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G = S3×PSU3(𝔽2)  order 432 = 24·33

Direct product of S3 and PSU3(𝔽2)

direct product, non-abelian, soluble, monomial, rational

Aliases: S3×PSU3(𝔽2), C33⋊(C2×Q8), (S3×C32)⋊Q8, C33⋊C2⋊Q8, C32⋊C4.5D6, C32⋊3(S3×Q8), C33⋊Q8⋊3C2, C33⋊C4.C22, C3⋊1(C2×PSU3(𝔽2)), (C3×PSU3(𝔽2))⋊2C2, (S3×C32⋊C4).1C2, (S3×C3⋊S3).2C22, C3⋊S3.2(C22×S3), (C3×C3⋊S3).2C23, (C3×C32⋊C4).4C22, SmallGroup(432,742)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊S3 — S3×PSU3(𝔽2)
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — S3×C3⋊S3 — S3×C32⋊C4 — S3×PSU3(𝔽2)
Lower central
C33 — C3×C3⋊S3 — S3×PSU3(𝔽2)
Upper central
C1

Generators and relations for S3×PSU3(𝔽2)
 G = < a,b,c,d,e,f | a3=b2=c3=d3=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fdf-1=cd=dc, ece-1=d-1, fcf-1=c-1d, ede-1=c, fef-1=e-1 >

Subgroups: 848 in 90 conjugacy classes, 28 normal (13 characteristic)
C1, C2, C3, C3, C4, C22, S3, S3, C6, C2×C4, Q8, C32, C32, Dic3, C12, D6, C2×Q8, C3×S3, C3⋊S3, C3⋊S3, C3×C6, Dic6, C4×S3, C3×Q8, C33, C32⋊C4, C32⋊C4, S32, C2×C3⋊S3, S3×Q8, S3×C32, C3×C3⋊S3, C33⋊C2, PSU3(𝔽2), PSU3(𝔽2), C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, S3×C3⋊S3, C2×PSU3(𝔽2), S3×C32⋊C4, C3×PSU3(𝔽2), C33⋊Q8, S3×PSU3(𝔽2)
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C22×S3, S3×Q8, PSU3(𝔽2), C2×PSU3(𝔽2), S3×PSU3(𝔽2)

Character table of S3×PSU3(𝔽2)

 class 12A2B2C3A3B3C4A4B4C4D4E4F6A6B12A12B12C
 size 1392728161818185454541824363636
ρ1111111111111111111    trivial
ρ21111111-1-11-11-111-1-11    linear of order 2
ρ31-11-11111-1-1-1111-1-11-1    linear of order 2
ρ41-11-1111111-1-1-11-1111    linear of order 2
ρ51111111-11-1-1-11111-1-1    linear of order 2
ρ61-11-1111-11-111-11-11-1-1    linear of order 2
ρ71-11-1111-1-111-111-1-1-11    linear of order 2
ρ811111111-1-11-1-111-11-1    linear of order 2
ρ92020-12-1222000-10-1-1-1    orthogonal lifted from S3
ρ102020-12-1-2-22000-1011-1    orthogonal lifted from D6
ρ112020-12-12-2-2000-101-11    orthogonal lifted from D6
ρ122020-12-1-22-2000-10-111    orthogonal lifted from D6
ρ1322-2-2222000000-22000    symplectic lifted from Q8, Schur index 2
ρ142-2-22222000000-2-2000    symplectic lifted from Q8, Schur index 2
ρ1540-40-24-200000020000    symplectic lifted from S3×Q8, Schur index 2
ρ168-8008-1-100000001000    orthogonal lifted from C2×PSU3(𝔽2)
ρ1788008-1-10000000-1000    orthogonal lifted from PSU3(𝔽2)
ρ1816000-8-2100000000000    orthogonal faithful

Permutation representations of S3×PSU3(𝔽2)
►On 24 points - transitive group 24T1335
Generators in S24
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)

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

(1 19 14)(2 15 20)(3 16 17)(4 18 13)(5 10 21)(7 23 12)

(1 19 14)(2 20 15)(3 16 17)(4 13 18)(6 11 22)(8 24 9)

(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
 
G:=sub<Sym(24)| (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24), (1,19,14)(2,15,20)(3,16,17)(4,18,13)(5,10,21)(7,23,12), (1,19,14)(2,20,15)(3,16,17)(4,13,18)(6,11,22)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;
 
G:=Group( (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24), (1,19,14)(2,15,20)(3,16,17)(4,18,13)(5,10,21)(7,23,12), (1,19,14)(2,20,15)(3,16,17)(4,13,18)(6,11,22)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );
 
G=PermutationGroup([[(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,20),(14,17),(15,18),(16,19),(21,23),(22,24)], [(1,19,14),(2,15,20),(3,16,17),(4,18,13),(5,10,21),(7,23,12)], [(1,19,14),(2,20,15),(3,16,17),(4,13,18),(6,11,22),(8,24,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)]])
 
