Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×#543⋊C3 |
| Fitting, Sylow | C22×#543 ⋊ C3 |
| derived | #543 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | C22⋊A4 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Frattini | C22 | C22×C22⋊A4 |
| Fitting | C22×#543 | C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C24 | |
1890 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 1890 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #72227587, #72227601, #72227691, … |
| maximal normal subgroup of | #72227601, #72227691, C22×#2029⋊C3, … |
| maximal quotient of | C2×D3×#543⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085008);