Finite Group DatabasebetaFinite Group Databasebeta| order | 4000 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×D5⋊(D5⋊AGL(1,𝔽5)) |
| Fitting, socle | C2×C53 ⋊ C22×C4 |
| derived, Sylow | C53 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5⋊(D5⋊AGL(1,𝔽5)) |
| commutator, hypocenter | C53 | C23×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C53 | C22×C4 |
| prime radical | C2×D53 | C2 |
| cosocle | C5⋊(C5⋊D5) | C24 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C2 | |
1332 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 1332 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,5,3,4)(6,7,10,8,9)(11,12,15,14,13)(16,17);(2,4)(3,5)(12,13)(14,15);(2,3,4,5)(7,8,9,10)(12,14,13,15);(12,13)(14,15)88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽5)×D5⋊AGL(1,𝔽5) |
| maximal normal subgroup of | C2×AGL(1,𝔽5)×D5⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,5,3,4)(6,7,10,8,9)(11,12,15,14,13)(16,17), (2,4)(3,5)(12,13)(14,15), (2,3,4,5)(7,8,9,10)(12,14,13,15), (12,13)(14,15));