non-abelian, soluble, monomial
Aliases: C2.PSU3(𝔽2), (C3×C6).Q8, C32⋊C4⋊2C4, C3⋊S3.3D4, C32⋊2(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
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) |
►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 4)(2 3)(5 7)(6 8)(9 18)(10 19)(11 20)(12 17)(13 21)(14 22)(15 23)(16 24)
(1 21 23)(2 24 22)(3 16 14)(4 13 15)(6 10 12)(8 19 17)
(1 21 23)(2 22 24)(3 14 16)(4 13 15)(5 11 9)(7 20 18)
(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 4 7)(2 8 3 6)(9 15 20 21)(10 22 17 16)(11 13 18 23)(12 24 19 14)
G:=sub<Sym(24)| (1,4)(2,3)(5,7)(6,8)(9,18)(10,19)(11,20)(12,17)(13,21)(14,22)(15,23)(16,24), (1,21,23)(2,24,22)(3,16,14)(4,13,15)(6,10,12)(8,19,17), (1,21,23)(2,22,24)(3,14,16)(4,13,15)(5,11,9)(7,20,18), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5,4,7)(2,8,3,6)(9,15,20,21)(10,22,17,16)(11,13,18,23)(12,24,19,14)>;
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,18)(10,19)(11,20)(12,17)(13,21)(14,22)(15,23)(16,24), (1,21,23)(2,24,22)(3,16,14)(4,13,15)(6,10,12)(8,19,17), (1,21,23)(2,22,24)(3,14,16)(4,13,15)(5,11,9)(7,20,18), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5,4,7)(2,8,3,6)(9,15,20,21)(10,22,17,16)(11,13,18,23)(12,24,19,14) );
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8),(9,18),(10,19),(11,20),(12,17),(13,21),(14,22),(15,23),(16,24)], [(1,21,23),(2,24,22),(3,16,14),(4,13,15),(6,10,12),(8,19,17)], [(1,21,23),(2,22,24),(3,14,16),(4,13,15),(5,11,9),(7,20,18)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,5,4,7),(2,8,3,6),(9,15,20,21),(10,22,17,16),(11,13,18,23),(12,24,19,14)]])
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
Export
Subgroup lattice of C2.PSU3(𝔽2) in TeX
Character table of C2.PSU3(𝔽2) in TeX