Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #1201 |
| automorphism | Aut(G) = C23×D3×AGL(1,𝔽5) (order 960) · Inn(G) = D3×D5 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#1202 |
| Hall | C32×C5 ⋊ D4 |
| Sylow | C32 ⋊ D20 |
| Sylow | C5 ⋊ C3×#67 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×D5 |
| commutator, cosocle | C2×C3×C5 | C22×C3 |
| Frattini | C2 | C3×D3×D5 |
| Fitting, socle | C2×C32×C5 | C22 |
| hypocenter | C3×C5 | C3×D4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2×C3 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,4,5,3)(6,7,8)(10,11)(12,15,14,13);(2,3)(4,5)(9,10,11)(13,15)63 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1202, #10624028, #10624030, … |
| maximal normal subgroup of | D3×#1202, #10624028, #10624030, … |
| maximal quotient of | C3×#6653, C3×#6657, C3×#6658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 62);