Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | A4×#575 |
| unambiguous semidirect product | (C22×#575)⋊C3 |
| Fitting, Sylow | C22×#575 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D4×A4 |
| commutator, cosocle | C22×C4 | C24×C3 |
| Frattini | C4 | C24×A4 |
| Fitting | C22×#575 | C3 |
| socle | C23 | C22×C3×D4 |
| hypercenter | #575 | A4 |
| hypocenter | C22 | C3×#575 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #575 | |
2319 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 2319 conjugacy classes of subgroups — too many to draw. Only its 237 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#575 |
| maximal normal subgroup of | S4×#575 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085264);