Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32894⋊C3 |
| Fitting, Sylow | #32894 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | AΣL(1,𝔽64) |
| commutator, cosocle | #585 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #32894 | C3 |
| hypercenter | D4 | #120⋊C3 |
| hypocenter | H(5,𝔽2) | C3×D4 |
| LCS | G ▷ #585 ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
678 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|f^2=i,g^2=i,c^a=cefh,d^a=ci,e^a=cdefgh,f^a=fh,g^a=f,h^a=fghi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,f^c=fi,h^c=hi,g^d=gi,f^e=fi34 conjugacy classes — view the full character table.
| maximal subgroup of | #72233783, #72236755, #72236763, … |
| maximal normal subgroup of | #72233783, #72236755, #72236763, … |
| maximal quotient of | C2×#32894⋊C3, #72341916, #72341934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083980);