Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #560⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C22×#560⋊C3 |
| unambiguous semidirect product | (#126⋊C3)⋊C23 |
| Fitting, Sylow | C22×#560 ⋊ C3 |
| derived | C42 ⋊ C24×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #560⋊C3 |
| commutator, cosocle, hypocenter | C42 | C24×C3 |
| Frattini | C22 | C24×A4 |
| Fitting | C22×#560 | C3 |
| socle | C24 | C22×A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C22 | |
1936 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1936 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213963, A4×#560⋊C3, #72219746, … |
| maximal normal subgroup of | #72213963, A4×#560⋊C3, #72219746, … |
| maximal quotient of | #72214043, C22×#2842⋊C3, #72227331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085024);