Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C16 · Schur multiplier (H2) = C2×C3 |
| direct product | A4×#3708 |
| unambiguous semidirect product | (C22×#3708)⋊C3 |
| derived | C22×C32 ⋊ C3×C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | A4×#157 |
| commutator, hypocenter | C22×C32 | C3×C16 |
| Fitting | C22×C32×C4 | C3×C4 |
| socle | C23×C32 | C3×C8 |
| prime radical | C2×C32×A4 | C8 |
| cosocle | C22×C3⋊(C3⋊C8) | C2×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C4 | |
142 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31906);