Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of AGL(1,𝔽16) |
| automorphism | Aut(G) = AΓL(1,𝔽16) (order 960) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C1 |
| affine general linear | AGL(1,𝔽16) |
| unambiguous semidirect product | (C22⋊A4)⋊C5 |
| unambiguous semidirect product | (C24⋊C5)⋊C3 |
| Fitting, socle, derived, Sylow | C24 ⋊ C3×C5 |
| Hall | C24⋊C5 ⋊ C3 |
| Hall | C22⋊A4 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C24 | C3×C5 |
| LCS | G ▷ C24 | |
| UCS | C1 | |
14 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,16,11,5,15,13,7,4,12,9,6,3,8,14,10);(1,16)(2,15)(3,7)(4,10)(5,9)(6,12)(8,14)(11,13)Polycyclic presentation:
3,5,2^4|c^a=cdf,d^a=cdef,e^a=cde,f^a=def,c^b=cdef,d^b=cde,e^b=def,f^b=ce16 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 191);