Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4⋊#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×A4⋊#622 |
| unambiguous semidirect product | (C2×A4)⋊#622 |
| derived | C32×A4 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4⋊#157 |
| commutator, hypocenter | C32×A4 | C2×C8 |
| Frattini | C2 | C2×A4⋊#157 |
| Fitting, socle | C24×C32 | Dic3 |
| prime radical | C22×C32×A4 | C4 |
| cosocle | A4×C3⋊Dic3 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C22 | |
350 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46723);