Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = C22×AGL(1,𝔽5) (order 80) · Inn(G) = AGL(1,𝔽5) · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = D3×#13107464 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×#170 |
| derived, Sylow | C5 ⋊ C3×C8 |
| Hall | C3×C5 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C3×C8 |
| Frattini | C2 | C3×AGL(1,𝔽5) |
| Fitting, socle, prime radical | C2×C3×C5 | C4 |
| cosocle | Dic5 | C2×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,5,4,3)(6,10,12,7,9,13,8,11)(14,15,16);(1,2,3,5,4)30 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#170, #6738, #6739, … |
| maximal normal subgroup of | D3×#170, #6738, #6739, … |
| maximal quotient of | C3×#674, C2×C3×#170, C9×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 6);