Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| Fitting | C3×#3458 ⋊ C2 |
| Sylow | C3 ⋊ #62843 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C25×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Fitting | C3×#3458 | C2 |
| prime radical | C23×D12 | C22 |
| hypercenter | #3458 | D3 |
| hypocenter | C3 | #62843 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #3458 | |
7130 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|a^2=g,b^2=g,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh,i^c=i^2,i^d=i^2126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1078585);