Finite Group DatabasebetaFinite Group Databasebeta| order | 105 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3×C4 (order 48) · Inn(G) = C1 · Out(G) = C22×C3×C4 |
| holomorph | G ⋊ Aut(G) = D3×AGL(1,𝔽5)×AGL(1,𝔽7) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C105 |
| abelian | C3×C5×C7 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 105 |
A minimal faithful permutation representation — 1 generator on 15 points:
(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(105, 2);