Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C4⋊#5450 |
| unambiguous semidirect product | #45479⋊C3 |
| Fitting, Sylow | #45479 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#5450 |
| commutator, cosocle | C25 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #45479 | C3 |
| hypercenter | C4 | #5450 |
| hypocenter | C24 | C2×C3×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ C4 | |
584 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=g,d^2=gh,b^a=bef,c^a=ceghi,d^a=defg,e^a=efh,f^a=ei,g^a=i,h^a=ghi,i^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fg,d^c=di,e^c=egi,f^c=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084446);