Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #14333⋊C3 |
| Fitting, Sylow | #14333 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#126⋊C3 |
| commutator, hypocenter | #95 | C2×C3×C4 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #14333 | C3 |
| prime radical | C2×#95⋊C3 | C22 |
| cosocle | C2×#95 | C22×C3 |
| hypercenter | C2×C4 | #126⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
239 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 239 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,d^2=f,e^2=ghi,b^a=bch,c^a=begi,d^a=dfg,e^a=efgh,g^a=gh,h^a=fg,i^a=fgi,c^b=cf,d^b=dg,d^c=dh,e^d=ef48 conjugacy classes — view the full character table.
| maximal subgroup of | #72222703, #72222829, #72222844 |
| maximal normal subgroup of | #72222703, #72222829, #72222844 |
| maximal quotient of | #72222699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083624);