Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×#82127 |
| Fitting | C23×C3×C22⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2410 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C23×C3×C22⋊C4 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C23×C22⋊C4 | D3 |
| hypocenter | C3 | C2×#2410 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×C22⋊C4 | |
8730 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
156 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67×#120, C2×D4×#882, C2×D3×#2410, … |
| maximal normal subgroup of | C2×#67×#120, C2×D4×#882, C2×D3×#2410, … |
| maximal quotient of | #72270011 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 336980);