Finite Group DatabasebetaFinite Group Databasebeta| order | 3 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2 (order 2) · Inn(G) = C1 · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = D3 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| alternating | A3 |
| cyclic | C3 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
2 conjugacy classes of subgroups.
| minimal degree | μ(G) = 3 |
| transitive degree | μt(G) = 3 |
A minimal faithful permutation representation — 1 generator on 3 points:
(1,2,3)| C1 | C2 | C3 | |
|---|---|---|---|
| size | 1 | 1 | 1 |
| order | 1 | 3 | 3 |
| χ1 | 1 | 1 | 1 |
| χ2 | 1 | ||
| χ3 | 1 |
Here i is the imaginary unit (i² = −1).
| center of | C3×D3, H(3,𝔽3), 3−1+2, … |
| derived subgroup of | D3, Dic3, C2×D3, … |
| Frattini subgroup of | C9, D9, C2×C9, … |
| Fitting subgroup of | D3, C3×A5, C3×S5, … |
| socle of | D3, C9, D9, … |
| maximal subgroup of | D3, C2×C3, C9, … |
| maximal normal subgroup of | D3, C2×C3, C9, … |
| maximal quotient of | C2×C3, C9, C32, … |
| outer automorphism group of | D7, D9, #261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3, 1);