Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C4⋊(#120⋊C3) |
| unambiguous semidirect product | (C2×#2847)⋊C3 |
| Fitting, Sylow | C2×#2847 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#120⋊C3 |
| commutator, cosocle | C25 | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#2847 | C3 |
| socle | C24 | C22×A4 |
| hypercenter | C2×C4 | #120⋊C3 |
| hypocenter | C24 | C2×C3×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1592 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 1592 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) = 48 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#120⋊C3, C2×#11713271, C2×#11713243, … |
| maximal normal subgroup of | C2×D4×#120⋊C3, C2×#11713271, C2×#11713243, … |
| maximal quotient of | #72236859, C2×#14694⋊C3, C4×C4⋊(#120⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084400);