Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C33 |
| direct product | C3×SL(2,𝔽3)2 |
| unambiguous semidirect product | (Q8×SL(2,𝔽3))⋊C32 |
| unambiguous semidirect product | (Q8⋊SL(2,𝔽3))⋊C32 |
| Fitting | C3×Q82 ⋊ C32 |
| derived, Sylow | Q82 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C3 | A42 |
| commutator, cosocle, hypocenter | Q82 | C33 |
| Frattini | C22 | C3×A42 |
| Fitting | C3×Q82 | C32 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C22×C3 | |
376 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32236);