Finite Group DatabasebetaFinite Group Databasebeta| order | 36300 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ((C11⋊(C11⋊AGL(1,𝔽5)))⋊C3)⋊C5 |
| unambiguous semidirect product | #27130893⋊C3 |
| Fitting, socle, derived | C5×C112 ⋊ C3×C4×C5 |
| Hall | #27130893 ⋊ C3 |
| Hall | C5×(C112⋊C3)⋊C5 ⋊ C4 |
| Hall | C5×#10597243 ⋊ C3×C4 |
| Hall | C112⋊C3 ⋊ C5×AGL(1,𝔽5) |
| Sylow | C112 ⋊ C3×C5×AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C112 | C3×C4×C5 |
| prime radical | D5×(C112⋊C3)⋊C5 | C2 |
| cosocle | D5×C112 | C2×C3×C5 |
| LCS | G ▷ C5×C112 | |
| UCS | C1 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 165 |
Polycyclic presentation:
2,3,5,2,5,11^2|a^2=d,e^a=e^2,f^a=f^10,g^a=g^10,f^b=f^7g^2,g^b=f^10g^3,f^c=f^3,g^c=g^3,e^d=e^491 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36300, 479);