Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×A4⋊((C22⋊C9)⋊C2) |
| derived | A4×C22⋊C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | A4×C22⋊C9 | C2×C3 |
| Frattini | C3 | C3×A4⋊S4 |
| Fitting, socle | C24×C32 | C3⋊D3 |
| LCS | G ▷ A4×C22⋊C9 | |
| UCS | C1 ◁ C3 | |
441 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 441 conjugacy classes of subgroups — too many to draw. Only its 44 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:
(1,4,2,3)(5,8,6,7)(9,10,11)(13,15)(14,18)(16,19)(17,20);(2,3,4)(6,8,7)(12,17,15,14,19,16,18,13,20);(6,7,8)90 conjugacy classes — view the full character table.
| maximal quotient of | #72276569 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2,3)(5,8,6,7)(9,10,11)(13,15)(14,18)(16,19)(17,20), (2,3,4)(6,8,7)(12,17,15,14,19,16,18,13,20), (6,7,8));