Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4×PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | S4×PSU(3,𝔽2) |
| Fitting, socle | C22×C32 ⋊ D3×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | A4×C3⋊D3 | C23 |
| Fitting, socle | C22×C32 | D3×Q8 |
| prime radical | C3⋊D3×S4 | C22 |
| hypocenter | C32×A4 | C2×Q8 |
| LCS | G ▷ A4×C3⋊D3 ▷ C32×A4 | |
| UCS | C1 | |
353 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
30 conjugacy classes — view the full character table.
| maximal subgroup of | S4×PSU(3,𝔽2)⋊C3, S4×AΓL(1,𝔽9), C2×S4×PSU(3,𝔽2) |
| maximal normal subgroup of | S4×PSU(3,𝔽2)⋊C3, S4×AΓL(1,𝔽9), C2×S4×PSU(3,𝔽2) |
| maximal quotient of | C2×S4×PSU(3,𝔽2), GL(2,𝔽3)×PSU(3,𝔽2) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47848);