Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #157 |
| automorphism | Aut(G) = C2×C4×AΓL(1,𝔽9) (order 1152) · Inn(G) = #157 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C5×#157 |
| Fitting, socle | C2×C32×C5 ⋊ C4 |
| derived, Sylow | C32 ⋊ C2×C4×C5 |
| Hall | C32×C5 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | #157 |
| commutator, hypocenter | C32 | C2×C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C4 |
| prime radical | C2×C5×C3⋊D3 | C2 |
| cosocle | C3⋊D3 | C22×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C5 | |
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:
(3,4,5,6,7)(8,9)(10,11,12,13);(1,2)(8,12,10)60 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×#157, #10623985, #10623986, … |
| maximal normal subgroup of | Dic5×#157, #10623985, #10623986, … |
| maximal quotient of | C5×#3787, C5×#3788, C4×C5×#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 148);