Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| Fitting | C3×#3026 ⋊ C2 |
| Sylow | C3 ⋊ #60398 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D3×#120 |
| commutator, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C3×#3026 | C2 |
| socle | C2×C3 | C22×#120 |
| hypercenter | #3026 | D3 |
| hypocenter | C3 | #60398 |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ #3026 | |
4836 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|e^2=h,b^a=bf,c^a=cg,d^a=dh,g^b=gh,f^c=fh,i^e=i^287 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#60398 |
| maximal normal subgroup of | D3×#60398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1036309);