Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽5) (order 480) · Inn(G) = ΓL(1,𝔽16) · Out(G) = C23 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×C3⋊#170 |
| unambiguous semidirect product | C3⋊(C3×#170) |
| derived | C3×C5 ⋊ C3×C8 |
| Hall | C32×C5 ⋊ C8 |
| Sylow | C32 ⋊ #170 |
| Sylow | C5 ⋊ C3×C3⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C3×C8 |
| Frattini | C2 | C3×ΓL(1,𝔽16) |
| Fitting, socle, prime radical | C2×C32×C5 | C4 |
| cosocle | C3×Dic5 | C2×C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C3 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,5,4,3)(6,7,8)(10,11)(12,16,18,13,15,19,14,17);(1,2,5,3,4)(9,10,11)54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×#170, C3×#6738, C3×#6739, … |
| maximal normal subgroup of | C3×D3×#170, C3×#6738, C3×#6739, … |
| maximal quotient of | C3×C3⋊#674, C2×C3×C3⋊#170, C3×C9⋊#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 53);