Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×#543⋊C3 |
| unambiguous semidirect product | (C4×#543)⋊C3 |
| Fitting, Sylow | C4×#543 ⋊ C3 |
| derived | #543 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | C22⋊A4 |
| commutator, hypocenter | #543 | C3×C4 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | C4×#543 | C3 |
| prime radical | C2×#543⋊C3 | C2 |
| cosocle | C2×#543 | C2×C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22×C4 | |
582 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 53 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 | AGL(1,𝔽5)×#543⋊C3, C4×#11710730, C4×#2032⋊C3 |
| maximal normal subgroup of | C4×#2032⋊C3 |
| maximal quotient of | AGL(1,𝔽5)×#543⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084163);