Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C42 |
| direct product | C2×D3×H(3,𝔽4) |
| Fitting | C2×C3×H(3,𝔽4) ⋊ C2 |
| Sylow | C3 ⋊ C22×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C24×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C2×C3×H(3,𝔽4) | C2 |
| socle | C23×C3 | C25 |
| hypercenter | C2×H(3,𝔽4) | D3 |
| hypocenter | C3 | C22×H(3,𝔽4) |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×H(3,𝔽4) | |
10246 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
114 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#51068, C2×D3×#3532, C2×D3×#2199, … |
| maximal normal subgroup of | D3×#51068, C2×D3×#3532, C2×D3×#2199, … |
| maximal quotient of | C2×D3×#2679, C2×D3×#2684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076439);