Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D9×#983 |
| Fitting | C2×Q8×C9 ⋊ C2×D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D9×S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C23 |
| Frattini, socle | C2×C3 | C2×D3×S4 |
| Fitting | C2×Q8×C9 | C2×D3 |
| hypercenter | C22 | D9×S4 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
485 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 485 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30187);