Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C5⋊((C22⋊C9)⋊C2) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊C9)⋊Dic5 |
| derived | C5×C22⋊C9 ⋊ C4 |
| Sylow | C5 ⋊ (C22⋊C9)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C5⋊((C22⋊C9)⋊C2) |
| commutator, hypocenter | C5×C22⋊C9 | C4 |
| Frattini | C2×C3 | C5⋊S4 |
| Fitting, socle, prime radical | C23×C3×C5 | D3 |
| cosocle | C2×C5×C22⋊C9 | C2 |
| LCS | G ▷ C5×C22⋊C9 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2^2,3,5,3,2^2|a^2=b,c^3=e,c^a=c^2e^2,d^a=d^4,e^a=e^2,f^a=g,g^a=f,f^c=g,g^c=fg66 conjugacy classes — view the full character table.
| maximal subgroup of | #13085154, Dic5×(C22⋊C9)⋊C2, D5×(C22⋊C9)⋊C4, … |
| maximal normal subgroup of | #13085154, Dic5×(C22⋊C9)⋊C2, D5×(C22⋊C9)⋊C4, … |
| maximal quotient of | (C22⋊C9)⋊#168, #13083943, #13083944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 107);