Finite Group DatabasebetaFinite Group Databasebeta| order | 40 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = D4×AGL(1,𝔽5) (order 160) · Inn(G) = D5 · Out(G) = C2×D4 |
| holomorph | G ⋊ Aut(G) = #72213689 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×Dic5 |
| derived, Sylow | C5 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D5 |
| commutator, hypocenter | C5 | C2×C4 |
| Frattini | C2 | C2×D5 |
| Fitting, socle, prime radical | C22×C5 | C2 |
| cosocle | C2×C5 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C22 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2)(3,4,5,6)(8,9)(10,11);(1,2)(7,8,10,11,9)16 conjugacy classes — view the full character table.
| maximal subgroup of | C4×Dic5, #683, C4⋊Dic5, … |
| maximal normal subgroup of | C4×Dic5, #683, C4⋊Dic5, … |
| maximal quotient of | C2×#168, #681, C4×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(40, 7);