Finite Group DatabasebetaFinite Group Databasebeta| order | 7 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3 (order 6) · Inn(G) = C1 · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽7) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C7 |
| 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) = 7 |
| transitive degree | μt(G) = 7 |
A minimal faithful permutation representation — 1 generator on 7 points:
(1,2,3,4,5,6,7)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | |
|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| order | 1 | 7 | 7 | 7 | 7 | 7 | 7 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | ζ | ζ2 | ζ3 | ζ4 | ζ5 | ζ6 |
| χ3 | 1 | ζ2 | ζ4 | ζ6 | ζ | ζ3 | ζ5 |
| χ4 | 1 | ζ3 | ζ6 | ζ2 | ζ5 | ζ | ζ4 |
| χ5 | 1 | ζ4 | ζ | ζ5 | ζ2 | ζ6 | ζ3 |
| χ6 | 1 | ζ5 | ζ3 | ζ | ζ6 | ζ4 | ζ2 |
| χ7 | 1 | ζ6 | ζ5 | ζ4 | ζ3 | ζ2 | ζ |
Here ζ is a primitive root of unity of order 7 (ζ7 = 1).
| center of | D3×C7, C7×D5, C7×A4, … |
| derived subgroup of | D7, ΓL(1,𝔽8), Dic7, … |
| Frattini subgroup of | C49, D49, C2×C49, … |
| Fitting subgroup of | D7, ΓL(1,𝔽8), AGL(1,𝔽7), … |
| socle of | D7, ΓL(1,𝔽8), AGL(1,𝔽7), … |
| maximal subgroup of | D7, C2×C7, ΓL(1,𝔽8), … |
| maximal normal subgroup of | D7, C2×C7, ΓL(1,𝔽8), … |
| maximal quotient of | C2×C7, C3×C7, C5×C7, … |
| outer automorphism group of | #1181, D43⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7, 1);