Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C213 |
| direct product | C22×#5247 |
| Sylow | C3 ⋊ C22×#379 |
| subgroup | G/H | |
|---|---|---|
| center | C25 | C22×D3 |
| commutator | C22×C3 | C24×C4 |
| Frattini | C23 | C24×D3 |
| Fitting, prime radical | C27×C3 | C2 |
| socle | C25×C3 | C23 |
| cosocle | C23×C3 | C25 |
| hypercenter | C27 | D3 |
| hypocenter | C3 | C22×#379 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C25 ◁ C27 | |
33214 conjugacy classes of subgroups; only the 3307 normal subgroups are shown.
This group has 33214 conjugacy classes of subgroups — too many to draw. Only its 3307 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C22×D3×#379, C22×#84052, #72271133, … |
| maximal normal subgroup of | C22×D3×#379, C22×#84052, #72271133, … |
| maximal quotient of | #72271502 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 327823);