Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | D3×#3584 |
| Fitting | C3×#3584 ⋊ C2 |
| Sylow | C3 ⋊ C2×#3584 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D3×D4 |
| commutator, cosocle | C3×C4 | C26 |
| Frattini | C4 | C25×D3 |
| Fitting | C3×#3584 | C2 |
| socle | C2×C3 | C24×D4 |
| hypercenter | #3584 | D3 |
| hypocenter | C3 | C2×#3584 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ #3584 | |
10858 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | #72222879, D3×#60184, D3×#62636, … |
| maximal normal subgroup of | D3×#60184, D3×#62636, D3×#33388, … |
| maximal quotient of | #72222879 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083395);