Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C23×C3×AGL(1,𝔽5) (order 480) · Inn(G) = D5 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C4×C9×D5 |
| Fitting | C4×C5×C9 ⋊ C2 |
| derived, Sylow | C5 ⋊ C2×C4×C9 |
| Hall | C5×C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | D5 |
| commutator, hypocenter | C5 | C2×C4×C9 |
| Frattini | C2×C3 | C2×C3×D5 |
| Fitting | C4×C5×C9 | C2 |
| socle, cosocle | C2×C3×C5 | C22×C3 |
| prime radical | C2×C3×D5 | C2×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C4×C9 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3)(4,5)(6,13,12,8,7,14,11,10,9);(1,2,4,5,3)(15,16,17,18)144 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C9⋊C8, #10623547, C8×C9×D5, … |
| maximal normal subgroup of | D5×C9⋊C8, #10623547, C8×C9×D5, … |
| maximal quotient of | C8×C9×D5, C9×#676, C4×C9×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 16);