Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #60276⋊C3 |
| Fitting, Sylow | #60276 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C3×C8 |
| Frattini | C2×C4 | C23×A4 |
| Fitting | #60276 | C3 |
| socle | C22 | C22×C4×A4 |
| prime radical | C22×#41⋊C3 | C4 |
| cosocle | C4×Q8 | C23×C3 |
| hypercenter | #431 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C42 ◁ #431 | |
546 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 546 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|b^2=h,f^2=g,h^2=i,c^a=cdfgi,d^a=ce,c^b=cg,e^b=eg,d^c=dg152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084677);