Finite Group DatabasebetaFinite Group Databasebeta| order | 40 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = C2×AGL(1,𝔽5) (order 40) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = #13107464 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C5 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C8 |
| Fitting, socle, prime radical | C2×C5 | C4 |
| cosocle | Dic5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,3,4,5)(6,7,8,10,9,11,12,13);(1,2,3,5,4)Polycyclic presentation:
2^3,5|a^2=b,b^2=c,d^a=d^2,d^b=d^4| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 |
| order | 1 | 2 | 5 | 10 | 4 | 4 | 8 | 8 | 8 | 8 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | -1 | -1 | ||||
| χ4 | 1 | 1 | 1 | 1 | -1 | -1 | ||||
| χ5 | 1 | -1 | 1 | -1 | ζ | −ζ | ζ3 | −ζ3 | ||
| χ6 | 1 | -1 | 1 | -1 | −ζ | ζ | −ζ3 | ζ3 | ||
| χ7 | 1 | -1 | 1 | -1 | ζ3 | −ζ3 | ζ | −ζ | ||
| χ8 | 1 | -1 | 1 | -1 | −ζ3 | ζ3 | −ζ | ζ | ||
| χ9 | 4 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ10 | 4 | -4 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 8 (ζ8 = 1); i = ζ2.
| maximal subgroup of | D5⋊C8, #700, C2×#170, … |
| maximal normal subgroup of | D5⋊C8, #700, C2×#170, … |
| maximal quotient of | #674, C2×#170, C3×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(40, 3);