Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽5) (order 480) · Inn(G) = D15 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic15 |
| unambiguous semidirect product | C3⋊(C3×Dic5) |
| derived | C3×C5 ⋊ C3×C4 |
| Hall | C32×C5 ⋊ C4 |
| Sylow | C32 ⋊ Dic5 |
| Sylow | C5 ⋊ C3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D15 |
| commutator, hypocenter | C3×C5 | C3×C4 |
| Frattini | C2 | C3×D15 |
| Fitting, socle, prime radical | C2×C32×C5 | C2 |
| cosocle | C2×C3×C5 | C2×C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C3 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,3)(4,5)(6,7,8)(10,11)(12,15,14,13);(1,2,4,5,3)(9,10,11)54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×Dic3, C3×D3×Dic5, C3×#1201, … |
| maximal normal subgroup of | C3×D5×Dic3, C3×D3×Dic5, C3×#1201, … |
| maximal quotient of | C3×C3⋊#168, C2×C3×Dic15, C3×Dic45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 15);