Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#82688 |
| Fitting | C22×C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2688 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C22×C3×#121 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C22×#121 | D3 |
| hypocenter | C3 | C2×#2688 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×#121 | |
3650 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | #72253631, #72253645, C2×#11427646, … |
| maximal normal subgroup of | #72253631, #72253645, C2×#11427646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 804401);