Finite Group DatabasebetaFinite Group Databasebeta| order | 200 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = S4×GL(2,𝔽5) (order 11520) · Inn(G) = C22 · Out(G) = D3×GL(2,𝔽5) |
| homology | Abelianization (H1) = C22×C52 · Schur multiplier (H2) = C5 |
| direct product | C52×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C52 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C52 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5)(11,12,13,14)(15,18,17,16);(6,7,8,9,10)(11,15,13,17)(12,16,14,18)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(200, 39);