Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C23×C9×SL(2,𝔽3) |
| Fitting | C23×Q8×C9 ⋊ C3 |
| derived | Q8 ⋊ C23×C3×C9 |
| Sylow | C23×Q8 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C9 | A4 |
| commutator, hypocenter | Q8 | C23×C3×C9 |
| Frattini | C2×C3 | C23×C3×A4 |
| Fitting | C23×Q8×C9 | C3 |
| socle | C24×C3 | C3×A4 |
| prime radical | C23×C3×SL(2,𝔽3) | C3 |
| cosocle | C3×Q8 | C23×C32 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C24×C9 | |
1172 conjugacy classes of subgroups; only the 361 normal subgroups are shown.
This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 361 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30888);