Finite Group DatabasebetaFinite Group Databasebeta| order | 1320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #63143⋊C5 |
| Hall | #63143 ⋊ C5 |
| Hall | C3×C11⋊C5 ⋊ Q8 |
| Hall | C11⋊C5 ⋊ Dic6 |
| Hall | C3×C11 ⋊ C5×Q8 |
| Sylow | C11 ⋊ C5×Dic6 |
| Sylow | C3 ⋊ (C11⋊C5)⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×AGL(1,𝔽11) |
| commutator, Fitting, socle, cosocle | C2×C3×C11 | C22×C5 |
| prime radical | C2×C3×C11⋊C5 | C22 |
| hypocenter | C3×C11 | C5×Q8 |
| LCS | G ▷ C2×C3×C11 ▷ C3×C11 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 264 |
Polycyclic presentation:
2^2,5,2,3,11|a^2=d,b^2=d,b^a=bd,f^a=f^10,e^b=e^2,f^c=f^951 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C11⋊C5)⋊Q8, #72216874, Dic6×AGL(1,𝔽11), … |
| maximal normal subgroup of | D3×(C11⋊C5)⋊Q8, #72216874, Dic6×AGL(1,𝔽11), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1320, 21);