Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C52 · Schur multiplier (H2) = C4×C52 |
| direct product | C42×H(3,𝔽5) |
| unambiguous semidirect product | (C42×C52)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C52 |
| commutator | C5 | C42×C52 |
| Frattini, socle, cosocle | C22×C5 | C22×C52 |
| prime radical | C22×H(3,𝔽5) | C22 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ G | |
225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4)(9,10,12,16,23)(11,14,18,25,30)(13,19,28,33,29)(15,21,27,24,31)(17,26,32,22,20);(5,6,7,8)(10,13,20,21,14)(12,17,18,19,27)(16,24,28,25,26)(23,30,31,32,33)464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 211);