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) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C32×D5 |
| unambiguous semidirect product | C5⋊(C22×C32) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C32) |
| Fitting, socle | C2×C32×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C22×C32 |
| Hall | C32×C5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D5 |
| commutator, cosocle, hypocenter | C5 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C32 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(3,4,5)(10,11)(12,13);(1,2)(6,7,8)(9,10,12,13,11)72 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×Dic3, C3×#1201, C3×#1202, … |
| maximal normal subgroup of | C3×D5×Dic3, C3×#1201, C3×#1202, … |
| maximal quotient of | C32×Dic10, C32×C4×D5, C32×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 32);