Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#73166 |
| Fitting, Sylow | C2×#2276 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C42⋊C3 |
| commutator, cosocle, hypocenter | #95 | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#2276 | C3 |
| socle | C24 | C22×A4 |
| prime radical | C2×#95⋊C3 | C22 |
| hypercenter | C2×Q8 | C42⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
699 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 699 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #72268659 |
| maximal normal subgroup of | #72268659 |
| maximal quotient of | C2×#13750⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084133);