Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D9)⋊AGL(1,𝔽5) |
| unambiguous semidirect product | (C2×AGL(1,𝔽5))⋊D9 |
| Hall | C5×C9 ⋊ C22⋊C4 |
| Sylow | C9 ⋊ C22⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ (C2×D9)⋊C4 |
102 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 102 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2^4,3,5,3|a^2=c,e^3=g,b^a=bd,f^a=f^2,e^b=e^2g^2,g^b=g^2,f^c=f^454 conjugacy classes — view the full character table.
| maximal subgroup of | #13085170, #13085172, AGL(1,𝔽5)×D36, … |
| maximal normal subgroup of | #13085170, #13085172, AGL(1,𝔽5)×D36, … |
| maximal quotient of | #13083912, #13083913, #13083916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 101);