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G = U2(𝔽3)  order 96 = 25·3

Unitary group on 𝔽32

non-abelian, soluble

Aliases: U2(𝔽3), C4.5S4, Q8.Dic3, SL2(𝔽3)⋊2C4, C4○D4.1S3, C4.A4.2C2, C2.3(A4⋊C4), SmallGroup(96,67)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — U2(𝔽3)
C1C2Q8SL2(𝔽3)C4.A4 — U2(𝔽3)
SL2(𝔽3) — U2(𝔽3)
C1C4

Generators and relations for U2(𝔽3)
 G = < a,b,c,d,e | a4=d3=1, b2=c2=a2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a2b, dbd-1=a2bc, ebe-1=bc, dcd-1=b, ece-1=a2c, ede-1=d-1 >

6C2
4C3
3C4
3C22
6C4
6C4
4C6
3C2×C4
3D4
6C8
6C2×C4
4C12
3C42
3M4(2)
4C3⋊C8
3C4≀C2

Character table of U2(𝔽3)

 class 12A2B34A4B4C4D4E4F4G68A8B12A12B
 size 116811666668121288
ρ11111111111111111    trivial
ρ2111111-1-1-1-111-1-111    linear of order 2
ρ311-11-1-1-iii-i11-ii-1-1    linear of order 4
ρ411-11-1-1i-i-ii11i-i-1-1    linear of order 4
ρ5222-12200002-100-1-1    orthogonal lifted from S3
ρ622-2-1-2-200002-10011    symplectic lifted from Dic3, Schur index 2
ρ72-20-12i-2i-1+i1+i-1-i1-i0100-ii    complex faithful
ρ82-20-1-2i2i1+i-1+i1-i-1-i0100i-i    complex faithful
ρ92-20-12i-2i1-i-1-i1+i-1+i0100-ii    complex faithful
ρ102-20-1-2i2i-1-i1-i-1+i1+i0100i-i    complex faithful
ρ1133-10331111-10-1-100    orthogonal lifted from S4
ρ1233-1033-1-1-1-1-101100    orthogonal lifted from S4
ρ133310-3-3i-i-ii-10-ii00    complex lifted from A4⋊C4
ρ143310-3-3-iii-i-10i-i00    complex lifted from A4⋊C4
ρ154-4014i-4i00000-100i-i    complex faithful
ρ164-401-4i4i00000-100-ii    complex faithful

Permutation representations of U2(𝔽3)
On 24 points - transitive group 24T138
Generators in S24
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)
(1 4 5 8)(2 3 6 7)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 20 22 24)
(1 3 5 7)(2 8 6 4)(9 23 13 19)(10 20 14 24)(11 17 15 21)(12 22 16 18)
(1 15 18)(2 19 16)(3 9 20)(4 21 10)(5 11 22)(6 23 12)(7 13 24)(8 17 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,138);

U2(𝔽3) is a maximal subgroup of
CU2(𝔽3)  C8.5S4  U2(𝔽3)⋊C2  Q8.4S4  Q8.5S4  D4.S4  D4.3S4  C3⋊U2(𝔽3)  GL2(𝔽5)  C52U2(𝔽3)  C5⋊U2(𝔽3)
U2(𝔽3) is a maximal quotient of
C2.U2(𝔽3)  C12.9S4  C3⋊U2(𝔽3)  C52U2(𝔽3)  C5⋊U2(𝔽3)

Matrix representation of U2(𝔽3) in GL2(𝔽5) generated by

20
02
,
22
03
,
30
42
,
02
24
,
44
41
G:=sub<GL(2,GF(5))| [2,0,0,2],[2,0,2,3],[3,4,0,2],[0,2,2,4],[4,4,4,1] >;

U2(𝔽3) in GAP, Magma, Sage, TeX

{\rm U}_2({\mathbb F}_3)
% in TeX

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

G:=SmallGroup(96,67);
// by ID

G=gap.SmallGroup(96,67);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,12,295,146,579,447,117,364,286,202,88]);
// Polycyclic

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

Export

Subgroup lattice of U2(𝔽3) in TeX
Character table of U2(𝔽3) in TeX

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