Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#85697 |
| Fitting | C23×C3×C22⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ C2×#3490 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C24×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C23×C3×C22⋊C4 | C2 |
| socle | C23×C3 | C25 |
| hypercenter | C23×C22⋊C4 | D3 |
| hypocenter | C3 | C2×#3490 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×C22⋊C4 | |
9502 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×#3490, C2×#11703804, C2×#11703805, … |
| maximal normal subgroup of | C2×D3×#3490, C2×#11703804, C2×#11703805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1077481);