Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #60388⋊C3 |
| Fitting, Sylow | #60388 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | A4×#120 |
| commutator, cosocle | C2×#41 | C23×C3 |
| Frattini | C2×C4 | C23×A4 |
| Fitting | #60388 | C3 |
| hypercenter | #456 | A4 |
| hypocenter | Q8 | C3×#120 |
| LCS | G ▷ C2×#41 ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #456 | |
1039 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 1039 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|g^2=i,b^a=bfgi,c^a=cefh,d^a=dfgi,e^a=fh,f^a=efgh,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226022, #72226230, #72226260, … |
| maximal normal subgroup of | #72226230, #72226260, #72226285 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084712);