Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C52 |
| direct product | M16×H(3,𝔽5) |
| unambiguous semidirect product | (C52×M16)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C5 | C22×C52 |
| commutator, socle | C2×C5 | C2×C4×C52 |
| prime radical | C22×H(3,𝔽5) | C4 |
| LCS | G ▷ C2×C5 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ G | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8)(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);(2,6)(4,8)(10,13,20,21,14)(12,17,18,19,27)(16,24,28,25,26)(23,30,31,32,33)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 223);