Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | A4⋊#401 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×S4 |
| commutator, prime radical, cosocle | C23×A4 | C23 |
| Fitting | C22×#95 | D3 |
| socle | C25 | C22×D3 |
| hypercenter | #95 | S4 |
| hypocenter | A4 | #401 |
| LCS | G ▷ C23×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ #95 | |
655 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 655 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3,2^2|a^2=f,b^2=d,c^2=e,b^a=be,c^a=cdf,g^a=g^2,h^a=hi,c^b=cef,h^g=i,i^g=hi64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086159);