Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C5 |
| unambiguous semidirect product | (C2×AGL(1,𝔽5))⋊(C5⋊D5) |
| Sylow | C53 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽5)×C5⋊D5 |
| commutator, Fitting, socle | C2×C53 | C2×C4 |
| prime radical | C2×D5×C5⋊D5 | C2 |
| cosocle | C2×C52×D5 | C22 |
| hypocenter | C53 | C22⋊C4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 | |
332 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^4,5^3|b^2=c,b^a=bd,f^a=f^4,g^a=g^4,e^b=e^2,e^c=e^4134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 685);