Finite Group DatabasebetaFinite Group Databasebeta| order | 31122 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#23780563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C13×C19 | C2×C3 |
| LCS | G ▷ C3×C7×C13×C19 | |
| UCS | C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 5187 |
A minimal faithful permutation representation — 2 generators on 42 points:
(2,3)(5,6,7)(8,10,9)(12,16,17,13,14,18)(15,21,22,19,20,23)(25,26,28)(27,30,31)(29,33,32)(34,38,36)(35,39,37)(40,42,41);(1,2,3)(4,5,6,8,7,9,10)(11,12,15,17,16,20,22,23,21,14,18,19,13)(24,25,27,31,36,41,38,26,29,34,39,28,32,37,30,35,40,42,33)Polycyclic presentation:
2,3^2,7,13,19|c^a=c^2,e^a=e^12,d^b=d^2,e^b=e^9,f^b=f^7939 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31122, 456);