Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊AGL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊AGL(2,𝔽3) |
| unambiguous semidirect product | (PSU(3,𝔽2)⋊C3)⋊D3 |
| Fitting, socle | C33 ⋊ GL(2,𝔽3) |
| derived | C3×PSU(3,𝔽2)⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×PSU(3,𝔽2)⋊C3 | C2 |
| Fitting, socle | C33 | GL(2,𝔽3) |
| LCS | G ▷ C3×PSU(3,𝔽2)⋊C3 | |
| UCS | C1 | |
151 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2,3^2,2^3,3^2|d^2=f,e^2=f,b^a=b^2,c^a=c^2,d^a=ef,e^a=df,g^a=g^2h,d^b=e,e^b=def,g^b=gh,e^d=ef,g^d=gh,h^d=gh^2,g^e=h^2,h^e=g,g^f=g^2,h^f=h^221 conjugacy classes — view the full character table.
| maximal subgroup of | #72210405, #72213290, D3×AGL(2,𝔽3), … |
| maximal normal subgroup of | D3×AGL(2,𝔽3), C3×C3⋊AGL(2,𝔽3), C2×C3⋊AGL(2,𝔽3), … |
| maximal quotient of | #72213290, #72213922, C3×C3⋊AGL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3488);