Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | SL(2,𝔽3)×#104 |
| unambiguous semidirect product | (Q8×#104)⋊C3 |
| Fitting, Sylow | Q8×#104 ⋊ C3 |
361 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
98 conjugacy classes — view the full character table.
| maximal subgroup of | #104×PSU(3,𝔽2)⋊C3, SL(2,𝔽3)×GU(2,𝔽3) |
| maximal quotient of | #104×PSU(3,𝔽2)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083876);