Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #157 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C16 · Schur multiplier (H2) = C3 |
| direct product | C5×#3708 |
| derived, Sylow | C32 ⋊ C5×C16 |
| Hall | C32×C5 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | #157 |
| commutator, hypocenter | C32 | C5×C16 |
| Frattini | C4 | C5×#157 |
| Fitting | C32×C4×C5 | C4 |
| socle, prime radical | C2×C32×C5 | C8 |
| cosocle | C3⋊(C3⋊C8) | C2×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4×C5 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#63725, #13085378, D5×#3708, … |
| maximal normal subgroup of | C5×#63725, #13085378, D5×#3708, … |
| maximal quotient of | C5×#63540, C2×C5×#3708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 192);