 — C3×C32⋊C4 — F9⋊S3
Upper central
C1

Generators and relations for F9⋊S3
 G = < a,b,c,d,e | a3=b3=c8=d3=e2=1, cac-1=ebe=ab=ba, ad=da, eae=a-1, cbc-1=a, bd=db, cd=dc, ece=c3, ede=d-1 >

C1
9C2
36C2
C3
4C3
8C3
9C4
54C4
54C22
9C6
12S3
12S3
12S3
24S3
36C6
C32
4C32
8C32
9C8
27Q8
27D4
9C12
18D6
18Dic3
36D6
C3⋊S3
4C3⋊S3
12C3×S3
12C3×S3
12C3×S3
24C3×S3
C33
27SD16
9Dic6
9D12
9C24
C32⋊C4
6C32⋊C4
6S32
12S32
C3×C3⋊S3
4C3×C3⋊S3
9C24⋊C2
F9
3S3≀C2
3PSU3(𝔽2)
C3×C32⋊C4
2C32⋊4D6
2C33⋊C4
3AΓL1(𝔽9)
C33⋊Q8
C3×F9
C32⋊2D12
F9⋊S3

Character table of F9⋊S3

 class 12A2B3A3B3C4A4B6A6B8A8B12A12B24A24B24C24D
 size 193628161810818721818181818181818
ρ1111111111111111111    trivial
ρ211-1111111-1-1-111-1-1-1-1    linear of order 2
ρ31111111-111-1-111-1-1-1-1    linear of order 2
ρ411-11111-11-111111111    linear of order 2
ρ5220-12-120-1022-1-1-1-1-1-1    orthogonal lifted from S3
ρ6220-12-120-10-2-2-1-11111    orthogonal lifted from D6
ρ7220222-202000-2-20000    orthogonal lifted from D4
ρ8220-12-1-20-100011√3√3-√3-√3    orthogonal lifted from D12
ρ9220-12-1-20-100011-√3-√3√3√3    orthogonal lifted from D12
ρ102-2022200-20√-2-√-200√-2-√-2-√-2√-2    complex lifted from SD16
ρ112-2022200-20-√-2√-200-√-2√-2√-2-√-2    complex lifted from SD16
ρ122-20-12-10010-√-2√-2-√3√3-ζ83ζ3+ζ8ζ3+ζ8-ζ87ζ3+ζ85ζ3+ζ85-ζ87ζ32+ζ85ζ32+ζ85-ζ83ζ32+ζ8ζ32+ζ8    complex lifted from C24⋊C2
ρ132-20-12-10010-√-2√-2√3-√3-ζ83ζ32+ζ8ζ32+ζ8-ζ87ζ32+ζ85ζ32+ζ85-ζ87ζ3+ζ85ζ3+ζ85-ζ83ζ3+ζ8ζ3+ζ8    complex lifted from C24⋊C2
ρ142-20-12-10010√-2-√-2-√3√3-ζ87ζ3+ζ85ζ3+ζ85-ζ83ζ3+ζ8ζ3+ζ8-ζ83ζ32+ζ8ζ32+ζ8-ζ87ζ32+ζ85ζ32+ζ85    complex lifted from C24⋊C2
ρ152-20-12-10010√-2-√-2√3-√3-ζ87ζ32+ζ85ζ32+ζ85-ζ83ζ32+ζ8ζ32+ζ8-ζ83ζ3+ζ8ζ3+ζ8-ζ87ζ3+ζ85ζ3+ζ85    complex lifted from C24⋊C2
ρ1680-28-1-1000100000000    orthogonal lifted from AΓL1(𝔽9)
ρ178028-1-1000-100000000    orthogonal lifted from AΓL1(𝔽9)
ρ181600-8-21000000000000    orthogonal faithful

Permutation representations of F9⋊S3
►On 24 points - transitive group 24T1333
Generators in S24
(1 15 21)(2 22 16)(4 24 10)(5 17 11)(6 12 18)(8 14 20)

(1 15 21)(2 16 22)(3 23 9)(5 17 11)(6 18 12)(7 13 19)

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

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

(2 4)(3 7)(6 8)(9 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)
 
G:=sub<Sym(24)| (1,15,21)(2,22,16)(4,24,10)(5,17,11)(6,12,18)(8,14,20), (1,15,21)(2,16,22)(3,23,9)(5,17,11)(6,18,12)(7,13,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,15,21)(2,16,22)(3,9,23)(4,10,24)(5,11,17)(6,12,18)(7,13,19)(8,14,20), (2,4)(3,7)(6,8)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)>;
 
