Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3×C4×D4 (order 192) · Inn(G) = C1 · Out(G) = C2×C3×C4×D4 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C4×C5×C9 |
| multiplicative | (ℤ/407)ˣ, (ℤ/475)ˣ, (ℤ/543)ˣ, (ℤ/675)ˣ, (ℤ/724)ˣ, (ℤ/814)ˣ, (ℤ/950)ˣ, (ℤ/1086)ˣ, (ℤ/1350)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C3 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | C2×C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(7,8,9,10,11,12,13,14,15)(16,17,18,19,20);(5,6)360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 30);