Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of SO(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×SO(3,𝔽7) |
| unambiguous semidirect product | GL(3,𝔽2)⋊(C2×C5) |
| socle | C5×GL(3,𝔽2) ⋊ C2 |
| derived | GL(3,𝔽2) ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C5 | SO(3,𝔽7) |
| commutator, cosocle, hypocenter | GL(3,𝔽2) | C2×C5 |
| Frattini | C1 | G |
| socle | C5×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C5 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3,4,5)(6,9,11,13,10,7,12,8);(6,8,13,11,10,9,12)45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 925);