Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×#3586⋊C3 |
| unambiguous semidirect product | (C2×#3586)⋊C3 |
| Fitting, Sylow | C2×#3586 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4×A4 |
| commutator, cosocle | #41 | C24×C3 |
| Frattini | C4 | C24×A4 |
| Fitting | C2×#3586 | C3 |
| hypercenter | C2×AGL(1,ℤ/8) | A4 |
| hypocenter | Q8 | C22×C3×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8) | |
2244 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 2244 conjugacy classes of subgroups — too many to draw. Only its 237 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
86 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215685, C2×#11714992 |
| maximal normal subgroup of | C2×#11714992 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085276);