Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C23×#857 |
| Fitting | C23×C42 ⋊ D3 |
| derived | C42⋊C3 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #857 |
| commutator, cosocle, hypocenter | C42⋊C3 | C24 |
| Frattini | C22 | C23×S4 |
| Fitting | C23×C42 | D3 |
| socle | C25 | S4 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C23 | |
2596 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 2596 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11710948, C2×#11711335, C2×#11711328, … |
| maximal normal subgroup of | C2×#11711335, C2×#11711328, C22⋊C4×#857, … |
| maximal quotient of | C2×#11712792, #41×#857, #72226084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088991);