Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×A4×ΓL(1,𝔽16) |
| Fitting, socle | C22×C32×C5 ⋊ C3×C4 |
| derived | C22×C3×C5 ⋊ C32×C4 |
| Sylow | C5 ⋊ C3×Dic3×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4×ΓL(1,𝔽16) |
| commutator, hypocenter | C22×C3×C5 | C32×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C3×C4 |
| prime radical | C32×D5×A4 | C2 |
| cosocle | C22×C3×D5 | C2×C32 |
| LCS | G ▷ C22×C3×C5 | |
| UCS | C1 ◁ C3 | |
272 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,3,4,5)(10,11)(13,14,15);(1,2,5,3,4)(6,7,8)(9,10,11)(12,15)(13,14)108 conjugacy classes — view the full character table.
| maximal subgroup of | A4×3−1+2⋊AGL(1,𝔽5), C3×D3×A4×AGL(1,𝔽5), C3×S4×ΓL(1,𝔽16), … |
| maximal normal subgroup of | C3×D3×A4×AGL(1,𝔽5), C3×S4×ΓL(1,𝔽16), C3×ΓL(1,𝔽16)⋊S4 |
| maximal quotient of | A4×3−1+2⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,5)(10,11)(13,14,15), (1,2,5,3,4)(6,7,8)(9,10,11)(12,15)(13,14));