Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11722945 |
| Sylow | C33 ⋊ C2×C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D33 |
| commutator | C2×C33 | C23×C4 |
| Frattini | C22 | C2×D33 |
| Fitting, socle, prime radical | C23×C33 | C23 |
| cosocle | C22×C33 | C24 |
| hypocenter | C33 | C2×C4×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
1570 conjugacy classes of subgroups; only the 324 normal subgroups are shown.
This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 324 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 38340);