Finite Group DatabasebetaFinite Group Databasebeta| order | 36300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#14559097⋊C5) |
| unambiguous semidirect product | (C3⋊#14559097)⋊C5 |
| Fitting, socle, derived | C3×C5×C112 ⋊ C4×C5 |
| Hall | C3×C5×#10597243 ⋊ C4 |
| Hall | C5×#10597243 ⋊ Dic3 |
| Hall | C3×C112 ⋊ C5×AGL(1,𝔽5) |
| Sylow | C112 ⋊ C5×ΓL(1,𝔽16) |
| Sylow | C3 ⋊ #14559097⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C112 | C4×C5 |
| prime radical | C3×C5⋊#11925892 | C2 |
| cosocle | C3×C5⋊(C11⋊D11) | C2×C5 |
| LCS | G ▷ C3×C5×C112 | |
| UCS | C1 | |
172 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 330 |
Polycyclic presentation:
2,5,2,3,5,11^2|a^2=c,d^a=d^2,e^a=e^2,f^a=f^5g^7,g^a=fg^6,f^b=f^3,g^b=g^3,e^c=e^4,f^c=f^10,g^c=g^10135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36300, 577);