Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3 |
| direct product | D5×C3⋊#5935 |
| Fitting | C3×C5×H(5,𝔽2) ⋊ C2×D3 |
| derived | C3×C5×#997 ⋊ C22 |
| Sylow | C5 ⋊ C2×C3⋊#5935 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×C5×#997 | C22 |
| Fitting | C3×C5×H(5,𝔽2) | C2×D3 |
| socle | C2×C3×C5 | C2×#1020 |
| LCS | G ▷ C3×C5×#997 | |
| UCS | C1 ◁ C2 | |
1540 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 1540 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(1,3)(2,8)(4,7)(5,6)(10,11)(12,13)(14,15,16);(2,5)(3,7)(6,8)(9,10,12,13,11)(15,16);(2,3,8)(5,6,7)96 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D5×#5935, #72231175, AGL(1,𝔽5)×C3⋊#5935 |
| maximal normal subgroup of | D3×D5×#5935, #72231175, AGL(1,𝔽5)×C3⋊#5935 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3)(2,8)(4,7)(5,6)(10,11)(12,13)(14,15,16), (2,5)(3,7)(6,8)(9,10,12,13,11)(15,16), (2,3,8)(5,6,7));