Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×AGL(1,𝔽5)×S4 |
| Fitting, socle | C22×C32×C5 ⋊ C4×D3 |
| derived | C5×A4 ⋊ C2×C32×C4 |
| Sylow | C5 ⋊ C32×C4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | AGL(1,𝔽5)×S4 |
| commutator, hypocenter | C5×A4 | C2×C32×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C4×D3 |
| prime radical | C32×D5×S4 | C2 |
| cosocle | D5×A4 | C22×C32 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C32 | |
768 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 180 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,4,3)(6,7,8,9)(13,14,15);(2,4)(5,6,9,7,8)(10,11,12)225 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×S4×3−1+2, C3×D3×AGL(1,𝔽5)×S4 |
| maximal normal subgroup of | AGL(1,𝔽5)×S4×3−1+2, C3×D3×AGL(1,𝔽5)×S4 |
| maximal quotient of | AGL(1,𝔽5)×S4×3−1+2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,7,8,9)(13,14,15), (2,4)(5,6,9,7,8)(10,11,12));