Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #35498⋊C3 |
| Fitting, Sylow | #35498 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3444⋊C3 |
| commutator, cosocle, hypocenter | #2276 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #35498 | C3 |
| prime radical | #73286 | C2 |
| LCS | G ▷ #2276 | |
| UCS | C1 ◁ C2 | |
287 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 19 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=gh,e^2=gh,f^2=g,b^a=bg,c^a=cdfg,d^a=ce,e^a=efgi,f^a=ehi,g^a=gh,h^a=g,c^b=cg,d^b=dh,d^c=di,f^c=fi,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72239921 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084270);