Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | A4×#525 |
| unambiguous semidirect product | (C22×#525)⋊C3 |
| Fitting, Sylow | C22×#525 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×A4 |
| commutator, socle, cosocle | C24 | C24×C3 |
| Frattini | C22 | C24×A4 |
| Fitting | C22×#525 | C3 |
| hypercenter | #525 | A4 |
| hypocenter | C22 | C3×#525 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C2×C4 ◁ #525 | |
2939 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#5912, A4×#3003, #72226307, … |
| maximal normal subgroup of | A4×#3003, #72226307, #72228056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084993);