G:=Group( (1,15,21)(2,22,16)(4,24,10)(5,17,11)(6,12,18)(8,14,20), (1,15,21)(2,16,22)(3,23,9)(5,17,11)(6,18,12)(7,13,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,15,21)(2,16,22)(3,9,23)(4,10,24)(5,11,17)(6,12,18)(7,13,19)(8,14,20), (2,4)(3,7)(6,8)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24) );
 
G=PermutationGroup([[(1,15,21),(2,22,16),(4,24,10),(5,17,11),(6,12,18),(8,14,20)], [(1,15,21),(2,16,22),(3,23,9),(5,17,11),(6,18,12),(7,13,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,15,21),(2,16,22),(3,9,23),(4,10,24),(5,11,17),(6,12,18),(7,13,19),(8,14,20)], [(2,4),(3,7),(6,8),(9,19),(10,22),(11,17),(12,20),(13,23),(14,18),(15,21),(16,24)]])
 
G:=TransitiveGroup(24,1333);
 

►On 27 points - transitive group 27T143
Generators in S27
(1 6 10)(2 15 19)(3 22 26)(4 7 5)(8 9 11)(12 17 18)(13 16 14)(20 23 21)(24 25 27)

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

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

(2 3)(5 7)(6 10)(9 11)(12 25)(13 20)(14 23)(15 26)(16 21)(17 24)(18 27)(19 22)
 
G:=sub<Sym(27)| (1,6,10)(2,15,19)(3,22,26)(4,7,5)(8,9,11)(12,17,18)(13,16,14)(20,23,21)(24,25,27), (1,7,11)(2,16,12)(3,23,27)(4,9,10)(5,8,6)(13,18,19)(14,17,15)(20,25,26)(21,24,22), (4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27), (1,2,3)(4,13,20)(5,14,21)(6,15,22)(7,16,23)(8,17,24)(9,18,25)(10,19,26)(11,12,27), (2,3)(5,7)(6,10)(9,11)(12,25)(13,20)(14,23)(15,26)(16,21)(17,24)(18,27)(19,22)>;
 
G:=Group( (1,6,10)(2,15,19)(3,22,26)(4,7,5)(8,9,11)(12,17,18)(13,16,14)(20,23,21)(24,25,27), (1,7,11)(2,16,12)(3,23,27)(4,9,10)(5,8,6)(13,18,19)(14,17,15)(20,25,26)(21,24,22), (4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27), (1,2,3)(4,13,20)(5,14,21)(6,15,22)(7,16,23)(8,17,24)(9,18,25)(10,19,26)(11,12,27), (2,3)(5,7)(6,10)(9,11)(12,25)(13,20)(14,23)(15,26)(16,21)(17,24)(18,27)(19,22) );
 
G=PermutationGroup([[(1,6,10),(2,15,19),(3,22,26),(4,7,5),(8,9,11),(12,17,18),(13,16,14),(20,23,21),(24,25,27)], [(1,7,11),(2,16,12),(3,23,27),(4,9,10),(5,8,6),(13,18,19),(14,17,15),(20,25,26),(21,24,22)], [(4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27)], [(1,2,3),(4,13,20),(5,14,21),(6,15,22),(7,16,23),(8,17,24),(9,18,25),(10,19,26),(11,12,27)], [(2,3),(5,7),(6,10),(9,11),(12,25),(13,20),(14,23),(15,26),(16,21),(17,24),(18,27),(19,22)]])
 
G:=TransitiveGroup(27,143);
 

Matrix representation of F9⋊S3 ►in GL10(𝔽73)

1000000000
0100000000
00000720100
00100720000
00000721000
00000720001
00000720010
00010720000
00001720000
00000720000
,
1000000000
0100000000
00000100720
00000000720
00100000720
00001000720
00000001720
00000000721
00010000720
00000010720
,
661400000000
59700000000
00000000720
00000000072
00000072000
00007200000
00072000000
00720000000
00000720000
00000007200
,
07200000000
17200000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
17200000000
07200000000
00720000000
00000007200
00000000720
00000000072
00000072000
00072000000
00007200000
00000720000

G:=sub<GL(10,GF(73))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,72,72,72,72,72,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,72,72,72,72,72,72,72,72,0,0,0,0,0,0,0,1,0,0],[66,59,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0] >;
 

F9⋊S3 in GAP, Magma, Sage, TeX 

F_9\rtimes S_3
 
% in TeX
 
G:=Group("F9:S3");
 
// GroupNames label
 
G:=SmallGroup(432,740);
 
// by ID
 
G=gap.SmallGroup(432,740);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,197,92,254,58,1131,998,165,5381,348,1363,530,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^3=e^2=1,c*a*c^-1=e*b*e=a*b=b*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a,b*d=d*b,c*d=d*c,e*c*e=c^3,e*d*e=d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of F9⋊S3 in TeX
Character table of F9⋊S3 in TeX

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