Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| Fitting, socle, derived | C3×C5×C31 ⋊ C3×C4×C5 |
| Hall | C3×C5×(C31⋊C3)⋊C5 ⋊ C4 |
| Hall | C5×ΓL(1,𝔽32) ⋊ C3×Dic3 |
| Hall | C3×C31⋊C3 ⋊ C5×AGL(1,𝔽5) |
| Sylow | C31 ⋊ C3×C5×ΓL(1,𝔽16) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C31 | C3×C4×C5 |
| prime radical | C3×D5×(C31⋊C3)⋊C5 | C2 |
| cosocle | C3×D5×C31 | C2×C3×C5 |
| LCS | G ▷ C3×C5×C31 | |
| UCS | C1 | |
124 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,3,5,2,3,5,31|a^2=d,e^a=e^2,f^a=f^2,g^a=g^30,g^b=g^5,g^c=g^16,f^d=f^4147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 421);