Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of SL(2,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C7 · Schur multiplier (H2) = C1 |
| direct product | C7×SL(2,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C2×C7 | A5 |
| commutator, hypocenter | SL(2,𝔽5) | C7 |
| Frattini, cosocle | C2 | C7×A5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2×C7 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,11,15)(10,13,16,12)(14,19,20,24)(17,21,18,22)(23,27,28,31)(25,29,26,30);(8,10,14)(9,12,17)(11,16,20)(13,18,15)(19,23,27)(21,25,29)(22,26,30)(24,28,31)63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 13);