Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #4380 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×#4380 |
| Fitting, socle | C2×C32×C5 ⋊ C4 |
| derived, Hall | C32×C5 ⋊ C2×C4 |
| Sylow | C32 ⋊ C2×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C2×#157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4380 |
| commutator, hypocenter | C32×C5 | C2×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C4 |
| prime radical | C2×C3⋊D15 | C2 |
| cosocle | C3⋊D15 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 13 points:
(4,5,6,7)(8,9)(10,11,12,13);(1,2)(3,4,5,7,6)(8,12,10)30 conjugacy classes — view the full character table.
| maximal subgroup of | #10623991, #10623992, #10623993, … |
| maximal normal subgroup of | #10623991, #10623992, #10623993, … |
| maximal quotient of | #10624113, #10624114, C4×#4380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 150);