Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#1452 ⋊ C2 |
| Sylow | C3 ⋊ #17319 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C23×D3 |
| commutator, socle | C22×C3 | C22×C42 |
| Fitting | C3×#1452 | C2 |
| prime radical | C23×D3 | C42 |
| cosocle | C3×C42 | C24 |
| hypercenter | #1452 | D3 |
| hypocenter | C3 | #17319 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ #1452 | |
834 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=e,b^2=f,e^2=g,f^2=h,b^a=bh,d^a=dg,c^b=cg,i^d=i^2144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 135173);