Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #61903⋊C3 |
| Fitting, Sylow | #61903 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4×A4 |
| commutator, cosocle | C4×Q8 | C23×C3 |
| Frattini | C2×C4 | C23×A4 |
| Fitting | #61903 | C3 |
| hypercenter | #482 | A4 |
| hypocenter | Q8 | C3×AGL(1,ℤ/8) |
| LCS | G ▷ C4×Q8 ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #482 | |
474 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 65 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|c^2=g,d^2=g,e^2=gi,f^2=i,h^2=i,c^a=cegi,e^a=cfi,c^b=cg,d^b=dh,f^b=fg,h^b=hi,e^c=egi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084872);