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## G = C2.PSU3(𝔽2)  order 144 = 24·32

### The central extension by C2 of PSU3(𝔽2)

Aliases: C2.PSU3(𝔽2), (C3×C6).Q8, C32⋊C42C4, C3⋊S3.3D4, C322(C4⋊C4), C3⋊S3.4(C2×C4), (C2×C32⋊C4).2C2, (C2×C3⋊S3).3C22, SmallGroup(144,120)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C2.PSU3(𝔽2)
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — 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=ad2, 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=ad-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 -i i -1 1 i -i -1 linear of order 4 ρ6 1 -1 1 -1 1 i i 1 -1 -i -i -1 linear of order 4 ρ7 1 -1 1 -1 1 i -i -1 1 -i i -1 linear of order 4 ρ8 1 -1 1 -1 1 -i -i 1 -1 i i -1 linear of order 4 ρ9 2 -2 -2 2 2 0 0 0 0 0 0 -2 orthogonal lifted from D4 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 24 points - transitive group 24T258
Generators in S24
(1 3)(2 4)(5 7)(6 8)(9 22)(10 23)(11 24)(12 21)(13 18)(14 19)(15 20)(16 17)
(2 17 14)(4 16 19)(5 10 21)(6 11 22)(7 23 12)(8 24 9)
(1 13 20)(3 18 15)(5 21 10)(6 11 22)(7 12 23)(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 8)(2 5)(3 6)(4 7)(9 13 24 20)(10 17 21 14)(11 15 22 18)(12 19 23 16)

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

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

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

G:=TransitiveGroup(24,258);

On 24 points - transitive group 24T259
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 23)
(1 16 14)(3 19 17)(4 18 20)(5 10 12)(7 22 24)(8 21 23)
(2 13 15)(3 19 17)(4 20 18)(6 11 9)(7 22 24)(8 23 21)
(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 8 5 4)(2 3 6 7)(9 24 13 19)(10 20 14 21)(11 22 15 17)(12 18 16 23)

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

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

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

G:=TransitiveGroup(24,259);

C2.PSU3(𝔽2) is a maximal subgroup of
C2.AΓL1(𝔽9)  F9⋊C4  C4.3PSU3(𝔽2)  C4×PSU3(𝔽2)  C4⋊PSU3(𝔽2)  C62⋊Q8  C6.PSU3(𝔽2)  C6.2PSU3(𝔽2)
C2.PSU3(𝔽2) is a maximal quotient of
C4.4PSU3(𝔽2)  C4.PSU3(𝔽2)  C4.2PSU3(𝔽2)  C62.Q8  C62.2Q8  C2.SU3(𝔽2)  C6.PSU3(𝔽2)  C6.2PSU3(𝔽2)

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
,
 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 -1 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 0 1 0 0 0 0 0 0 -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 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 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 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 0 0 0 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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[0,-1,0,0,0,0,0,0,1,-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,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C2.PSU3(𝔽2) in GAP, Magma, Sage, TeX

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

G:=Group("C2.PSU(3,2)");
// GroupNames label

G:=SmallGroup(144,120);
// by ID

G=gap.SmallGroup(144,120);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,73,79,3364,730,256,4613,587,881]);
// Polycyclic

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

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