Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of C3⋊ΓL(1,𝔽16) |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(2,𝔽3) (order 8640) · Inn(G) = G · Out(G) = GL(2,𝔽3) |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊ΓL(1,𝔽16) |
| Fitting, socle, derived, Hall | C32×C5 ⋊ C4 |
| Sylow | C32 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ C3⋊Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5 | C4 |
| prime radical, cosocle | C32×D5 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 3 generators on 11 points:
(2,3,4,5)(7,8)(10,11);(1,2,5,3,4)(9,10,11);(6,7,8)Polycyclic presentation:
2^2,3^2,5|a^2=b,c^a=c^2,d^a=d^2,e^a=e^3,e^b=e^421 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 22);