Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | D3×#1619 |
| Fitting | C3×#1619 ⋊ C2 |
| Sylow | C3 ⋊ C2×#1619 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×#120 |
| commutator, cosocle | C3×C42 | C24 |
| Frattini | C42 | C23×D3 |
| Fitting | C3×#1619 | C2 |
| socle | C22×C3 | C2×#120 |
| hypercenter | #1619 | D3 |
| hypocenter | C3 | C2×#1619 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1619 | |
1838 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#18524, #72219772 |
| maximal normal subgroup of | D3×#18524 |
| maximal quotient of | D3×#12310, D3×#12314, #72219772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 145169);