Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×C4×D3 (order 192) · Inn(G) = D3 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C3×C5×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C3×C5×C8) |
| derived | C3 ⋊ C3×C5×C8 |
| Hall | C32×C5 ⋊ C8 |
| Sylow | C32 ⋊ C5×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4×C5 | D3 |
| commutator, hypocenter | C3 | C3×C5×C8 |
| Frattini | C4 | C3×C5×D3 |
| Fitting | C32×C4×C5 | C2 |
| socle, prime radical | C2×C32×C5 | C4 |
| cosocle | C3×C4 | C2×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C4×C5 | |
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:
(1,2,3)(5,6)(7,14,13,12,10,11,9,8)(15,16,17,18,19);(4,5,6)180 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×C3⋊C8, C3×#6648, C3×#6649, … |
| maximal normal subgroup of | C3×D5×C3⋊C8, C3×#6648, C3×#6649, … |
| maximal quotient of | C3×C5×C3⋊C16, C2×C3×C5×C3⋊C8, C3×C5×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 34);