Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #60471⋊C3 |
| Fitting, Sylow | #60471 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | A4×#121 |
| commutator, cosocle | C2×#41 | C23×C3 |
| Frattini | C2×C4 | C23×A4 |
| Fitting | #60471 | C3 |
| hypercenter | #473 | A4 |
| hypocenter | Q8 | C3×#121 |
| LCS | G ▷ C2×#41 ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #473 | |
600 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|c^2=gi,g^2=i,h^2=i,b^a=beh,d^a=defgh,e^a=efg,f^a=eghi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,h^d=hi,f^e=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084763);