Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #3774 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#3776 |
| Hall | C32×C5 ⋊ Q16 |
| Sylow | C32 ⋊ C5×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C22×C5 |
| Frattini | C2 | C5×GO+(4,𝔽2) |
| Fitting, socle, prime radical | C2×C32×C5 | D4 |
| hypocenter | C32 | C5×Q16 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C5 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3776, #13088465, #13088468, … |
| maximal normal subgroup of | D5×#3776, #13088465, #13088468, … |
| maximal quotient of | C5×#63740, C5×#63743, C2×C5×#3776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 471);