Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#11724029 |
| derived | (C3×A4)⋊A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | A4⋊S4 |
| commutator, cosocle, hypocenter | (C3×A4)⋊A4 | C22 |
| Frattini | C3 | C2×A4⋊S4 |
| Fitting, socle | C25×C3 | C3⋊D3 |
| LCS | G ▷ (C3×A4)⋊A4 | |
| UCS | C1 ◁ C2×C3 | |
674 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 674 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72213251, C2×#13180034, C22×#11724029 |
| maximal normal subgroup of | C2×#13180034, C22×#11724029 |
| maximal quotient of | C2×#13164078, C22×#11724029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46112);