Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1020⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#1020⋊C2 |
| Fitting | C4×#120 ⋊ D3 |
| derived | C22⋊A4 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #1020⋊C2 |
| commutator, hypocenter | C22⋊A4 | C22×C4 |
| Frattini, socle | C23 | C22×S4 |
| Fitting | C4×#120 | D3 |
| prime radical | C2×#1020⋊C2 | C2 |
| cosocle | C2×C22⋊A4 | C23 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C4 | |
858 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 858 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087219);