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G = S3×F9  order 432 = 24·33

Direct product of S3 and F9

direct product, metabelian, soluble, monomial, A-group

Aliases: S3×F9, C33⋊(C2×C8), C3⋊F9⋊3C2, C3⋊1(C2×F9), (S3×C32)⋊C8, C33⋊C2⋊C8, (C3×F9)⋊2C2, C32⋊3(S3×C8), C33⋊C4.C4, C32⋊C4.4D6, (S3×C3⋊S3).C4, C3⋊S3.1(C4×S3), (S3×C32⋊C4).2C2, (C3×C32⋊C4).5C22, (C3×C3⋊S3).(C2×C4), SmallGroup(432,736)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for S3×F9
 G = < a,b,c,d,e | a3=b2=c3=d3=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 568 in 58 conjugacy classes, 18 normal (all characteristic)
C1, C2, C3, C3, C4, C22, S3, S3, C6, C8, C2×C4, C32, C32, Dic3, C12, D6, C2×C8, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C33, C32⋊C4, C32⋊C4, S32, C2×C3⋊S3, S3×C8, S3×C32, C3×C3⋊S3, C33⋊C2, F9, F9, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, S3×C3⋊S3, C2×F9, C3×F9, C3⋊F9, S3×C32⋊C4, S3×F9
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C4×S3, S3×C8, F9, C2×F9, S3×F9

Character table of S3×F9

 class 12A2B2C3A3B3C4A4B4C4D6A6B8A8B8C8D8E8F8G8H12A12B24A24B24C24D
 size 1392728169927271824999927272727181818181818
ρ1111111111111111111111111111    trivial
ρ21111111111111-1-1-1-1-1-1-1-111-1-1-1-1    linear of order 2
ρ31-11-111111-1-11-11111-1-1-1-1111111    linear of order 2
ρ41-11-111111-1-11-1-1-1-1-1111111-1-1-1-1    linear of order 2
ρ51-11-1111-1-1111-1i-i-ii-ii-ii-1-1i-i-ii    linear of order 4
ρ61111111-1-1-1-111i-i-iii-ii-i-1-1i-i-ii    linear of order 4
ρ71111111-1-1-1-111-iii-i-ii-ii-1-1-iii-i    linear of order 4
ρ81-11-1111-1-1111-1-iii-ii-ii-i-1-1-iii-i    linear of order 4
ρ91-1-11111i-ii-i-1-1ζ83ζ85ζ8ζ87ζ87ζ85ζ83ζ8i-iζ83ζ85ζ8ζ87    linear of order 8
ρ101-1-11111-ii-ii-1-1ζ8ζ87ζ83ζ85ζ85ζ87ζ8ζ83-iiζ8ζ87ζ83ζ85    linear of order 8
ρ1111-1-1111i-i-ii-11ζ83ζ85ζ8ζ87ζ83ζ8ζ87ζ85i-iζ83ζ85ζ8ζ87    linear of order 8
ρ1211-1-1111-iii-i-11ζ8ζ87ζ83ζ85ζ8ζ83ζ85ζ87-iiζ8ζ87ζ83ζ85    linear of order 8
ρ131-1-11111-ii-ii-1-1ζ85ζ83ζ87ζ8ζ8ζ83ζ85ζ87-iiζ85ζ83ζ87ζ8    linear of order 8
ρ141-1-11111i-ii-i-1-1ζ87ζ8ζ85ζ83ζ83ζ8ζ87ζ85i-iζ87ζ8ζ85ζ83    linear of order 8
ρ1511-1-1111-iii-i-11ζ85ζ83ζ87ζ8ζ85ζ87ζ8ζ83-iiζ85ζ83ζ87ζ8    linear of order 8
ρ1611-1-1111i-i-ii-11ζ87ζ8ζ85ζ83ζ87ζ85ζ83ζ8i-iζ87ζ8ζ85ζ83    linear of order 8
ρ172020-12-12200-1022220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ182020-12-12200-10-2-2-2-20000-1-11111    orthogonal lifted from D6
ρ192020-12-1-2-200-10-2i2i2i-2i000011i-i-ii    complex lifted from C4×S3
ρ202020-12-1-2-200-102i-2i-2i2i000011-iii-i    complex lifted from C4×S3
ρ2120-20-12-12i-2i00102ζ832ζ852ζ82ζ870000-iiζ87ζ8ζ85ζ83    complex lifted from S3×C8
ρ2220-20-12-1-2i2i00102ζ82ζ872ζ832ζ850000i-iζ85ζ83ζ87ζ8    complex lifted from S3×C8
ρ2320-20-12-12i-2i00102ζ872ζ82ζ852ζ830000-iiζ83ζ85ζ8ζ87    complex lifted from S3×C8
ρ2420-20-12-1-2i2i00102ζ852ζ832ζ872ζ80000i-iζ8ζ87ζ83ζ85    complex lifted from S3×C8
ρ2588008-1-100000-100000000000000    orthogonal lifted from F9
ρ268-8008-1-100000100000000000000    orthogonal lifted from C2×F9
ρ2716000-8-2100000000000000000000    orthogonal faithful

