Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#3459⋊C3 |
| unambiguous semidirect product | (C2×#3459)⋊C3 |
| Fitting, Sylow | C2×#3459 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Frattini | C22 | C22×C22⋊A4 |
| Fitting | C2×#3459 | C3 |
| socle | C23 | C2×C22⋊A4 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22×C4 | |
1536 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 65 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.
| derived subgroup of | #72268143, #72268161, #72271326, … |
| maximal subgroup of | C2×#72260605, C2×#62844⋊C3, #72268058, … |
| maximal normal subgroup of | C2×#62844⋊C3, #72268058, #72268129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085013);