Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #157 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C26×C3 |
| direct product | C23×C3×#622 |
| derived | C32 ⋊ C23×C3×C8 |
| Sylow | C33 ⋊ C23×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C3 | #157 |
| commutator, hypocenter | C32 | C23×C3×C8 |
| Frattini | C2 | C23×C3×#157 |
| Fitting, socle, prime radical | C24×C33 | C4 |
| cosocle | C3⋊Dic3 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C24×C3 | |
1416 conjugacy classes of subgroups; only the 472 normal subgroups are shown.
This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 472 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47568);