Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #14331⋊C3 |
| Fitting, Sylow | #14331 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#126⋊C3 |
| commutator, hypocenter | #95 | C2×C3×C4 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #14331 | C3 |
| prime radical | C2×#396⋊C3 | C2 |
| cosocle | C2×#95 | C22×C3 |
| hypercenter | C2×C4 | #126⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
285 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 33 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=i,c^2=g,e^2=ghi,b^a=cefgi,c^a=bcefg,e^a=efgh,g^a=fh,h^a=fgh,i^a=hi,c^b=cf,d^b=dg,d^c=dh,e^d=ef48 conjugacy classes — view the full character table.
| maximal subgroup of | #72222701, #72222823, #72222837 |
| maximal normal subgroup of | #72222701, #72222823, #72222837 |
| maximal quotient of | #72222696 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083623);