Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#5934 |
| Fitting | C32×H(5,𝔽2) ⋊ D3 |
| derived | #997 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C32 | #1020 |
| commutator, cosocle, hypocenter | #997 | C2×C32 |
| Frattini | C2 | C32×#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C32 | |
676 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
117 conjugacy classes — view the full character table.
| maximal subgroup of | 3−1+2×#5934, C3×D3×#5934 |
| maximal normal subgroup of | 3−1+2×#5934, C3×D3×#5934 |
| maximal quotient of | 3−1+2×#5934 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47481);