Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #32869⋊C3 |
| Fitting, Sylow | #32869 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #2823⋊C3 |
| commutator, cosocle | #376 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #32869 | C3 |
| socle | C23 | D4×A4 |
| prime radical | #95⋊C3 | D4 |
| hypercenter | C4 | #560⋊C3 |
| hypocenter | #95 | C3×D4 |
| LCS | G ▷ #376 ▷ #95 | |
| UCS | C1 ◁ C2 ◁ C4 | |
137 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=fgi,c^2=i,d^2=i,e^2=gi,h^2=i,b^a=bei,c^a=cfg,d^a=dg,e^a=bgi,f^a=g,g^a=fg,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=efg,h^c=hi,e^d=ef,h^d=hi30 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083966);