Finite Group DatabasebetaFinite Group Databasebeta| order | 4000 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×D52×AGL(1,𝔽5) |
| Fitting, socle | C2×C53 ⋊ C22×C4 |
| derived, Sylow | C53 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D52×AGL(1,𝔽5) |
| commutator, hypocenter | C53 | C23×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C53 | C22×C4 |
| prime radical | C2×D53 | C2 |
| cosocle | C52×D5 | C24 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C2 | |
1136 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 80 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,4,5,3)(6,7,10,8,9)(11,12,14,15,13)(16,17);(2,3)(4,5)(12,13)(14,15);(7,8,9,10);(12,13)(14,15)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽5)×D5⋊AGL(1,𝔽5), AGL(1,𝔽5)×#87886 |
| maximal normal subgroup of | C2×AGL(1,𝔽5)×D5⋊AGL(1,𝔽5), AGL(1,𝔽5)×#87886 |
| maximal quotient of | AGL(1,𝔽5)×#87886, C2×AGL(1,𝔽5)×#6036 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,5,3)(6,7,10,8,9)(11,12,14,15,13)(16,17), (2,3)(4,5)(12,13)(14,15), (7,8,9,10), (12,13)(14,15));