Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #17307⋊C3 |
| Fitting, derived, Sylow | #17307 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #564⋊C3 |
| commutator, Fitting, cosocle, hypocenter | #17307 | C3 |
| Frattini, socle | C24 | C22⋊A4 |
| LCS | G ▷ #17307 | |
| UCS | C1 ◁ C22 | |
204 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 19 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=fg,c^2=fghi,d^2=h,e^2=f,b^a=eh,c^a=bcdefgh,d^a=bchi,e^a=befg,f^a=gi,g^a=fgi,h^a=fhi,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=efg,e^d=ef26 conjugacy classes — view the full character table.
| maximal subgroup of | #72231842, #72231843, #72231844, … |
| maximal normal subgroup of | #72231842, #72231843, #72231844, … |
| maximal quotient of | #72237342, #72299953 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083719);