Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C23×C3×#4183 |
| Fitting | C23×C3×AΣL(1,𝔽27) ⋊ C2 |
| derived | AΣL(1,𝔽27) ⋊ C24×C3 |
| Sylow | C3×AΣL(1,𝔽27) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | #4183 |
| commutator, cosocle, hypocenter | AΣL(1,𝔽27) | C24×C3 |
| Frattini | C32 | C23×C3×C3⋊D3 |
| Fitting | C23×C3×AΣL(1,𝔽27) | C2 |
| socle | C23×C32 | #272 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C23×C3 | |
4430 conjugacy classes of subgroups; only the 390 normal subgroups are shown.
This group has 4430 conjugacy classes of subgroups — too many to draw. Only its 390 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 18 points:
(5,6)(7,8,9)(10,13,15,14,17,18,12,16,11);(3,4)(13,16,17);(10,11)(12,15)(14,18)(16,17);(1,2)312 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,8,9)(10,13,15,14,17,18,12,16,11), (3,4)(13,16,17), (10,11)(12,15)(14,18)(16,17), (1,2));