Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #3688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#3688 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | (C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C2×C5 |
| Frattini | C2×C3 | C5×S4 |
| Fitting | C3×C5×Q8 | D3 |
| socle, prime radical | C2×C3×C5 | S4 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2×C5 | |
42 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | #13085146, #13085148, D5×#3688, … |
| maximal normal subgroup of | #13085146, #13085148, D5×#3688, … |
| maximal quotient of | C5×#63421, C2×C5×#3688 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 102);