Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of ((C24⋊C5)⋊C2)⋊C3 |
| automorphism | Aut(G) = AΓL(1,𝔽16) (order 960) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ((C24⋊C5)⋊C2)⋊C3 |
| unambiguous semidirect product | AGL(1,𝔽16)⋊C2 |
| Fitting, socle | C24 ⋊ C3×D5 |
| derived | C24⋊C5 ⋊ C2×C3 |
| Hall | (C24⋊C5)⋊C2 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24⋊C5 | C2×C3 |
| Fitting, socle | C24 | C3×D5 |
| LCS | G ▷ C24⋊C5 | |
| UCS | C1 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,14,13,15,8,2)(3,7,6,10,5,16)(4,11,12);(2,8,5,6,15)(3,7,9,11,14)(4,10,12,13,16)Polycyclic presentation:
2,3,5,2^4|c^a=c^4,d^a=de,f^a=deg,g^a=df,d^b=eg,e^b=dfg,f^b=de,g^b=ef,d^c=df,e^c=eg,f^c=dfg,g^c=de14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1188);