Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C2×C3×#519 ⋊ C2 |
| Sylow | C3 ⋊ #62653 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C25×D3 |
| commutator | C2×C3 | C25×C4 |
| Fitting | C2×C3×#519 | C2 |
| socle, cosocle | C22×C3 | C26 |
| prime radical | C22×#1007 | C2 |
| hypercenter | C2×#519 | D3 |
| hypocenter | C3 | #62653 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#519 | |
10930 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=h,c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg,i^e=i^2198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076189);