Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of GL(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C2 |
| direct product | C5×GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C5 | GL(3,𝔽2) |
| commutator, hypocenter | GL(3,𝔽2) | C5 |
| Frattini | C1 | G |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C5 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 35 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4,5)(8,9)(10,11);(6,7,8)(9,10,12)30 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 134);