Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C32 |
| direct product | C23×#4169⋊C3 |
| Fitting | C23×#6891 ⋊ C2 |
| derived | C3×C9 ⋊ C24×C32 |
| Sylow | #6891 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #4169⋊C3 |
| commutator, cosocle, hypocenter | C3×C9 | C24×C32 |
| Frattini | C32 | C23×C32×D3 |
| Fitting | C23×#6891 | C2 |
| socle | C23×C32 | C32×D3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C23 | |
3604 conjugacy classes of subgroups; only the 642 normal subgroups are shown.
This group has 3604 conjugacy classes of subgroups — too many to draw. Only its 642 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(8,11,12)(10,14,15)(16,19,21,20,23,24,18,22,17);(7,8,10)(9,11,14,13,12,15)(16,17)(18,21)(20,24)(22,23);(3,4);(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)(8,11,12)(10,14,15)(16,19,21,20,23,24,18,22,17), (7,8,10)(9,11,14,13,12,15)(16,17)(18,21)(20,24)(22,23), (3,4), (1,2));