Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D3×#3292 |
| Fitting | C3×#3292 ⋊ C2 |
| Sylow | C3 ⋊ C2×#3292 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D42 |
| commutator, cosocle | C2×C3×C4 | C25 |
| Frattini | C2×C4 | C24×D3 |
| Fitting | C3×#3292 | C2 |
| socle | C2×C3 | C2×D42 |
| hypercenter | #3292 | D3 |
| hypocenter | C3 | C2×#3292 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3292 | |
2766 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
87 conjugacy classes — view the full character table.
| maximal subgroup of | #72214341, #72214946, D3×#61110, … |
| maximal normal subgroup of | D3×#61110, D3×#12310, D3×#12366, … |
| maximal quotient of | #72214341, #72214946, C2×D3×#3292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1045789);