Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11709219 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| Fitting, derived, Sylow | #13827 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #13827 | C3 |
| Frattini, socle | C24 | C22⋊A4 |
| LCS | G ▷ #13827 | |
| UCS | C1 | |
588 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=i,b^a=cdefg,c^a=bcefi,d^a=efhi,e^a=defgi,f^a=fg,g^a=f,h^a=gi,i^a=fghi,c^b=cf,d^b=dg,e^b=efg,d^c=dfg,e^c=ef28 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083600);