Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#85310 |
| Fitting | C23×C4⋊C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Frattini | C22 | C23×S4 |
| Fitting | C23×C4⋊C4 | D3 |
| socle | C25 | C22×D3 |
| hypercenter | C2×C4⋊C4 | S4 |
| hypocenter | A4 | C2×#121 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
2433 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 2433 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×C4⋊S4, C2×#72245603, C2×S4×#121, … |
| maximal normal subgroup of | C2×D4×C4⋊S4, C2×S4×#121, C2×#11714546 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088721);