Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = C4×D4×AGL(1,𝔽5) (order 640) · Inn(G) = AGL(1,𝔽5) · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×D5⋊C8 |
| unambiguous semidirect product | D5⋊(C5×C8) |
| derived | C5 ⋊ C2×C5×C8 |
| Sylow | C52 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C2×C5×C8 |
| Frattini | C2 | C2×C5×AGL(1,𝔽5) |
| Fitting | C4×C52 | C4 |
| socle | C2×C52 | C2×C4 |
| prime radical | C2×C5×D5 | C4 |
| cosocle | Dic5 | C22×C5 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C4×C5 | |
47 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)(6,7,8,10,9)(11,13,15,14)(12,16,18,17);(7,8,9,10)(11,18,14,16,15,12,13,17)100 conjugacy classes — view the full character table.
| maximal subgroup of | #11716731, #11716733, #11716735, … |
| maximal normal subgroup of | #11716731, #11716733, #11716735, … |
| maximal quotient of | C5×D5⋊C16, C5×#3990, C4×C5×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 135);