G:=TransitiveGroup(24,1335);
 

►On 27 points - transitive group 27T135
Generators in S27
(1 2 3)(4 11 23)(5 8 20)(6 9 21)(7 10 22)(12 17 25)(13 18 26)(14 19 27)(15 16 24)

(2 3)(4 11)(5 8)(6 9)(7 10)(16 24)(17 25)(18 26)(19 27)

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

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

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

(4 17 6 19)(5 16 7 18)(8 24 10 26)(9 27 11 25)(12 21 14 23)(13 20 15 22)
 
G:=sub<Sym(27)| (1,2,3)(4,11,23)(5,8,20)(6,9,21)(7,10,22)(12,17,25)(13,18,26)(14,19,27)(15,16,24), (2,3)(4,11)(5,8)(6,9)(7,10)(16,24)(17,25)(18,26)(19,27), (1,13,15)(2,18,16)(3,26,24)(4,17,5)(6,7,19)(8,11,25)(9,10,27)(12,20,23)(14,21,22), (1,14,12)(2,19,17)(3,27,25)(4,16,7)(5,18,6)(8,26,9)(10,11,24)(13,21,20)(15,22,23), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,17,6,19)(5,16,7,18)(8,24,10,26)(9,27,11,25)(12,21,14,23)(13,20,15,22)>;
 
G:=Group( (1,2,3)(4,11,23)(5,8,20)(6,9,21)(7,10,22)(12,17,25)(13,18,26)(14,19,27)(15,16,24), (2,3)(4,11)(5,8)(6,9)(7,10)(16,24)(17,25)(18,26)(19,27), (1,13,15)(2,18,16)(3,26,24)(4,17,5)(6,7,19)(8,11,25)(9,10,27)(12,20,23)(14,21,22), (1,14,12)(2,19,17)(3,27,25)(4,16,7)(5,18,6)(8,26,9)(10,11,24)(13,21,20)(15,22,23), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,17,6,19)(5,16,7,18)(8,24,10,26)(9,27,11,25)(12,21,14,23)(13,20,15,22) );
 
G=PermutationGroup([[(1,2,3),(4,11,23),(5,8,20),(6,9,21),(7,10,22),(12,17,25),(13,18,26),(14,19,27),(15,16,24)], [(2,3),(4,11),(5,8),(6,9),(7,10),(16,24),(17,25),(18,26),(19,27)], [(1,13,15),(2,18,16),(3,26,24),(4,17,5),(6,7,19),(8,11,25),(9,10,27),(12,20,23),(14,21,22)], [(1,14,12),(2,19,17),(3,27,25),(4,16,7),(5,18,6),(8,26,9),(10,11,24),(13,21,20),(15,22,23)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,17,6,19),(5,16,7,18),(8,24,10,26),(9,27,11,25),(12,21,14,23),(13,20,15,22)]])
 
G:=TransitiveGroup(27,135);
 

Matrix representation of S3×PSU3(𝔽2) ►in GL12(𝔽13)

010000000000
12120000000000
000100000000
00121200000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
1200000000000
110000000000
0012000000000
001100000000
0000120000000
0000012000000
0000001200000
0000000120000
0000000012000
0000000001200
0000000000120
0000000000012
,
100000000000
010000000000
001000000000
000100000000
0000000100012
0000000010012
0000100000012
0000001000012
0000000000112
0000000000012
0000010000012
0000000001012
,
100000000000
010000000000
001000000000
000100000000
0000011200000
0000001200001
0000001200010
0000001201000
0000001200100
0000001210000
0000001200000
0000101200000
,
001000000000
000100000000
1200000000000
0120000000000
000001000000
000000001000
000000000001
000010000000
000000010000
000000100000
000000000100
000000000010
,
800000000000
080000000000
005000000000
000500000000
0000000000120
0000000000012
0000120000000
0000000001200
0000001200000
0000012000000
0000000012000
0000000120000

G:=sub<GL(12,GF(13))| [0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,12,12,12,12,12,12],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,12,12,12,12,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[8,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0] >;
 

S3×PSU3(𝔽2) in GAP, Magma, Sage, TeX 

S_3\times {\rm PSU}_3({\mathbb F}_2)
 
% in TeX
 
G:=Group("S3xPSU(3,2)");
 
// GroupNames label
 
G:=SmallGroup(432,742);
 
// by ID
 
G=gap.SmallGroup(432,742);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,64,135,58,1411,858,165,1356,187,530,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^2=c^3=d^3=e^4=1,f^2=e^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*d*f^-1=c*d=d*c,e*c*e^-1=d^-1,f*c*f^-1=c^-1*d,e*d*e^-1=c,f*e*f^-1=e^-1>;
 
// generators/relations
 

Export

Character table of S3×PSU3(𝔽2) in TeX

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