Finite Group DatabasebetaFinite Group Databasebeta| order | 2 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C1 (order 1) · Inn(G) = C1 · Out(G) = C1 |
| holomorph | G ⋊ Aut(G) = C2 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| symmetric | S2 |
| affine general linear | AGL(1,𝔽2) |
| Borel | B(2,𝔽2) |
| cyclic | C2 |
| multiplicative | (ℤ/3)ˣ, (ℤ/4)ˣ, (ℤ/6)ˣ |
| 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) = 2 |
| transitive degree | μt(G) = 2 |
A minimal faithful permutation representation — 1 generator on 2 points:
(1,2)| C1 | C2 | |
|---|---|---|
| size | 1 | 1 |
| order | 1 | 2 |
| χ1 | 1 | 1 |
| χ2 | 1 | -1 |
| center of | D4, Q8, Dic3, … |
| derived subgroup of | D4, Q8, C22⋊C4, … |
| Frattini subgroup of | C4, C2×C4, D4, … |
| Fitting subgroup of | SL(2,𝔽5), C2×A5, #6728, … |
| socle of | C4, C8, D4, … |
| maximal subgroup of | C4, C22, D3, … |
| maximal normal subgroup of | C4, C22, C2×C3, … |
| maximal quotient of | C4, C22, D3, … |
| automorphism group of | C3, C4, C2×C3 |
| outer automorphism group of | C3, C4, C2×C3, … |
| holomorph of | C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2, 1);