Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C4 |
| direct product | C2×#82588 |
| Fitting | C22×C3×#120 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2433 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C22×C3×#120 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C22×#120 | D3 |
| hypocenter | C3 | C2×#2433 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×#120 | |
9422 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67×#120, C2×#10990485, C2×#10990476, … |
| maximal normal subgroup of | C2×#67×#120, C2×#10990485, C2×#10990476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 337385);