Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| isoclinism | isoclinic class of D52 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×D52 |
| unambiguous semidirect product | (C5×D5)⋊(C22×C5) |
| Fitting, socle | C2×C53 ⋊ C22 |
| derived | C52 ⋊ C23×C5 |
| Sylow | C53 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D52 |
| commutator, cosocle, hypocenter | C52 | C23×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C53 | C22 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C2×C5 | |
204 conjugacy classes of subgroups.
| transitive degree | μt(G) = 40 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#87831, C5×(C2×AGL(1,𝔽5))⋊D5, #14522707, … |
| maximal normal subgroup of | C5×#87831, C5×(C2×AGL(1,𝔽5))⋊D5, #14522707, … |
| maximal quotient of | C5×D5×Dic10, C5×#87870, C5×#87871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 193);