Finite Group DatabasebetaFinite Group Databasebeta| order | 21780 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C11 |
| unambiguous semidirect product | C3⋊((C11⋊(C11⋊AGL(1,𝔽5)))⋊C3) |
| unambiguous semidirect product | (C3⋊(C11⋊(C11⋊AGL(1,𝔽5))))⋊C3 |
| Fitting, socle, derived | C3×C5×C112 ⋊ C3×C4 |
| 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⋊((C11⋊Dic11)⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C112 | C3×C4 |
| prime radical | C3×D5×C112⋊C3 | C2 |
| cosocle | C3×D5×C112 | C2×C3 |
| LCS | G ▷ C3×C5×C112 | |
| UCS | C1 | |
104 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,2,3,5,11^2|a^2=c,d^a=d^2,e^a=e^2,f^a=f^10,g^a=g^10,f^b=f^7g^5,g^b=f^4g^3,e^c=e^4267 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21780, 550);