Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
| direct product | C22×D5×C3⋊S4 |
| Fitting, socle | C24×C3×C5 ⋊ C2×D3 |
| derived | C3×C5×A4 ⋊ C24 |
| Sylow | C5 ⋊ C23×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D5×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×C5×A4 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | C2×D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 ◁ C22 | |
3340 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 3340 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 4 generators on 16 points:
(3,4)(5,6,8)(12,13,15,16,14);(9,10,11)(13,14)(15,16);(6,7)(10,11);(1,2)144 conjugacy classes — view the full character table.
| maximal subgroup of | C22×D5×A4⋊S4 |
| maximal quotient of | C22×D5×A4⋊S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,6,8)(12,13,15,16,14), (9,10,11)(13,14)(15,16), (6,7)(10,11), (1,2));