Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
No aliases for this group.
| Fitting | C2×C3×#585 ⋊ C2 |
| Sylow | C3 ⋊ #63103 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C25×D3 |
| commutator, socle, cosocle | C2×C3 | C27 |
| Frattini | C2 | C26×D3 |
| Fitting | C2×C3×#585 | C2 |
| hypercenter | C2×#585 | D3 |
| hypocenter | C3 | #63103 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×#585 | |
59452 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|g^2=h,b^a=bh,f^a=fh,e^b=eh,d^c=dh,i^f=i^2198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083470);