Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #8546⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #8546⋊C3 |
| Fitting, Sylow | #8546 ⋊ C3 |
| derived | C22×C42 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C22×C42 | C3×C4 |
| Frattini | C25 | C2×A4 |
| Fitting | #8546 | C3 |
| socle | C22 | #379⋊C3 |
| prime radical | #85647 | C2 |
| cosocle | C22×#127 | C2×C3 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 | |
341 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=h,d^2=i,b^a=bcefh,c^a=dfi,d^a=cdehi,e^a=fi,f^a=efh,g^a=egi,h^a=i,i^a=hi,c^b=ce,d^b=df,e^b=eh,f^b=fi,g^c=gh,g^d=gi24 conjugacy classes — view the full character table.
| derived subgroup of | #72283950 |
| maximal subgroup of | #72283900, #72283914, #72283925, … |
| maximal normal subgroup of | #72283900, #72283914, #72283925 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083514);