Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | AGL(1,𝔽5)×#379⋊C3 |
| Fitting | C5×#379 ⋊ C3×C4 |
| derived | C24×C5 ⋊ C3×C42 |
| Hall | AGL(1,𝔽5)×#379 ⋊ C3 |
| Sylow | C5 ⋊ C4×#379⋊C3 |
2448 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 2448 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 60 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3,4,5)(12,16,13)(14,17,15);(1,2,5,3,4)(6,7,8,9)(10,17,16,11)(12,15,13,14)100 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×AΣL(1,𝔽64), AGL(1,𝔽5)×#1020⋊C4, AGL(1,𝔽5)×#72991, … |
| maximal normal subgroup of | AGL(1,𝔽5)×AΣL(1,𝔽64), AGL(1,𝔽5)×#1020⋊C4, AGL(1,𝔽5)×#72991 |
| maximal quotient of | #72258468 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,5)(12,16,13)(14,17,15), (1,2,5,3,4)(6,7,8,9)(10,17,16,11)(12,15,13,14));