Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #53065⋊C3 |
| Fitting, Sylow | #53065 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#5450 |
| commutator, cosocle, hypocenter | C22×Q8 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #53065 | C3 |
| hypercenter | C4 | #5450 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 | |
403 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=hi,f^2=ghi,b^a=bdfhi,c^a=cde,d^a=defgh,e^a=dghi,f^a=fg,g^a=gh,h^a=g,i^a=gi,c^b=cg,d^b=dh,e^b=egh,f^b=fghi,d^c=di,e^c=eg,e^d=eghi42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084511);