Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #5450 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #23306⋊C3 |
| Fitting, Sylow | #23306 ⋊ C3 |
| derived | C24 ⋊ C2×C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #5450 |
| commutator, hypocenter | C24 | C2×C3×C8 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #23306 | C3 |
| socle | C23 | C2×C4×A4 |
| prime radical | C2×#120⋊C3 | C4 |
| cosocle | C24×C4 | C22×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C4 | |
240 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 36 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=h,h^2=i,b^a=bdfg,c^a=cfg,d^a=e,e^a=de,f^a=g,g^a=fg,h^a=gh,c^b=cf,d^b=dg,e^b=efg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083892);