Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #63035⋊C3 |
| Fitting, Sylow | #63035 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C24×A4 |
| commutator, cosocle | C2×Q8 | C24×C3 |
| Fitting | #63035 | C3 |
| hypercenter | #555 | A4 |
| hypocenter | Q8 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #555 | |
1541 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 1541 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=i,d^2=i,e^2=h,f^2=h,g^2=h,b^a=bdei,c^a=cdei,d^a=chi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^f=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085216);