Finite Group DatabasebetaFinite Group Databasebeta| order | 36300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #20300037⋊C5 |
| Fitting, socle | C5×C112 ⋊ C5×Dic3 |
| derived | C5×C112⋊C3 ⋊ C4×C5 |
| Hall | C5×(C112⋊C3)⋊C5 ⋊ C4 |
| Hall | C5×#10597243 ⋊ Dic3 |
| Hall | C112⋊C3 ⋊ C5×AGL(1,𝔽5) |
| Sylow | C112 ⋊ C5×ΓL(1,𝔽16) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C5×C112⋊C3 | C4×C5 |
| Fitting, socle | C5×C112 | C5×Dic3 |
| prime radical | D5×(C112⋊C3)⋊C5 | C2 |
| cosocle | D5×C112⋊C3 | C2×C5 |
| LCS | G ▷ C5×C112⋊C3 | |
| UCS | C1 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 165 |
Polycyclic presentation:
2,5,2,3,5,11^2|a^2=c,d^a=d^2,e^a=e^2,f^a=fg^4,g^a=g^10,f^b=f^4,g^b=g^4,e^c=e^4,f^d=f^3g^9,g^d=fg^767 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36300, 478);