Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×Q8⋊SL(2,𝔽3) |
| Fitting, Sylow | C22×Q82 ⋊ C3 |
| derived | Q82 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | C22⋊A4 |
| commutator, cosocle, hypocenter | Q82 | C22×C3 |
| Frattini | C22 | C22×C22⋊A4 |
| Fitting | C22×Q82 | C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C24 | |
1561 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1561 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#85554, C2×#11714277, #72268123, … |
| maximal normal subgroup of | C22×#85554, C2×#11714277, #72268125, … |
| maximal quotient of | C22×#11709340 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085070);