Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
| direct product | AGL(1,𝔽5)×#4625 |
| derived | C5×C22⋊A4 ⋊ C42 |
| Sylow | C5 ⋊ C4×#4625 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C5×C22⋊A4 | C42 |
| Frattini | C23 | AGL(1,𝔽5)×S4 |
| Fitting | C5×#120 | C4×D3 |
| socle | C22×C5 | C4×SL(2,ℤ/4) |
| prime radical, cosocle | D5×#120⋊C3 | C22 |
| LCS | G ▷ C5×C22⋊A4 | |
| UCS | C1 | |
552 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,3,4,5)(6,19,17)(7,20,12)(8,18,11)(9,13,16)(10,21,15);(1,2,5,3,4)(7,11,12,19)(8,15,16,17)(9,10)(13,20,21,14)55 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#11714270, AGL(1,𝔽5)×#4625⋊C2 |
| maximal normal subgroup of | AGL(1,𝔽5)×#4625⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,5)(6,19,17)(7,20,12)(8,18,11)(9,13,16)(10,21,15), (1,2,5,3,4)(7,11,12,19)(8,15,16,17)(9,10)(13,20,21,14));