Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| isoclinism | isoclinic class of D5×AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×Dic5×AGL(1,𝔽5) |
| derived | C52 ⋊ C42×C5 |
| Sylow | C53 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D5×AGL(1,𝔽5) |
| commutator, hypocenter | C52 | C42×C5 |
| Frattini | C2 | C5×D5×AGL(1,𝔽5) |
| Fitting, socle | C2×C53 | C2×C4 |
| prime radical | C2×C52×D5 | C22 |
| cosocle | C2×C5×D5 | C22×C5 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C2×C5 | |
172 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
200 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#6036, C5×AGL(1,𝔽5)×C22⋊D5, AGL(1,𝔽5)×#6038, … |
| maximal normal subgroup of | AGL(1,𝔽5)×#6036, C5×AGL(1,𝔽5)×C22⋊D5, AGL(1,𝔽5)×#6038, … |
| maximal quotient of | C2×C5×Dic5×AGL(1,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 634);