Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32895⋊C3 |
| Fitting, Sylow | #32895 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | AΣL(1,𝔽64) |
| commutator, cosocle | #585 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #32895 | C3 |
| hypercenter | Q8 | #120⋊C3 |
| hypocenter | H(5,𝔽2) | C3×D4 |
| LCS | G ▷ #585 ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
655 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 655 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|d^2=i,f^2=i,g^2=i,b^a=bfg,c^a=dfg,d^a=cef,e^a=cdfhi,f^a=gi,g^a=fh,h^a=fgi,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 | #72236743, #72236755, #72236770, … |
| maximal normal subgroup of | #72236743, #72236755, #72236770, … |
| maximal quotient of | C2×#32895⋊C3, #72341821, #72341912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083981);