Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C23×A4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C2×S4 |
| commutator, cosocle | C2×A4 | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C25×C4 | D3 |
| socle | C26 | C2×D3 |
| prime radical | C24×A4 | C22 |
| hypercenter | C23×C4 | S4 |
| hypocenter | A4 | C23×Q8 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C24 ◁ C23×C4 | |
6209 conjugacy classes of subgroups; only the 661 normal subgroups are shown.
This group has 6209 conjugacy classes of subgroups — too many to draw. Only its 661 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090144);