Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | D3×#2321 |
| Fitting | C3×#2321 ⋊ C2 |
| Sylow | C3 ⋊ C2×#2321 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator | C22×C3 | C24×C4 |
| Fitting | C3×#2321 | C2 |
| socle, cosocle | C23×C3 | C25 |
| prime radical | C2×D3×#120 | C2 |
| hypercenter | #2321 | D3 |
| hypocenter | C3 | C2×#2321 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2321 | |
5094 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#35352, D3×#37644, D3×#38358, … |
| maximal normal subgroup of | D3×#35352, D3×#37644, D3×#38358, … |
| maximal quotient of | D3×#14357, D3×#13967, D3×#14299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 328278);