Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×GO+(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×A4×GO+(4,𝔽2) |
| unambiguous semidirect product | #157⋊(C22×A4) |
| unambiguous semidirect product | D32⋊(C22×A4) |
| unambiguous semidirect product | (C23×GO+(4,𝔽2))⋊C3 |
| Fitting, socle | C23×C32 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×GO+(4,𝔽2) |
| commutator, cosocle | C22×C3⋊D3 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | C3×D4 |
| hypocenter | C22×C32 | C2×C3×D4 |
| LCS | G ▷ C22×C3⋊D3 ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
1030 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×AΓL(1,𝔽9), C2×S4×GO+(4,𝔽2), C2×#13181779 |
| maximal normal subgroup of | C2×A4×AΓL(1,𝔽9), C2×S4×GO+(4,𝔽2), C2×#13181779 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47872);