Finite Group DatabasebetaFinite Group Databasebeta| order | 36900 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D123⋊#3874 |
| Fitting, socle | C3×C52×C41 ⋊ C2×D3 |
| derived, Hall | C3×C41×C52⋊C3 ⋊ C22 |
| Hall | C52×C41 ⋊ D32 |
| Hall | C3×C52⋊C3 ⋊ C2×D41 |
| Sylow | C41 ⋊ D3×#3874 |
| Sylow | C52 ⋊ (C3⋊D3)⋊D41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C41×C52⋊C3 | C22 |
| Fitting, socle | C3×C52×C41 | C2×D3 |
| LCS | G ▷ C3×C41×C52⋊C3 | |
| UCS | C1 | |
128 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1845 |
Polycyclic presentation:
2^2,3^2,41,5^2|d^a=d^2,e^a=e^40,c^b=c^2,d^b=d^2,f^b=f^2g^4,g^b=f^3g^3,f^c=f^2g^4,g^c=f^2g^2519 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36900, 414);