Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C4×D3 (order 96) · Inn(G) = D3 · Out(G) = C22×C4 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×#10595100 |
| homology | Abelianization (H1) = C5×C8 · Schur multiplier (H2) = C1 |
| direct product | C5×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C5×C8) |
| derived, Sylow | C3 ⋊ C5×C8 |
| Hall | C3×C5 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | D3 |
| commutator, hypocenter | C3 | C5×C8 |
| Frattini | C4 | C5×D3 |
| Fitting | C3×C4×C5 | C2 |
| socle, prime radical | C2×C3×C5 | C4 |
| cosocle | C3×C4 | C2×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C5 | |
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)(4,9,6,11,7,5,10,8)(12,13,14,15,16);(1,2,3)60 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C3⋊C8, #6648, #6649, … |
| maximal normal subgroup of | D5×C3⋊C8, #6648, #6649, … |
| maximal quotient of | C5×C3⋊C16, C2×C5×C3⋊C8, C5×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 1);