Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×C4×S4 (order 576) · Inn(G) = C22 · Out(G) = C2×C3×C4×D3 |
| homology | Abelianization (H1) = C22×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×Q8×C9 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C9 | C22 |
| commutator | C2 | C22×C5×C9 |
| Frattini, cosocle | C2×C3 | C22×C3×C5 |
| socle, prime radical | C2×C3×C5 | C22×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C9 ◁ G | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4,5)(6,7,8,9)(10,13,12,11)(14,15,16,17,18,19,20,21,22);(6,10,8,12)(7,11,9,13)225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 32);