Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×D20 |
| unambiguous semidirect product | C4⋊(C32×D5) |
| Fitting | C32×C4×C5 ⋊ C2 |
| Hall | C32×C5 ⋊ D4 |
| Sylow | C5 ⋊ C32×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C2×D5 |
| commutator, cosocle | C2×C5 | C22×C32 |
| Frattini | C2 | C2×C32×D5 |
| Fitting | C32×C4×C5 | C2 |
| socle | C2×C32×C5 | C22 |
| hypercenter | C32×C4 | D5 |
| hypocenter | C5 | C32×D4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C32 ◁ C32×C4 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(7,8,9,10)(11,12,13,14,15);(4,5,6)(8,10)(12,15)(13,14)117 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#6652, C3×#6653, C3×#6655, … |
| maximal normal subgroup of | C3×#6652, C3×#6653, C3×#6655, … |
| maximal quotient of | C32×#677, C32×D40, C32×Dic20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 92);