Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #20508⋊C3 |
| Fitting, Sylow | #20508 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×C42⋊C3 |
| commutator, cosocle | #376 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #20508 | C3 |
| socle | C23 | D4×A4 |
| hypercenter | D8 | C42⋊C3 |
| hypocenter | #95 | C3×D4 |
| LCS | G ▷ #376 ▷ #95 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
251 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 26 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=g,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bdghi,c^a=cdefgh,d^a=efg,e^a=degi,g^a=ghi,h^a=g,c^b=cf,d^b=di,e^b=ei,f^b=fi,f^c=fi,e^d=ei48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083829);