Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | D3×#3407 |
| Fitting | C3×#3407 ⋊ C2 |
| Sylow | C3 ⋊ C2×#3407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C24×D3 |
| commutator, socle | C2×C3 | C24×C8 |
| Fitting | C3×#3407 | C2 |
| prime radical | C2×D3×#41 | C4 |
| cosocle | C3×C8 | C25 |
| hypercenter | #3407 | D3 |
| hypocenter | C3 | C2×#3407 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ #3407 | |
2074 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
204 conjugacy classes — view the full character table.
| maximal subgroup of | #72222733, D3×#33443, D3×#33329, … |
| maximal normal subgroup of | D3×#33443, D3×#33329, D3×#33404, … |
| maximal quotient of | #72222733, D3×#33326 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076034);