Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C33 |
| direct product | #120×C3⋊(C3⋊D3) |
| Fitting | C33×#120 ⋊ C2 |
| Sylow | C33 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×#120 | C2 |
| hypercenter | #120 | C3⋊(C3⋊D3) |
| hypocenter | C33 | C2×#120 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ #120 | |
9268 conjugacy classes of subgroups; only the 807 normal subgroups are shown.
This group has 9268 conjugacy classes of subgroups — too many to draw. Only its 807 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
210 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊D3×#120, C3⋊(C3⋊D3)×#1020, #72220851 |
| maximal normal subgroup of | D3×C3⋊D3×#120, #72220851 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 45777);