Finite Group DatabasebetaFinite Group Databasebeta| order | 4800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C22×A5⋊AGL(1,𝔽5) |
| Fitting | C22×C5 ⋊ A5⋊C4 |
| socle | C22×C5×A5 ⋊ C4 |
| derived | C5×A5 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A5⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×A5 | C22×C4 |
| Frattini | C1 | G |
| Fitting | C22×C5 | A5⋊C4 |
| socle | C22×C5×A5 | C4 |
| prime radical | C22×D5×A5 | C2 |
| cosocle | D5×A5 | C23 |
| LCS | G ▷ C5×A5 | |
| UCS | C1 ◁ C22 | |
1080 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 3 generators on 14 points:
(5,8,6,9)(10,13,12,11);(1,2)(5,7,6,9,8)(11,13)(12,14);(1,2)(3,4)76 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#72219548 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,8,6,9)(10,13,12,11), (1,2)(5,7,6,9,8)(11,13)(12,14), (1,2)(3,4));