Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×C5×D4 |
| unambiguous semidirect product | C4⋊(C2×C32×C5) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32×C5 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C32×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C5 ◁ G | |
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)(13,15)225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 116);