Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×C3⋊(((C22⋊C9)⋊C2)⋊C3) |
| derived | C3×C22⋊C9 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(((C22⋊C9)⋊C2)⋊C3) |
| commutator, cosocle, hypocenter | C3×C22⋊C9 | C23×C3 |
| Frattini | C3 | C22×C3×C3⋊S4 |
| Fitting | C24×C33 | D3 |
| socle | C24×C32 | C3×D3 |
| LCS | G ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C22 | |
1686 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1686 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 20 points:
(3,4)(5,6)(7,8)(9,10,11)(13,16,20)(15,19,17);(1,2)(6,7,8)(12,16,15,14,20,17,18,13,19);(6,7)(10,11)(13,15)(14,18)(16,19)(17,20)156 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,6)(7,8)(9,10,11)(13,16,20)(15,19,17), (1,2)(6,7,8)(12,16,15,14,20,17,18,13,19), (6,7)(10,11)(13,15)(14,18)(16,19)(17,20));