Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
| direct product | S4×#127 |
| Fitting | C22×#127 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Fitting | C22×#127 | D3 |
| socle | C24 | C23×D3 |
| hypercenter | #127 | S4 |
| hypocenter | A4 | C2×#127 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ #127 | |
2899 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 2899 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | #72225946, S4×#127⋊C3, D42×S4, … |
| maximal normal subgroup of | S4×#127⋊C3, D42×S4, S4×#453, … |
| maximal quotient of | #127×GL(2,𝔽3), #72225946, #72259039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088789);