Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| Fitting | C3×#2276 ⋊ C2 |
| Sylow | C3 ⋊ C4×#520 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator | C2×C3 | C23×C42 |
| Fitting | C3×#2276 | C2 |
| socle, cosocle | C23×C3 | C25 |
| prime radical | C22×#239 | C22 |
| hypercenter | #2276 | D3 |
| hypocenter | C3 | C4×#520 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2276 | |
3698 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=g,b^2=h,c^2=f,e^2=g,b^a=bf,d^a=df,c^b=cf,i^c=i^2,i^d=i^2,i^e=i^2204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 327810);