Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×#2021 ⋊ C2 |
| Sylow | C3 ⋊ #25587 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×#120 |
| commutator, cosocle | C24×C3 | C24 |
| Frattini | C24 | C23×D3 |
| Fitting | C3×#2021 | C2 |
| socle | C22×C3 | C2×#120 |
| hypercenter | #2021 | D3 |
| hypocenter | C3 | #25587 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2021 | |
1332 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|c^2=f,d^2=eh,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^c=gh,i^d=i^269 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 222955);