Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×A4⋊SL(2,ℤ/4) |
| unambiguous semidirect product | A4⋊(C3×SL(2,ℤ/4)) |
| derived | A42 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | A4⋊S4 |
| commutator, hypocenter | A42 | C3×C4 |
| Frattini | C2 | C3×A4⋊S4 |
| Fitting, socle | C25×C3 | C3⋊D3 |
| prime radical | C2×C3×A42 | C2 |
| cosocle | C2×A42 | C2×C3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2×C3 | |
466 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | #72242208, C3×C4×A4⋊S4, #72242699, … |
| maximal normal subgroup of | C3×C4×A4⋊S4, C3×#11923523 |
| maximal quotient of | #72242208, #72242697, #72244840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46400);