Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×AGL(2,𝔽4) |
| Fitting, socle | C25 ⋊ C3×A5 |
| derived | #11757162 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(2,𝔽4) |
| commutator, hypocenter | #11757162 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25 | C3×A5 |
| cosocle | C24 | C2×C3×A5 |
| LCS | G ▷ #11757162 | |
| UCS | C1 ◁ C2 | |
246 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 246 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) = 30 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,4,9,8,13,6)(5,18,17,14,11,15)(7,10)(12,16);(3,9,11,18,10,14,15,5,6,7,4,8,17,13,12)38 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AGL(2,𝔽4), C2×AΓL(2,𝔽4), C4×AGL(2,𝔽4), … |
| maximal normal subgroup of | C2×AΓL(2,𝔽4), C4×AGL(2,𝔽4), #72277485 |
| maximal quotient of | D3×AGL(2,𝔽4), C4×AGL(2,𝔽4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,9,8,13,6)(5,18,17,14,11,15)(7,10)(12,16), (3,9,11,18,10,14,15,5,6,7,4,8,17,13,12));