Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×#3500⋊C3 |
| unambiguous semidirect product | (C2×#3500)⋊C3 |
| Fitting, Sylow | C2×#3500 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×A4 |
| commutator, cosocle | C2×Q8 | C24×C3 |
| Frattini | C22 | C24×A4 |
| Fitting | C2×#3500 | C3 |
| hypercenter | C2×#124 | A4 |
| hypocenter | Q8 | C22×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
2394 conjugacy classes of subgroups; only the 245 normal subgroups are shown.
This group has 2394 conjugacy classes of subgroups — too many to draw. Only its 245 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085103);