Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting, Sylow | #62652 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C24×A4 |
| commutator, hypocenter | Q8 | C23×C3×C4 |
| Fitting | #62652 | C3 |
| prime radical | C22×#994 | C2 |
| cosocle | C2×Q8 | C24×C3 |
| hypercenter | #519 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #519 | |
2364 conjugacy classes of subgroups; only the 357 normal subgroups are shown.
This group has 2364 conjugacy classes of subgroups — too many to draw. Only its 357 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=i,b^a=bcdehi,c^a=fi,d^a=efh,e^a=df,f^a=cdefi,c^b=ch,g^b=gh,f^c=fh,e^d=eh134 conjugacy classes — view the full character table.
| maximal subgroup of | #72226033, #72226116, #72226247, … |
| maximal normal subgroup of | #72226033, #72226116, #72226247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084982);