Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | SL(2,𝔽3)×#99 |
| unambiguous semidirect product | (Q8×#99)⋊C3 |
| Fitting, Sylow | Q8×#99 ⋊ C3 |
444 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 444 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 |
77 conjugacy classes — view the full character table.
| maximal subgroup of | #99×PSU(3,𝔽2)⋊C3, SL(2,𝔽3)×B(4,𝔽2), #72241524, … |
| maximal normal subgroup of | SL(2,𝔽3)×B(4,𝔽2), #72241524, #99×GL(2,𝔽3) |
| maximal quotient of | #99×PSU(3,𝔽2)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083871);