Finite Group DatabasebetaFinite Group Databasebeta| order | 40 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C4×GL(3,𝔽2) (order 672) · Inn(G) = C1 · Out(G) = C4×GL(3,𝔽2) |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×AGL(3,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C23×C5 |
| multiplicative | (ℤ/88)ˣ, (ℤ/132)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2)(7,8,9,10,11);(3,4);(5,6)40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(40, 14);