Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #62788⋊C3 |
| Fitting, Sylow | #62788 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×A4 |
| commutator, cosocle | C2×Q8 | C24×C3 |
| Frattini, socle | C22 | C24×A4 |
| Fitting | #62788 | C3 |
| hypercenter | #525 | A4 |
| hypocenter | Q8 | C2×C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ #525 | |
1594 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|d^2=i,e^2=i,g^2=h,d^a=deh,e^a=dfhi,c^b=ch,d^b=di,f^b=fi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085090);