Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C25×C33 |
| direct product | C22×C32×C22⋊A4 |
| Fitting, socle | C26×C32 ⋊ C3 |
| derived | C24 ⋊ C22×C33 |
| Sylow | C26 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | C22⋊A4 |
| commutator, cosocle, hypocenter | C24 | C22×C33 |
| Frattini | C1 | G |
| Fitting, socle | C26×C32 | C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C22×C32 | |
6560 conjugacy classes of subgroups; only the 320 normal subgroups are shown.
This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 320 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47932);