Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C4×D4 (order 64) · Inn(G) = C22 · Out(G) = C22×C4 |
| holomorph | G ⋊ Aut(G) = D3×AGL(1,𝔽5)×#453 |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×D4 |
| unambiguous semidirect product | C4⋊(C2×C3×C5) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C3×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ G | |
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);(10,12)75 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 32);