Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #32878⋊C3 |
| Fitting, Sylow | #32878 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | AΣL(1,𝔽64) |
| commutator, cosocle | C24×C4 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #32878 | C3 |
| socle | C23 | D4×A4 |
| hypercenter | D4 | #120⋊C3 |
| hypocenter | C24 | C3×D8 |
| LCS | G ▷ C24×C4 ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
1085 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 1085 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|f^2=i,b^a=bg,c^a=cgh,d^a=deg,e^a=dgh,g^a=gh,h^a=g,c^b=cf,d^b=dg,e^b=eh,f^b=fi,f^c=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72233784, #72252246, #72252248, … |
| maximal normal subgroup of | #72233784, #72252246, #72252248, … |
| maximal quotient of | C2×#32878⋊C3, #72341925, #72341934 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083972);