Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | SL(2,𝔽3)×A5⋊C4 |
| derived | Q8×A5 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | A4×S5 |
| commutator, hypocenter | Q8×A5 | C3×C4 |
| Fitting | C2×Q8 | C3×S5 |
| socle | C22×A5 | C2×A4 |
| prime radical | C2×SL(2,𝔽3)×A5 | C2 |
| cosocle | C2×Q8×A5 | C2×C3 |
| LCS | G ▷ Q8×A5 | |
| UCS | C1 ◁ C22 | |
491 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2)(3,4)(5,7)(6,8)(10,15,17,11,13,16)(12,14);(1,3,2,4)(6,7,8,9)(10,13,11,15)(12,16,14,17)98 conjugacy classes — view the full character table.
| maximal quotient of | SL(2,𝔽3)×GL(2,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4)(5,7)(6,8)(10,15,17,11,13,16)(12,14), (1,3,2,4)(6,7,8,9)(10,13,11,15)(12,16,14,17));