Finite Group DatabasebetaFinite Group Databasebeta| order | 820 |
| isoclinism | isoclinic class of D41⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic41⋊C5 |
| derived, Sylow | C41 ⋊ C4×C5 |
| Hall | C41⋊C5 ⋊ C4 |
| Hall | Dic41 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D41⋊C5 |
| commutator, hypocenter | C41 | C4×C5 |
| Fitting, socle, cosocle | C2×C41 | C2×C5 |
| prime radical | C2×C41⋊C5 | C2 |
| LCS | G ▷ C41 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
| transitive degree | μt(G) = 164 |
Polycyclic presentation:
2,5,2,41|a^2=c,d^a=d^40,d^b=d^1028 conjugacy classes — view the full character table.
| maximal subgroup of | #66493⋊C5, (C41⋊C5)⋊Q8, C4×D41⋊C5, … |
| maximal normal subgroup of | #66493⋊C5, (C41⋊C5)⋊Q8, C4×D41⋊C5, … |
| maximal quotient of | #66491⋊C5, C2×Dic41⋊C5, C3×Dic41⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(820, 1);