Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #62635⋊C3 |
| Fitting, Sylow | #62635 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | A4×D8 |
| commutator, cosocle | #131 | C23×C3 |
| Frattini | C8 | C23×A4 |
| Fitting | #62635 | C3 |
| socle | C2 | C2×A4×D8 |
| hypercenter | #508 | A4 |
| hypocenter | Q8 | C2×C3×D8 |
| LCS | G ▷ #131 ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508 | |
473 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|f^2=i,g^2=hi,h^2=i,d^a=ei,e^a=defi,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084949);