Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #53304⋊C3 |
| Fitting, Sylow | #53304 ⋊ C3 |
| derived | #2810 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3567⋊C3 |
| commutator, cosocle, hypocenter | #2810 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #53304 | C3 |
| LCS | G ▷ #2810 | |
| UCS | C1 ◁ C2 | |
297 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 21 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|d^2=gh,e^2=h,f^2=gh,b^a=bcdfgi,c^a=bhi,d^a=degi,e^a=dg,f^a=efhi,g^a=h,h^a=gh,i^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fg,d^c=di,e^c=eg,f^c=fhi26 conjugacy classes — view the full character table.
| maximal subgroup of | #72239174 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084525);