Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×#5402 |
| Fitting | C24×C8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C4⋊S4 |
| commutator, cosocle | C4×A4 | C24 |
| Frattini | C4 | C23×S4 |
| Fitting | C24×C8 | D3 |
| socle | C25 | D12 |
| hypercenter | C22×C8 | S4 |
| hypocenter | A4 | C22×D8 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×C8 | |
1881 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 1881 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11713991, A4×#5402, C22×#85440, … |
| maximal normal subgroup of | C2×#11713991, A4×#5402, C22×#85440, … |
| maximal quotient of | C2×#11713991 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089134);