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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, C323(S3×Q8), C33⋊Q83C2, C33⋊C4.C22, C31(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

C1C32C3×C3⋊S3 — S3×PSU3(𝔽2)
C1C3C33C3×C3⋊S3S3×C3⋊S3S3×C32⋊C4 — S3×PSU3(𝔽2)
C33C3×C3⋊S3 — S3×PSU3(𝔽2)
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 [×3], C3, C3 [×2], C4 [×6], C22, S3, S3 [×4], C6 [×2], C2×C4 [×3], Q8 [×4], C32, C32 [×2], Dic3 [×3], C12 [×3], D6 [×2], C2×Q8, C3×S3 [×2], C3⋊S3, C3⋊S3 [×3], C3×C6, Dic6 [×3], C4×S3 [×3], C3×Q8, C33, C32⋊C4 [×3], C32⋊C4 [×3], S32, C2×C3⋊S3, S3×Q8, S3×C32, C3×C3⋊S3, C33⋊C2, PSU3(𝔽2), PSU3(𝔽2) [×3], C2×C32⋊C4 [×3], C3×C32⋊C4 [×3], C33⋊C4 [×3], S3×C3⋊S3, C2×PSU3(𝔽2), S3×C32⋊C4 [×3], C3×PSU3(𝔽2), C33⋊Q8 [×3], S3×PSU3(𝔽2)
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], 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 16 8)(5 17 9)(6 18 10)(7 19 11)(12 22 25)(13 23 26)(14 20 27)(15 21 24)
(2 3)(4 8)(5 9)(6 10)(7 11)(12 22)(13 23)(14 20)(15 21)
(1 25 27)(2 12 14)(3 22 20)(4 21 5)(6 7 23)(8 15 9)(10 11 13)(16 24 17)(18 19 26)
(1 26 24)(2 13 15)(3 23 21)(4 20 7)(5 22 6)(8 14 11)(9 12 10)(16 27 19)(17 25 18)
(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 21 6 23)(5 20 7 22)(8 15 10 13)(9 14 11 12)(16 24 18 26)(17 27 19 25)

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

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

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

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

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Character table of S3×PSU3(𝔽2) in TeX

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