Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×C5×#157 |
| Fitting, socle | C22×C32×C5 ⋊ C4 |
| derived, Sylow | C32 ⋊ C22×C4×C5 |
| Hall | C32×C5 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | #157 |
| commutator, hypocenter | C32 | C22×C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C4 |
| prime radical | C22×C5×C3⋊D3 | C2 |
| cosocle | C3⋊D3 | C23×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C5 | |
172 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | #13085518, C5×#63737, #13085612, … |
| maximal normal subgroup of | #13085518, C5×#63737, #13085612, … |
| maximal quotient of | C2×C5×#3787, C2×C5×#3788, C5×#64283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 825);