Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of #4380 |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(1,𝔽9) (order 1440) · Inn(G) = G · Out(G) = C8 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, socle, derived, Hall | C32×C5 ⋊ C4 |
| Sylow | C32 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ #157 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5 | C4 |
| prime radical, cosocle | C3⋊D15 | 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)(3,5,6,4)(8,9,10,11);(1,3,6)(2,5,4)(7,8,9,11,10)Polycyclic presentation:
2^2,3^2,5|a^2=b,c^a=cd^2,d^a=c^2d^2,e^a=e^2,c^b=c^2,d^b=d^2,e^b=e^415 conjugacy classes — view the full character table.
| maximal subgroup of | D5⋊#157, C2×#4380, C3×#4380, … |
| maximal normal subgroup of | D5⋊#157, C2×#4380, C3×#4380, … |
| maximal quotient of | #67127, C2×#4380, #10587383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 25);