Finite Group DatabasebetaFinite Group Databasebeta| order | 36300 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ((C11⋊Dic55)⋊C3)⋊C5 |
| unambiguous semidirect product | #27130851⋊C3 |
| derived | C5×C112 ⋊ C3×C4×C5 |
| Hall | #27130851 ⋊ C3 |
| Hall | C5×(C112⋊C3)⋊C5 ⋊ C4 |
| Hall | C5×#10597243 ⋊ C3×C4 |
| Hall | C112⋊C3 ⋊ C5×Dic5 |
| Sylow | C112 ⋊ C3×C5×Dic5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C5⋊(((C11⋊D11)⋊C3)⋊C5) |
| commutator, hypocenter | C5×C112 | C3×C4×C5 |
| Fitting, socle, cosocle | C2×C5×C112 | C2×C3×C5 |
| prime radical | C2×C5×(C112⋊C3)⋊C5 | C2 |
| LCS | G ▷ C5×C112 | |
| UCS | C1 ◁ C2 | |
128 conjugacy classes of subgroups.
| transitive degree | μt(G) = 660 |
Polycyclic presentation:
2,3,5^2,11^2,2|a^2=g,d^a=d^4,e^a=e^10,f^a=f^10,e^b=e^3f^9,f^b=ef^7,e^c=e^3,f^c=f^3160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36300, 357);