Permutation representations of S3×F9
►On 24 points - transitive group 24T1336
Generators in S24
(1 14 22)(2 15 23)(3 16 24)(4 9 17)(5 10 18)(6 11 19)(7 12 20)(8 13 21)

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

(2 15 23)(3 16 24)(4 17 9)(6 19 11)(7 20 12)(8 13 21)

(1 14 22)(3 16 24)(4 9 17)(5 18 10)(7 20 12)(8 21 13)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
 
G:=sub<Sym(24)| (1,14,22)(2,15,23)(3,16,24)(4,9,17)(5,10,18)(6,11,19)(7,12,20)(8,13,21), (1,5)(2,6)(3,7)(4,8)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (2,15,23)(3,16,24)(4,17,9)(6,19,11)(7,20,12)(8,13,21), (1,14,22)(3,16,24)(4,9,17)(5,18,10)(7,20,12)(8,21,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
 
G:=Group( (1,14,22)(2,15,23)(3,16,24)(4,9,17)(5,10,18)(6,11,19)(7,12,20)(8,13,21), (1,5)(2,6)(3,7)(4,8)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (2,15,23)(3,16,24)(4,17,9)(6,19,11)(7,20,12)(8,13,21), (1,14,22)(3,16,24)(4,9,17)(5,18,10)(7,20,12)(8,21,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
 
G=PermutationGroup([[(1,14,22),(2,15,23),(3,16,24),(4,9,17),(5,10,18),(6,11,19),(7,12,20),(8,13,21)], [(1,5),(2,6),(3,7),(4,8),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(2,15,23),(3,16,24),(4,17,9),(6,19,11),(7,20,12),(8,13,21)], [(1,14,22),(3,16,24),(4,9,17),(5,18,10),(7,20,12),(8,21,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
 
G:=TransitiveGroup(24,1336);
 

►On 27 points - transitive group 27T138
Generators in S27
(1 2 3)(4 20 13)(5 21 14)(6 22 15)(7 23 16)(8 24 17)(9 25 18)(10 26 19)(11 27 12)

(2 3)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)

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

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

(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)| (1,2,3)(4,20,13)(5,21,14)(6,22,15)(7,23,16)(8,24,17)(9,25,18)(10,26,19)(11,27,12), (2,3)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27), (1,13,17)(2,4,8)(3,20,24)(5,11,10)(6,7,9)(12,19,14)(15,16,18)(21,27,26)(22,23,25), (1,14,18)(2,5,9)(3,21,25)(4,11,6)(7,8,10)(12,15,13)(16,17,19)(20,27,22)(23,24,26), (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( (1,2,3)(4,20,13)(5,21,14)(6,22,15)(7,23,16)(8,24,17)(9,25,18)(10,26,19)(11,27,12), (2,3)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27), (1,13,17)(2,4,8)(3,20,24)(5,11,10)(6,7,9)(12,19,14)(15,16,18)(21,27,26)(22,23,25), (1,14,18)(2,5,9)(3,21,25)(4,11,6)(7,8,10)(12,15,13)(16,17,19)(20,27,22)(23,24,26), (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([[(1,2,3),(4,20,13),(5,21,14),(6,22,15),(7,23,16),(8,24,17),(9,25,18),(10,26,19),(11,27,12)], [(2,3),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27)], [(1,13,17),(2,4,8),(3,20,24),(5,11,10),(6,7,9),(12,19,14),(15,16,18),(21,27,26),(22,23,25)], [(1,14,18),(2,5,9),(3,21,25),(4,11,6),(7,8,10),(12,15,13),(16,17,19),(20,27,22),(23,24,26)], [(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(27,138);
 

Matrix representation of S3×F9 ►in GL10(ℤ)

-1100000000
-1000000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
0100000000
1000000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
000-1100000
000-1000100
000-1000001
000-1000010
000-1010000
000-1000000
000-1001000
001-1000000
,
1000000000
0100000000
000000-1000
000001-1000
000100-1000
000010-1000
001000-1000
000000-1010
000000-1001
000000-1100
,
-1000000000
0-100000000
0000000100
0000010000
0000001000
0000000010
0001000000
0000000001
0010000000
0000100000

G:=sub<GL(10,Integers())| [-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
 

S3×F9 in GAP, Magma, Sage, TeX 

S_3\times F_9
 
% in TeX
 
G:=Group("S3xF9");
 
// GroupNames label
 
G:=SmallGroup(432,736);
 
// by ID
 
G=gap.SmallGroup(432,736);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,36,58,1131,718,165,348,691,530,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
 
// generators/relations
 

Export

Character table of S3×F9 in TeX

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