Finite Group DatabasebetaFinite Group Databasebeta| order | 200 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C4×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C2×D5 · Out(G) = C2×C42 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic10 |
| unambiguous semidirect product | C5⋊(C5×Q8) |
| Sylow | C52 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×D5 |
| commutator, cosocle | C2×C5 | C22×C5 |
| Frattini | C2 | C2×C5×D5 |
| Fitting | C4×C52 | C2 |
| socle, prime radical | C2×C52 | C22 |
| hypercenter | C4×C5 | D5 |
| hypocenter | C5 | C5×Q8 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5)(6,7,9,10,8)(11,13,14,18)(12,15,16,17);(7,8)(9,10)(11,12,14,16)(13,17,18,15)65 conjugacy classes — view the full character table.
| maximal subgroup of | #87772, #87774, #87775, … |
| maximal normal subgroup of | #87772, #87774, #87775, … |
| maximal quotient of | C5×#683, C5×C4⋊Dic5, C2×C5×Dic10, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(200, 27);