Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#5936 |
| derived | C22⋊A4 ⋊ C32×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | #1020 |
| commutator, hypocenter | C22⋊A4 | C32×C4 |
| Frattini | C2 | C32×#1020 |
| Fitting, socle | C25×C32 | D3 |
| prime radical | C2×C32×C22⋊A4 | C2 |
| cosocle | C2×C22⋊A4 | C2×C32 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2×C32 | |
1012 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47483);