Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C3×C43 |
| direct product | C42×C3⋊#157 |
| Fitting | C33×C42 ⋊ C4 |
| derived, Sylow | C33 ⋊ C43 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | C3⋊#157 |
| commutator, hypocenter | C33 | C43 |
| Frattini | C22 | C22×C3⋊#157 |
| Fitting | C33×C42 | C4 |
| socle | C22×C33 | C22×C4 |
| prime radical, cosocle | C22×C3×C3⋊D3 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C42 | |
792 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 792 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39392);