Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | D3×#1440 |
| Fitting | C3×#1440 ⋊ C2 |
| Sylow | C3 ⋊ C2×#1440 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C23×C3 | C23×C4 |
| Fitting | C3×#1440 | C2 |
| socle, cosocle | C24×C3 | C24 |
| prime radical | C25×D3 | C22 |
| hypercenter | #1440 | D3 |
| hypocenter | C3 | C2×#1440 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #1440 | |
4622 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#16249, D3×#16255, D3×#14357, … |
| maximal normal subgroup of | D3×#16249, D3×#16255, D3×#14357, … |
| maximal quotient of | D3×#11157, D3×#11183, D3×#11181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 91233);