Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| Fitting | C22×C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ #35347 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C22×C3×#121 | C2 |
| hypercenter | C22×#121 | D3 |
| hypocenter | C3 | #35347 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×#121 | |
5806 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,i^c=i^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#35347, #72253389, #72253393, … |
| maximal normal subgroup of | D3×#35347, #72253389, #72253393, … |
| maximal quotient of | #72253816, #72270857, #72270865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 348635);