Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C23×C5×D5 |
| unambiguous semidirect product | C5⋊(C24×C5) |
| unambiguous semidirect product | (C2×C5)⋊(C23×C5) |
| Fitting, socle | C23×C52 ⋊ C2 |
| derived | C5 ⋊ C24×C5 |
| Sylow | C52 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C5 | D5 |
| commutator, cosocle, hypocenter | C5 | C24×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C52 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23×C5 | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(5,6)(8,9)(10,11)(12,13,14,15,16);(5,6)(7,8,10,11,9);(3,4);(1,2)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#87778, D5×C22⋊Dic5, #11717094, … |
| maximal normal subgroup of | C2×#87778, D5×C22⋊Dic5, #11717094, … |
| maximal quotient of | C22×C5×Dic10, C22×C4×C5×D5, C22×C5×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 219);