Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×D4×C22⋊A4 |
| unambiguous semidirect product | C4⋊(C22×C22⋊A4) |
| Fitting, Sylow | C25×D4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊A4 |
| commutator, cosocle | C25 | C23×C3 |
| Frattini | C2 | C23×C22⋊A4 |
| Fitting | C25×D4 | C3 |
| socle | C26 | C22×C3 |
| hypercenter | C2×D4 | C22⋊A4 |
| hypocenter | C24 | C2×C3×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
18848 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 18848 conjugacy classes of subgroups — too many to draw. Only its 152 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.
| maximal subgroup of | C2×D4×#1020, #120×C22⋊A4, #72216669, … |
| maximal normal subgroup of | C2×D4×#1020, #120×C22⋊A4, #72216669, … |
| maximal quotient of | #72215685, #120×C22⋊A4, #72226313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085287);