Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of #4379 |
| automorphism | Aut(G) = AGL(1,𝔽5)×AΓL(1,𝔽9) (order 2880) · Inn(G) = G · Out(G) = #41 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, socle, derived, Hall | C32×C5 ⋊ C4 |
| Sylow | C32 ⋊ Dic5 |
| Sylow | C5 ⋊ #157 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5 | C4 |
| prime radical, cosocle | C5×C3⋊D3 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2)(3,4,6,5)(8,9)(10,11);(1,3,6)(7,8,10,11,9)Polycyclic presentation:
2^2,3^2,5|a^2=b,c^a=c^2d,d^a=cd,e^a=e^4,c^b=c^2,d^b=d^218 conjugacy classes — view the full character table.
| maximal subgroup of | #67195, D5×#157, D32⋊D5, … |
| maximal normal subgroup of | #67195, D5×#157, D32⋊D5, … |
| maximal quotient of | #67126, C2×#4379, #10587382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 24);