Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
| direct product | D3×#2619 |
| Fitting | C3×#2619 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2619 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C3×#2619 | C2 |
| hypercenter | #2619 | D3 |
| hypocenter | C3 | C2×#2619 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2619 | |
4342 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#37650, D3×#37297, D3×#37674, … |
| maximal normal subgroup of | D3×#37650, D3×#37297, D3×#37674, … |
| maximal quotient of | D3×#14257, D3×#15124 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 336572);