Finite Group DatabasebetaFinite Group Databasebeta| order | 200 |
| isoclinism | isoclinic class of #1035 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/25) (order 1000) · Inn(G) = #1035 · Out(G) = C2×C5 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C25 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1035 |
| commutator, hypocenter | C25 | C8 |
| Frattini, socle, prime radical | C2×C5 | AGL(1,𝔽5) |
| Fitting | C2×C25 | C4 |
| cosocle | Dic25 | C2 |
| LCS | G ▷ C25 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 2 generators on 33 points:
(2,4,5,10)(3,8,9,20)(6,14,15,19)(7,17,18,11)(12,22,23,25)(13,16,21,24)(26,27,28,30,29,31,32,33);(1,2,6,16,13,3,7,4,11,22,8,19,12,23,14,20,25,17,10,18,9,21,24,15,5)Polycyclic presentation:
2^3,5^2|a^2=b,b^2=c,d^5=e,d^a=d^2e,e^a=e^2,d^b=d^4e^4,e^b=e^420 conjugacy classes — view the full character table.
| maximal subgroup of | D25⋊C8, #87735, C2×#6016, … |
| maximal normal subgroup of | D25⋊C8, #87735, C2×#6016, … |
| maximal quotient of | #87709, C2×#6016, C3×#6016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(200, 3);