Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | D3×#1454 |
| Fitting | C3×#1454 ⋊ C2 |
| Sylow | C3 ⋊ C2×#1454 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C23×D3 |
| commutator, socle | C22×C3 | C22×C42 |
| Fitting | C3×#1454 | C2 |
| prime radical | C23×D3 | C42 |
| cosocle | C3×C42 | C24 |
| hypercenter | #1454 | D3 |
| hypocenter | C3 | C2×#1454 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ #1454 | |
802 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 135190);