Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| isoclinism | isoclinic class of #1042 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C22×C5×#1042 |
| unambiguous semidirect product | D5⋊(C2×C5×Dic5) |
| Fitting, socle | C22×C53 ⋊ C4 |
| derived | C52 ⋊ C22×C4×C5 |
| Sylow | C53 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | #1042 |
| commutator, hypocenter | C52 | C22×C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C53 | C4 |
| prime radical | C22×C52×D5 | C2 |
| cosocle | C5×D5 | C23×C5 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C22×C5 | |
340 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
260 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×#1042⋊D5, C2×C5×Dic5×AGL(1,𝔽5), C2×Dic5×#1042 |
| maximal normal subgroup of | C22×#1042⋊D5, C2×C5×Dic5×AGL(1,𝔽5), C2×Dic5×#1042 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 951);