Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#11722436 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×(C3⋊D3)⋊A4 |
| commutator, cosocle, hypocenter | C32×Q8 | C23×C3 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C2×C32×#41 | C2×C3 |
| socle | C22×C3 | C2×D3×A4 |
| hypercenter | C2×C4 | (C3⋊D3)⋊A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1099 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1099 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
86 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13164635 |
| maximal normal subgroup of | C2×#13164635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30692);