Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C23×#869 |
| Sylow | C3 ⋊ C23×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C25 | C22×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C23×C3×C42 | C2 |
| socle, prime radical | C25×C3 | C23 |
| hypercenter | C23×C42 | D3 |
| hypocenter | C3 | C23×C4⋊Q8 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C25 ◁ C23×C42 | |
12190 conjugacy classes of subgroups; only the 5451 normal subgroups are shown.
This group has 12190 conjugacy classes of subgroups — too many to draw. Only its 5451 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 768 |
240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076259);