Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #62845⋊C3 |
| Fitting, Sylow | #62845 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Fitting | #62845 | C3 |
| hypercenter | C4⋊C4 | C22⋊A4 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
1282 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 1282 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|d^2=h,e^2=h,b^a=bceh,c^a=bi,d^a=dghi,f^a=fghi,g^a=fi,c^b=ch,d^b=di,g^b=gi,f^c=fi,e^d=ei54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085120);