Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | D5×A5⋊C4 |
| derived | C5×A5 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×S5 |
| commutator, hypocenter | C5×A5 | C2×C4 |
| Fitting | C2×C5 | C2×S5 |
| socle, cosocle | C2×C5×A5 | C22 |
| prime radical | C2×D5×A5 | C2 |
| LCS | G ▷ C5×A5 | |
| UCS | C1 ◁ C2 | |
239 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 60 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,2,4)(5,8,6,9,7)(10,13,11,12);(5,6)(7,9)(10,14,11)56 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×A5⋊C4 |
| maximal normal subgroup of | AGL(1,𝔽5)×A5⋊C4 |
| maximal quotient of | D5×GL(2,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,2,4)(5,8,6,9,7)(10,13,11,12), (5,6)(7,9)(10,14,11));