Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C22×#123 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Fitting | C22×#123 | D3 |
| socle | C24 | C23×D3 |
| prime radical | C2×#5914 | C2 |
| hypercenter | #123 | S4 |
| hypocenter | A4 | #536 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ #123 | |
1533 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 1533 conjugacy classes of subgroups — too many to draw. Only its 125 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:
2^6,3,2^2|a^2=f,d^2=f,d^a=df,g^a=g^2,h^a=hi,c^b=cf,d^c=de,h^g=i,i^g=hi58 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088855);