Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of SL(2,𝔽5) |
| automorphism | Aut(G) = C2×S5 (order 240) · Inn(G) = A5 · Out(G) = C22 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| direct product | C2×SL(2,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C22 | A5 |
| commutator, hypocenter | SL(2,𝔽5) | C2 |
| Frattini, cosocle | C2 | C2×A5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C22 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 26 points:
(3,4,6)(5,7,9)(8,10,13)(11,14,18)(12,15,19)(16,20,17)(21,23,25)(22,24,26);(1,2)(3,4,5,7)(6,8,9,11)(10,12,14,17)(13,16,18,15)(19,21,20,22)(23,25,24,26)18 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 94);