Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | SL(2,𝔽3)×D36 |
| unambiguous semidirect product | C4⋊(D9×SL(2,𝔽3)) |
| Fitting | C4×Q8×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×A4×D9 |
| commutator, cosocle | C2×Q8×C9 | C22×C3 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C4×Q8×C9 | C2×C3 |
| hypercenter | C2×C4 | A4×D9 |
| hypocenter | Q8×C9 | C3×D4 |
| LCS | G ▷ C2×Q8×C9 ▷ Q8×C9 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
337 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13169);