Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33 · Schur multiplier (H2) = C23×C33 |
| direct product | C2×C32×#994 |
| Fitting | C2×C32×H(5,𝔽2) ⋊ C3 |
| Sylow | C2×H(5,𝔽2) ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C33 |
| Frattini | C2 | C23×C32×A4 |
| Fitting | C2×C32×H(5,𝔽2) | C3 |
| hypercenter | C2×C32×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×Q8 | |
1808 conjugacy classes of subgroups; only the 562 normal subgroups are shown.
This group has 1808 conjugacy classes of subgroups — too many to draw. Only its 562 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
342 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72245009, C2×C3×#10593663 |
| maximal normal subgroup of | C2×#72245009, C2×C3×#10593663 |
| maximal quotient of | C2×C3×#11722576 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47765);