Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C9 · Schur multiplier (H2) = C22 |
| direct product | C2×C5×Q8×C9 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C9 | C22 |
| commutator | C2 | C23×C5×C9 |
| Frattini, cosocle | C2×C3 | C23×C3×C5 |
| socle, prime radical | C22×C3×C5 | C22×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C9 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(11,12,13,14,15)(16,17,18,19,20,21,22,23,24);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 187);