Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C3⋊(A4⋊AGL(1,𝔽5)) |
| Fitting, socle | C22×C32×C5 ⋊ Dic3 |
| derived | C3×C5×A4 ⋊ C3×C4 |
| Sylow | C5 ⋊ C3×C3⋊SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(A4⋊AGL(1,𝔽5)) |
| commutator, hypocenter | C3×C5×A4 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | Dic3 |
| prime radical | C32×D5×A4 | C2 |
| cosocle | C3×D5×A4 | C2×C3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 ◁ C3 | |
272 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 15 points:
(1,4,3,2)(6,7,8,9)(10,11,12)(14,15);(5,6,9,7,8)(13,14,15);(2,3,4)90 conjugacy classes — view the full character table.
| maximal subgroup of | #72247686, C3×S4×ΓL(1,𝔽16), C3×D3×A4⋊AGL(1,𝔽5), … |
| maximal normal subgroup of | C3×S4×ΓL(1,𝔽16), C3×D3×A4⋊AGL(1,𝔽5), C3×AGL(1,𝔽5)×C3⋊S4 |
| maximal quotient of | #72247686 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(6,7,8,9)(10,11,12)(14,15), (5,6,9,7,8)(13,14,15), (2,3,4));