Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting, Sylow | #33999 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C42⋊C3 |
| commutator, cosocle, hypocenter | #95 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #33999 | C3 |
| hypercenter | #41 | C42⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C4 ◁ #41 | |
693 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 693 conjugacy classes of subgroups — too many to draw. Only its 66 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=h,c^2=i,d^2=h,f^2=g,b^a=bcdfghi,c^a=cdghi,d^a=cf,h^a=gi,i^a=ghi,c^b=cg,e^b=eg,d^c=dg72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084137);