Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C33 · Schur multiplier (H2) = C27×C33 |
| direct product | C24×C32×A4 |
| Fitting, socle | C26×C32 ⋊ C3 |
| derived | C22 ⋊ C24×C33 |
| Sylow | C26 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C32 | A4 |
| commutator, cosocle, hypocenter | C22 | C24×C33 |
| Frattini | C1 | G |
| Fitting, socle | C26×C32 | C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C24×C32 | |
9134 conjugacy classes of subgroups; only the 2278 normal subgroups are shown.
This group has 9134 conjugacy classes of subgroups — too many to draw. Only its 2278 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
576 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×A4×#1094, C32×A4×C22⋊A4, A4×#88907, … |
| maximal normal subgroup of | C22×A4×#1094, C32×A4×C22⋊A4, A4×#88907, … |
| maximal quotient of | C24×C3×#1094 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47933);