Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | A4×B(4,𝔽2) |
| Fitting, Sylow | C22×B(4,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | A4×#120 |
| commutator, cosocle | C25 | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C22×B(4,𝔽2) | C3 |
| socle | C23 | C3×#120 |
| hypercenter | B(4,𝔽2) | A4 |
| hypocenter | C22 | C3×B(4,𝔽2) |
| LCS | G ▷ C25 ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ B(4,𝔽2) | |
1499 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1499 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | S4×B(4,𝔽2), A4×B(4,𝔽2)⋊C3, A5×B(4,𝔽2), … |
| maximal normal subgroup of | S4×B(4,𝔽2), A4×B(4,𝔽2)⋊C3, #72267053, … |
| maximal quotient of | #72220576, SL(2,𝔽3)×B(4,𝔽2), A4×#2196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084716);