Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C23×C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ #62811 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C23×C3×#41 | C2 |
| hypercenter | C23×#41 | D3 |
| hypocenter | C3 | #62811 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C23×#41 | |
11534 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|f^2=h,b^a=bg,c^a=ch,e^a=eh,i^a=i^2,d^c=dh168 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#62811, #72253957, #72253971, … |
| maximal normal subgroup of | D3×#62811, #72253957, #72253971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1078108);