Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C22⋊#92566 |
| Fitting | C22×C53 ⋊ C22 |
| Sylow | C53 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#92566 |
| commutator, socle, cosocle | C2×C53 | C23 |
| Fitting | C22×C53 | C22 |
| hypercenter | C22 | #92566 |
| hypocenter | C53 | C2×D4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22 | |
384 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 40 |
Polycyclic presentation:
2^4,5^3|e^a=e^4,f^a=f^4,c^b=cd,e^b=e^4,g^b=g^4158 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#87886, #72259205, #72259206 |
| maximal normal subgroup of | D5×#87886, #72259205, #72259206 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 861);