Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | D3×#2845 |
| Fitting | C3×#2845 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2845 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C3×#2845 | C2 |
| prime radical | C23×D3 | C24 |
| hypercenter | #2845 | D3 |
| hypocenter | C3 | C2×#2845 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2845 | |
1742 conjugacy classes of subgroups; only the 579 normal subgroups are shown.
This group has 1742 conjugacy classes of subgroups — too many to draw. Only its 579 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 376155);