Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽5) (order 240) · Inn(G) = AGL(1,𝔽5) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×#170 |
| derived, Sylow | C5 ⋊ C8×C9 |
| Hall | C5×C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C8×C9 |
| Frattini | C2×C3 | C3×AGL(1,𝔽5) |
| Fitting | C2×C5×C9 | C4 |
| socle, prime radical | C2×C3×C5 | C3×C4 |
| cosocle | C3×Dic5 | C2×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C9 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 22 points:
(2,3,4,5)(6,7,8,10,9,11,12,13)(14,15,17,16,18,20,19,21,22);(1,2,3,5,4)90 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#170, #10623627, #10623628, … |
| maximal normal subgroup of | D9×#170, #10623627, #10623628, … |
| maximal quotient of | C9×#674, C2×C9×#170, C27×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 5);