Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#76764 |
| Fitting | C2×C3×#392 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2669 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C2×C3×#392 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C2×#392 | D3 |
| hypocenter | C3 | C2×#2669 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C2×#392 | |
5958 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11427639, C2×#11427637, C2×D3×#2669, … |
| maximal normal subgroup of | C2×#11427639, C2×#11427637, C2×D3×#2669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 804129);