Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of SL(2,𝔽5) |
| automorphism | Aut(G) = C2×S5 (order 240) · Inn(G) = A5 · Out(G) = C22 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| direct product | C3×SL(2,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C2×C3 | A5 |
| commutator, hypocenter | SL(2,𝔽5) | C3 |
| Frattini, cosocle | C2 | C3×A5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2×C3 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 27 points:
(4,6,10)(5,8,14)(7,12,19)(9,15,21)(11,17,16)(13,20,24)(18,23,26)(22,25,27);(1,2,3)(4,5,7,11)(6,9,12,18)(8,13,17,22)(10,16,19,14)(15,21,23,26)(20,24,25,27)27 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 51);