Finite Group DatabasebetaFinite Group Databasebeta| order | 4000 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C2 |
| direct product | #104×5−1+2 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C52×D4 |
| commutator | C4×C5 | C2×C4×C52 |
| Frattini, cosocle | C2×C4×C5 | C22×C52 |
| socle | C2×C5 | C52×C22⋊C4 |
| prime radical | C52×#41 | C2×C5 |
| LCS | G ▷ C4×C5 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×5−1+2 ◁ G | |
220 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2)(3,5)(4,6)(7,8)(10,12,15,20,28)(13,17,23,16,21)(18,25,22,29,24)(26,33,32,31,30);(2,5,6,7)(9,10,13,18,26,14,15,17,24,31,27,28,23,29,33,11,12,16,22,30,19,20,21,25,32)406 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | #104×D25⋊C5, #104×AGL(1,ℤ/25), #72275823 |
| maximal subgroup of | #104×D25⋊C5, #453×5−1+2, #72275823 |
| maximal normal subgroup of | #104×D25⋊C5, #453×5−1+2, #72275823 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,5)(4,6)(7,8)(10,12,15,20,28)(13,17,23,16,21)(18,25,22,29,24)(26,33,32,31,30), (2,5,6,7)(9,10,13,18,26,14,15,17,24,31,27,28,23,29,33,11,12,16,22,30,19,20,21,25,32));