Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of AGL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C22×AGL(2,𝔽3) |
| unambiguous semidirect product | (PSU(3,𝔽2)⋊C3)⋊C23 |
| unambiguous semidirect product | (C2×PSU(3,𝔽2)⋊C3)⋊C22 |
| Fitting, socle | C22×C32 ⋊ GL(2,𝔽3) |
| derived | PSU(3,𝔽2)⋊C3 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | AGL(2,𝔽3) |
| commutator, cosocle, hypocenter | PSU(3,𝔽2)⋊C3 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32 | GL(2,𝔽3) |
| LCS | G ▷ PSU(3,𝔽2)⋊C3 | |
| UCS | C1 ◁ C22 | |
592 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47845);