Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#73211 |
| Fitting, Sylow | C2×#2840 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5450 |
| commutator, cosocle | C25 | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#2840 | C3 |
| socle | C24 | C22×A4 |
| prime radical | C22×C22⋊A4 | C22 |
| hypocenter | C24 | C2×C3×Q8 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C22 | |
695 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 695 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72231801, #72231955 |
| maximal normal subgroup of | #72231955 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084416);