Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting, Sylow | C24×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C22⋊(C22⋊A4) |
| commutator, cosocle, hypocenter | C24×Q8 | C2×C3 |
| Frattini | C2 | C2×C22⋊(C22⋊A4) |
| Fitting | C24×#41 | C3 |
| socle | C25 | C2×A4 |
| LCS | G ▷ C24×Q8 | |
| UCS | C1 ◁ C4 | |
11070 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 11070 conjugacy classes of subgroups — too many to draw. Only its 97 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|d^2=i,b^a=cefgi,c^a=bcdhi,e^a=ghi,f^a=ehi,g^a=fhi,h^a=efgi,c^b=ci64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085294);