Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#5919 |
| Fitting | C24×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×S4 |
| commutator, cosocle | C2×A4 | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C24×Q8 | D3 |
| socle | C25 | C22×D3 |
| hypercenter | C22×Q8 | S4 |
| hypocenter | A4 | C22×#41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
5969 conjugacy classes of subgroups; only the 541 normal subgroups are shown.
This group has 5969 conjugacy classes of subgroups — too many to draw. Only its 541 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#85460, C2×#72236893, C22×#41×S4 |
| maximal normal subgroup of | C22×#85460, C22×#41×S4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090149);