Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 384 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C3×C4×C5×C7 |
| multiplicative | (ℤ/899)ˣ, (ℤ/923)ˣ, (ℤ/1055)ˣ, (ℤ/1075)ˣ, (ℤ/1225)ˣ, (ℤ/1263)ˣ, (ℤ/1684)ˣ, (ℤ/1798)ˣ, (ℤ/1846)ˣ, (ℤ/2110)ˣ, (ℤ/2150)ˣ, (ℤ/2450)ˣ, (ℤ/2526)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C22×C3×C5×C7 |
| socle, prime radical | C22×C3×C5×C7 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 840 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4)(7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21);(5,6)840 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 131);