Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
| direct product | D3×#2670 |
| Fitting | C3×#2670 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2670 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C3×#2670 | C2 |
| hypercenter | #2670 | D3 |
| hypocenter | C3 | C2×#2670 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2670 | |
5006 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#58991, D3×#43098, D3×#43095, … |
| maximal normal subgroup of | D3×#58991, D3×#43098, D3×#43095, … |
| maximal quotient of | D3×#16249, D3×#16255, D3×#15314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 804138);