Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | #157×SL(2,ℤ/4) |
| derived | C32×A4 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S4×#157 |
| commutator, hypocenter | C32×A4 | C42 |
| Fitting, socle | C23×C32 | C4×D3 |
| prime radical, cosocle | C2×A4×C3⋊D3 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
400 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46254);