Finite Group DatabasebetaFinite Group Databasebeta| order | 10980 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#11927414⋊C3) |
| unambiguous semidirect product | (C3⋊#11927414)⋊C3 |
| Fitting, socle, derived | C3×C5×C61 ⋊ C3×C4 |
| Hall | C3×C5×C61⋊C3 ⋊ C4 |
| Hall | C3×C61⋊C3 ⋊ AGL(1,𝔽5) |
| Hall | C5×C61 ⋊ C3×Dic3 |
| Sylow | C61 ⋊ C3×ΓL(1,𝔽16) |
| Sylow | C5 ⋊ C3⋊(Dic61⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C61 | C3×C4 |
| prime radical | C3×D5×C61⋊C3 | C2 |
| cosocle | C3×D5×C61 | C2×C3 |
| LCS | G ▷ C3×C5×C61 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,2,3,5,61|a^2=c,d^a=d^2,e^a=e^2,f^a=f^60,f^b=f^13,e^c=e^4147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10980, 134);