Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214×C4 |
| direct product | C23×#940 |
| Fitting | C24×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ C23×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C25 | C22×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C24×C3×D4 | C2 |
| socle | C25×C3 | C23 |
| hypercenter | C24×D4 | D3 |
| hypocenter | C3 | C23×#127 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C25 ◁ C24×D4 | |
24030 conjugacy classes of subgroups; only the 4939 normal subgroups are shown.
This group has 24030 conjugacy classes of subgroups — too many to draw. Only its 4939 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
192 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10826820, C2×#10826822, C23×D3×#127, … |
| maximal normal subgroup of | C2×#10826820, C2×#10826822, C23×D3×#127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076254);