Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4×(C22⋊C9)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | S4×(C22⋊C9)⋊C2 |
| unambiguous semidirect product | (A4×(C22⋊C9)⋊C2)⋊C2 |
| unambiguous semidirect product | (S4×C22⋊C9)⋊C2 |
| unambiguous semidirect product | (A4⋊((C22⋊C9)⋊C2))⋊C2 |
| derived | A4×C22⋊C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×C22⋊C9 | C22 |
| Frattini | C3 | S42 |
| Fitting, socle | C24×C3 | D32 |
| LCS | G ▷ A4×C22⋊C9 | |
| UCS | C1 | |
486 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 54 |
45 conjugacy classes — view the full character table.
| maximal subgroup of | S4×((C22⋊C9)⋊C2)⋊C3 |
| maximal normal subgroup of | S4×((C22⋊C9)⋊C2)⋊C3 |
| maximal quotient of | (C22⋊C9)⋊C22 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46098);