Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C23×C4×AGL(1,𝔽5) (order 640) · Inn(G) = D5 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×C8×D5 |
| Fitting | C52×C8 ⋊ C2 |
| derived | C5 ⋊ C2×C5×C8 |
| Sylow | C52 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C8 | D5 |
| commutator, hypocenter | C5 | C2×C5×C8 |
| Frattini | C4 | C2×C5×D5 |
| Fitting | C52×C8 | C2 |
| socle | C2×C52 | C2×C4 |
| prime radical | C2×C5×D5 | C4 |
| cosocle | C4×C5 | C22×C5 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C5×C8 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5)(7,8)(9,10);(6,7,9,10,8)(11,16,13,18,14,12,17,15)160 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#672, #11716584, C5×D5×C16, … |
| maximal normal subgroup of | D5×#672, #11716584, C5×D5×C16, … |
| maximal quotient of | C5×D5×C16, C5×#3930, C5×C8×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 76);