Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | D3×#1437 |
| Fitting | C3×#1437 ⋊ C2 |
| Sylow | C3 ⋊ C2×#1437 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C22×C3 | C22×C42 |
| Fitting | C3×#1437 | C2 |
| socle, cosocle | C24×C3 | C24 |
| prime radical | C25×D3 | C22 |
| hypercenter | #1437 | D3 |
| hypocenter | C3 | C2×#1437 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #1437 | |
5230 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#1437, D3×#15258, D3×#14357, … |
| maximal normal subgroup of | D3×#15258, D3×#14357, D3×#13967, … |
| maximal quotient of | C3⋊D3×#1437, D3×#10948, D3×#10937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90701);