Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = C2×D5 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D20 |
| unambiguous semidirect product | C4⋊(C3×D5) |
| Fitting | C3×C4×C5 ⋊ C2 |
| Hall | C3×C5 ⋊ D4 |
| Sylow | C5 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D5 |
| commutator, cosocle | C2×C5 | C22×C3 |
| Frattini | C2 | C2×C3×D5 |
| Fitting | C3×C4×C5 | C2 |
| socle | C2×C3×C5 | C22 |
| hypercenter | C3×C4 | D5 |
| hypocenter | C5 | C3×D4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3)(4,5,6,7)(8,9,10,11,12);(5,7)(9,12)(10,11)39 conjugacy classes — view the full character table.
| maximal subgroup of | #6652, #6653, #6655, … |
| maximal normal subgroup of | #6652, #6653, #6655, … |
| maximal quotient of | C3×#677, C3×D40, C3×Dic20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 18);