Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #60376⋊C3 |
| Fitting, Sylow | #60376 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4×A4 |
| commutator, cosocle | C2×#41 | C23×C3 |
| Frattini | C2×C4 | C23×A4 |
| Fitting | #60376 | C3 |
| hypercenter | #452 | A4 |
| hypocenter | Q8 | C3×#120 |
| LCS | G ▷ C2×#41 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #452 | |
1039 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1039 conjugacy classes of subgroups — too many to draw. Only its 81 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|d^2=i,g^2=i,b^a=bfghi,c^a=cfghi,e^a=fh,f^a=efg,c^b=cg,d^b=dh,g^b=gi,g^c=gi,f^e=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084706);