Finite Group DatabasebetaFinite Group Databasebeta| order | 21780 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3×C11 |
| unambiguous semidirect product | C3⋊#20300038 |
| Fitting, socle | C3×C5×C112 ⋊ Dic3 |
| derived, Hall | C3×C5×C112⋊C3 ⋊ C4 |
| Hall | C3×C112⋊C3 ⋊ AGL(1,𝔽5) |
| Hall | C5×C112 ⋊ C3⋊Dic3 |
| Sylow | C112 ⋊ C3⋊ΓL(1,𝔽16) |
| Sylow | C5 ⋊ C3⋊#13089721 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C3×C5×C112⋊C3 | C4 |
| Fitting, socle | C3×C5×C112 | Dic3 |
| prime radical, cosocle | C3×C5⋊((C11⋊D11)⋊C3) | C2 |
| LCS | G ▷ C3×C5×C112⋊C3 | |
| UCS | C1 | |
88 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3^2,5,11^2|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^5g^7,g^a=fg^6,e^b=e^4,f^b=f^10,g^b=g^10,f^c=f^3g,g^c=f^9g^7171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21780, 547);