Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D5×GL(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| direct product | D5×GL(3,𝔽2) |
| unambiguous semidirect product | C5⋊(C2×GL(3,𝔽2)) |
| Fitting, Sylow | C5 ⋊ C2×GL(3,𝔽2) |
| socle, derived | C5×GL(3,𝔽2) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C5×GL(3,𝔽2) | C2 |
| Fitting, cosocle | C5 | C2×GL(3,𝔽2) |
| LCS | G ▷ C5×GL(3,𝔽2) | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 35 |
24 conjugacy classes — view the full character table.
| maximal subgroup of | A5×GL(3,𝔽2), D5×AGL(3,𝔽2), AGL(1,𝔽5)×GL(3,𝔽2), … |
| maximal normal subgroup of | AGL(1,𝔽5)×GL(3,𝔽2), C2×D5×GL(3,𝔽2) |
| maximal quotient of | D5×SL(2,𝔽7), Dic5×GL(3,𝔽2), #72209750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 926);