Finite Group DatabasebetaFinite Group Databasebeta| order | 21 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3 (order 12) · Inn(G) = C1 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = D3×AGL(1,𝔽7) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C21 |
| abelian | C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 21 |
A minimal faithful permutation representation — 1 generator on 10 points:
(1,2,3)(4,5,6,7,8,9,10)21 conjugacy classes — view the full character table.
| center of | C3×D3×C7, C7×H(3,𝔽3), C7×3−1+2, … |
| derived subgroup of | D21, Dic21, D3×D7, … |
| Frattini subgroup of | C49⋊C9, C9×C49, D49⋊C9, … |
| Fitting subgroup of | D3×C7, C3×D7, D21, … |
| socle of | D3×C7, C3×D7, D21, … |
| maximal subgroup of | D3×C7, C3×D7, D21, … |
| maximal normal subgroup of | D3×C7, C3×D7, D21, … |
| maximal quotient of | C2×C3×C7, C7×C9, C32×C7, … |
| outer automorphism group of | D43, D49, C23⋊C49 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21, 2);