Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C23×AGL(1,𝔽5) (order 160) · Inn(G) = D5 · Out(G) = C24 |
| holomorph | G ⋊ Aut(G) = D3×#72213282 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×#168 |
| derived, Sylow | C5 ⋊ C3×C8 |
| Hall | C3×C5 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D5 |
| commutator, hypocenter | C5 | C3×C8 |
| Frattini | C4 | C3×D5 |
| Fitting | C3×C4×C5 | C2 |
| socle, prime radical | C2×C3×C5 | C4 |
| cosocle | C4×C5 | C2×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C3×C4 | |
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:
(2,3)(4,5)(6,10,12,7,9,13,8,11)(14,15,16);(1,2,4,5,3)48 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#674, C3⋊#674, D3×#168, … |
| maximal normal subgroup of | C3×#674, C3⋊#674, D3×#168, … |
| maximal quotient of | C3×#672, C2×C3×#168, C9×#168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 2);