Finite Group DatabasebetaFinite Group Databasebeta| order | 11520 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
| direct product | D3×AGL(1,𝔽5)×#1020 |
| Fitting, socle | C24×C3×C5 ⋊ C2×C4×D3 |
| derived | C3×C5×C22⋊A4 ⋊ C22×C4 |
| Sylow | C5 ⋊ C4×D3×#1020 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C3×C5×C22⋊A4 | C22×C4 |
| Fitting, socle | C24×C3×C5 | C2×C4×D3 |
| prime radical | D3×D5×#1020 | C2 |
| cosocle | C3×D5×C22⋊A4 | C23 |
| LCS | G ▷ C3×C5×C22⋊A4 | |
| UCS | C1 | |
10862 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 10862 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 120 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,3,4)(5,8,7)(9,10,11)(13,14,15,16);(1,4)(2,3)(10,11)(12,13,16,14,15);(2,4)(6,8)150 conjugacy classes — view the full character table.
| maximal quotient of | D3×AGL(1,𝔽5)×#5935, D3×AGL(1,𝔽5)×#5934 |
| automorphism group of | C22×D3×D5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4)(5,8,7)(9,10,11)(13,14,15,16), (1,4)(2,3)(10,11)(12,13,16,14,15), (2,4)(6,8));