Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C4×S4 (order 192) · Inn(G) = C22 · Out(G) = C2×C4×D3 |
| holomorph | G ⋊ Aut(G) = D3×AGL(1,𝔽5)×#5935 |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C3×C5 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C3×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3)(4,5,6,7,8)(9,10,11,12)(13,16,15,14);(9,13,11,15)(10,14,12,16)75 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 33);