Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3 |
| Fitting | C3×H(5,𝔽2) ⋊ C3×D3 |
| derived | #994 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | A4×S4 |
| commutator, cosocle, hypocenter | #994 | C2×C32 |
| Frattini | C2 | C3×A4×S4 |
| Fitting | C3×H(5,𝔽2) | C3×D3 |
| LCS | G ▷ #994 | |
| UCS | C1 ◁ C2×C3 | |
290 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
87 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#983⋊A4, C3×#11923454, C3×#11923456, … |
| maximal normal subgroup of | C3×#983⋊A4, C3×#11923454, C3×#11923456, … |
| maximal quotient of | #72253010, #72253013 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46381);