Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #72224455 |
| commutator, cosocle | C4×A6 | C22 |
| Frattini, Fitting, hypercenter | C4 | PΓL(2,𝔽9) |
| socle | C2×A6 | D4 |
| prime radical | #72224450 | C2 |
| hypocenter | A6 | SD16 |
| LCS | G ▷ C4×A6 ▷ C2×A6 ▷ A6 | |
| UCS | C1 ◁ C2 ◁ C4 | |
284 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 10 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) = 40 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,6,4,5,3,2,8,7)(9,16,12,18,11,13,15,14,17,10);(1,2,3,6)(4,5,8,7)(9,16,15,11)(12,17,13,18)34 conjugacy classes — view the full character table.
| maximal quotient of | ΓL(2,𝔽9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,6,4,5,3,2,8,7)(9,16,12,18,11,13,15,14,17,10), (1,2,3,6)(4,5,8,7)(9,16,15,11)(12,17,13,18));