Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×A4×SL(2,ℤ/4) |
| derived | C22×A4 ⋊ C3×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | A4×S4 |
| commutator, hypocenter | C22×A4 | C3×C4×C5 |
| Frattini | C2 | C5×A4×S4 |
| Fitting, socle | C25×C5 | C3×D3 |
| prime radical | C2×C5×A42 | C2 |
| cosocle | C23×A4 | C2×C3×C5 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C2×C5 | |
334 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 180 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,4,3)(5,6,8,7)(9,10,11,12)(13,14,15,16,17);(1,4)(2,3)(6,7,8)200 conjugacy classes — view the full character table.
| maximal subgroup of | D5×A4×SL(2,ℤ/4) |
| maximal normal subgroup of | D5×A4×SL(2,ℤ/4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4,3)(5,6,8,7)(9,10,11,12)(13,14,15,16,17), (1,4)(2,3)(6,7,8));