Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#2992 ⋊ C2 |
| Sylow | C3 ⋊ #60264 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C24×D3 |
| commutator, socle | C22×C3 | C24×C4 |
| Fitting | C3×#2992 | C2 |
| prime radical | D3×#121 | C4 |
| cosocle | C2×C3×C4 | C25 |
| hypercenter | #2992 | D3 |
| hypocenter | C3 | #60264 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #2992 | |
1872 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=g,d^2=f,g^2=h,b^a=bf,d^a=df,e^a=eh,c^b=cf,d^b=dh,i^e=i^2120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1034693);