Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D32×C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
| direct product | D32×C22⋊A4 |
| unambiguous semidirect product | (C3×D3)⋊(C2×C22⋊A4) |
| Fitting, socle, derived | C24×C32 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C24×C32 | C22×C3 |
| LCS | G ▷ C24×C32 | |
| UCS | C1 | |
4218 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 4218 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊D3×C22⋊A4, D32×A42, D32×#1020, … |
| maximal normal subgroup of | D32×A42, D32×#1020, D32×#120⋊C3, … |
| maximal quotient of | D3×C3⋊D3×C22⋊A4, #72225260, D32×#997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47902);