Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
| direct product | C23×#1020 |
| Fitting, socle | C27 ⋊ D3 |
| derived | C22⋊A4 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #1020 |
| commutator, cosocle, hypocenter | C22⋊A4 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C27 | D3 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C23 | |
12082 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 12082 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090234);