Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×GL(2,𝔽3) (order 1920) · Inn(G) = D5 · Out(G) = C22×GL(2,𝔽3) |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×Dic5 |
| derived, Sylow | C5 ⋊ C32×C4 |
| Hall | C32×C5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D5 |
| commutator, hypocenter | C5 | C32×C4 |
| Frattini | C2 | C32×D5 |
| Fitting, socle, prime radical | C2×C32×C5 | C2 |
| cosocle | C2×C5 | C2×C32 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C32 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(7,8)(9,10)(11,12,13,14,15);(4,5,6)(7,9,8,10)(12,15)(13,14)72 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#170, C3×C3⋊#170, C3⋊(C3⋊#170), … |
| maximal normal subgroup of | C32×#170, C3×C3⋊#170, C3⋊(C3⋊#170), … |
| maximal quotient of | C32×#168, C2×C32×Dic5, C3×C9×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 13);