Finite Group DatabasebetaFinite Group Databasebeta| order | 200 |
| isoclinism | isoclinic class of #6057 |
| automorphism | Aut(G) = #72210663 (order 2400) · Inn(G) = G · Out(G) = Dic3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, socle, Sylow | C52 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, prime radical, cosocle | C5⋊D5 | C22 |
| Fitting, socle, hypocenter | C52 | Q8 |
| LCS | G ▷ C5⋊D5 ▷ C52 | |
| UCS | C1 | |
21 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 10 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,3,9,4)(2,5,10,6);(1,2)(3,7,8,6)(4,10,9,5)Polycyclic presentation:
2^3,5^2|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^3,d^b=e,e^b=d^4,d^c=d^4,e^c=e^4| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
|---|---|---|---|---|---|---|---|---|
| size | 1 | 8 | 8 | 8 | 25 | 50 | 50 | 50 |
| order | 1 | 5 | 5 | 5 | 2 | 4 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 |
| χ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 |
| χ5 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 |
| χ6 | 8 | 3 | -2 | -2 | 0 | 0 | 0 | 0 |
| χ7 | 8 | -2 | 3 | -2 | 0 | 0 | 0 | 0 |
| χ8 | 8 | -2 | -2 | 3 | 0 | 0 | 0 | 0 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(200, 44);