Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C211 |
| direct product | C24×A4×AGL(1,𝔽5) |
| Fitting, socle | C26×C5 ⋊ C3×C4 |
| derived | C22×C5 ⋊ C24×C3×C4 |
| Hall | C26×AGL(1,𝔽5) ⋊ C3 |
| Sylow | C5 ⋊ C24×C4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | A4×AGL(1,𝔽5) |
| commutator, hypocenter | C22×C5 | C24×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C26×C5 | C3×C4 |
| prime radical | C24×D5×A4 | C2 |
| cosocle | C22×D5 | C25×C3 |
| LCS | G ▷ C22×C5 | |
| UCS | C1 ◁ C24 | |
41606 conjugacy classes of subgroups; only the 2244 normal subgroups are shown.
This group has 41606 conjugacy classes of subgroups — too many to draw. Only its 2244 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) = 240 |
A minimal faithful permutation representation — 5 generators on 17 points:
(7,8)(10,12,11)(13,14,17,15,16);(5,6)(9,12)(10,11);(3,4);(14,15,16,17);(1,2)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×A42×AGL(1,𝔽5) |
| maximal normal subgroup of | C22×A42×AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,8)(10,12,11)(13,14,17,15,16), (5,6)(9,12)(10,11), (3,4), (14,15,16,17), (1,2));