Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #62949⋊C3 |
| Fitting, Sylow | #62949 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C24×A4 |
| commutator, cosocle | C2×Q8 | C24×C3 |
| Fitting | #62949 | C3 |
| hypercenter | #539 | A4 |
| hypocenter | Q8 | C3×2−1+4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #539 | |
1361 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|c^2=h,d^2=h,e^2=i,f^2=hi,b^a=bcdghi,c^a=cdgi,d^a=cghi,c^b=ch,d^b=di,e^b=eh,g^b=gi,d^c=dh,f^e=fi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085154);