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G = C2.PSU3(𝔽2)  order 144 = 24·32

The central extension by C2 of PSU3(𝔽2)

non-abelian, soluble, monomial

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

C1C32C3⋊S3 — C2.PSU3(𝔽2)
C1C32C3⋊S3C2×C3⋊S3C2×C32⋊C4 — C2.PSU3(𝔽2)
C32C3⋊S3 — C2.PSU3(𝔽2)
C1C2

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 >

9C2
9C2
4C3
9C4
9C4
9C22
18C4
18C4
4C6
12S3
12S3
9C2×C4
9C2×C4
9C2×C4
12D6
9C4⋊C4
2C32⋊C4
2C32⋊C4

Character table of C2.PSU3(𝔽2)

 class 12A2B2C34A4B4C4D4E4F6
 size 119981818181818188
ρ1111111111111    trivial
ρ211111-11-1-1-111    linear of order 2
ρ311111-1-111-1-11    linear of order 2
ρ4111111-1-1-11-11    linear of order 2
ρ51-11-11-ii-11i-i-1    linear of order 4
ρ61-11-11ii1-1-i-i-1    linear of order 4
ρ71-11-11i-i-11-ii-1    linear of order 4
ρ81-11-11-i-i1-1ii-1    linear of order 4
ρ92-2-222000000-2    orthogonal lifted from D4
ρ1022-2-220000002    symplectic lifted from Q8, Schur index 2
ρ118-800-10000001    orthogonal faithful
ρ128800-1000000-1    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)

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00010000
00-1-10000
0000-1-100
00001000
00000001
000000-1-1
,
01000000
-1-1000000
00100000
00010000
0000-1-100
00001000
000000-1-1
00000010
,
00100000
00010000
10000000
-1-1000000
000000-10
00000011
0000-1000
00000-100
,
00001000
00000100
00000010
00000001
-10000000
11000000
00-100000
00110000

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

Export

Subgroup lattice of C2.PSU3(𝔽2) in TeX
Character table of C2.PSU3(𝔽2) in TeX

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