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## G = C2×PSU3(𝔽2)  order 144 = 24·32

### Direct product of C2 and PSU3(𝔽2)

Aliases: C2×PSU3(𝔽2), C3⋊S3⋊Q8, (C3×C6)⋊Q8, C32⋊(C2×Q8), C3⋊S3.2C23, C32⋊C4.3C22, (C2×C32⋊C4).5C2, (C2×C3⋊S3).7C22, SmallGroup(144,187)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C2×PSU3(𝔽2)
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — PSU3(𝔽2) — C2×PSU3(𝔽2)
 Lower central C32 — C3⋊S3 — C2×PSU3(𝔽2)
 Upper central C1 — C2

Generators and relations for C2×PSU3(𝔽2)
G = < a,b,c,d,e | a2=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=d-1 >

Character table of C2×PSU3(𝔽2)

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 size 1 1 9 9 8 18 18 18 18 18 18 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ3 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 1 -1 -1 -1 1 1 linear of order 2 ρ5 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ7 1 1 1 1 1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ8 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ9 2 2 -2 -2 2 0 0 0 0 0 0 2 symplectic lifted from Q8, Schur index 2 ρ10 2 -2 -2 2 2 0 0 0 0 0 0 -2 symplectic lifted from Q8, Schur index 2 ρ11 8 -8 0 0 -1 0 0 0 0 0 0 1 orthogonal faithful ρ12 8 8 0 0 -1 0 0 0 0 0 0 -1 orthogonal lifted from PSU3(𝔽2)

Permutation representations of C2×PSU3(𝔽2)
On 18 points - transitive group 18T64
Generators in S18
(1 2)(3 17)(4 18)(5 15)(6 16)(7 14)(8 11)(9 12)(10 13)
(1 5 3)(2 15 17)(4 14 10)(6 8 12)(7 13 18)(9 16 11)
(1 16 18)(2 6 4)(3 9 13)(5 11 7)(8 14 15)(10 17 12)
(1 2)(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(3 9 5 7)(4 8 6 10)(11 16 13 18)(12 15 14 17)

G:=sub<Sym(18)| (1,2)(3,17)(4,18)(5,15)(6,16)(7,14)(8,11)(9,12)(10,13), (1,5,3)(2,15,17)(4,14,10)(6,8,12)(7,13,18)(9,16,11), (1,16,18)(2,6,4)(3,9,13)(5,11,7)(8,14,15)(10,17,12), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (3,9,5,7)(4,8,6,10)(11,16,13,18)(12,15,14,17)>;

G:=Group( (1,2)(3,17)(4,18)(5,15)(6,16)(7,14)(8,11)(9,12)(10,13), (1,5,3)(2,15,17)(4,14,10)(6,8,12)(7,13,18)(9,16,11), (1,16,18)(2,6,4)(3,9,13)(5,11,7)(8,14,15)(10,17,12), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (3,9,5,7)(4,8,6,10)(11,16,13,18)(12,15,14,17) );

G=PermutationGroup([[(1,2),(3,17),(4,18),(5,15),(6,16),(7,14),(8,11),(9,12),(10,13)], [(1,5,3),(2,15,17),(4,14,10),(6,8,12),(7,13,18),(9,16,11)], [(1,16,18),(2,6,4),(3,9,13),(5,11,7),(8,14,15),(10,17,12)], [(1,2),(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(3,9,5,7),(4,8,6,10),(11,16,13,18),(12,15,14,17)]])

G:=TransitiveGroup(18,64);

On 24 points - transitive group 24T257
Generators in S24
(1 8)(2 7)(3 5)(4 6)(9 22)(10 23)(11 24)(12 21)(13 20)(14 17)(15 18)(16 19)
(1 23 21)(2 22 24)(4 19 17)(6 16 14)(7 9 11)(8 10 12)
(1 23 21)(2 24 22)(3 20 18)(5 13 15)(7 11 9)(8 10 12)
(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 5)(2 6)(3 8)(4 7)(9 19 11 17)(10 18 12 20)(13 23 15 21)(14 22 16 24)

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

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

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

G:=TransitiveGroup(24,257);

On 24 points - transitive group 24T260
Generators in S24
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(8 9 24)
(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)| (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (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( (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (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([[(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,20),(14,17),(15,18),(16,19),(21,23),(22,24)], [(2,15,20),(4,18,13),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(1,19,14),(3,16,17),(5,10,21),(6,22,11),(7,23,12),(8,9,24)], [(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(24,260);

C2×PSU3(𝔽2) is a maximal subgroup of   PSU3(𝔽2)⋊C4  C4⋊PSU3(𝔽2)  C62⋊Q8
C2×PSU3(𝔽2) is a maximal quotient of   C4.3PSU3(𝔽2)  C4⋊PSU3(𝔽2)  C62⋊Q8

Matrix representation of C2×PSU3(𝔽2) in GL8(ℤ)

 -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 -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 -1 0 0 0 0 0 0 0 0 -1
,
 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 -1 -1 -1 -1 -1 -1 -1 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 1 0 0 0 0 0 0 0 0 1 0 0 0
,
 0 0 1 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 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 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 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0 1 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 0 -1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 -1 0 0 -1 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 -1 0

G:=sub<GL(8,Integers())| [-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,-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,-1,0,0,0,0,0,0,0,0,-1],[0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0],[0,1,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,1,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,1,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,0],[-1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,0,0,-1,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,0,-1,0,0,1,-1,0,0,0,0] >;

C2×PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_2\times {\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C2xPSU(3,2)");
// GroupNames label

G:=SmallGroup(144,187);
// by ID

G=gap.SmallGroup(144,187);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,48,121,55,3364,730,142,4613,587,455]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*c*e^-1=b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^-1*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations

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