Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D5×AΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| Fitting, socle | C23×C5 ⋊ C2×ΓL(1,𝔽8) |
| derived | C5×AGL(1,𝔽8) ⋊ C2×C3 |
| Hall | D5×AGL(1,𝔽8) ⋊ C3 |
| Hall | C23×D5 ⋊ ΓL(1,𝔽8) |
| Sylow | C5 ⋊ C2×AΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C5×AGL(1,𝔽8) | C2×C3 |
| Fitting, socle | C23×C5 | C2×ΓL(1,𝔽8) |
| LCS | G ▷ C5×AGL(1,𝔽8) | |
| UCS | C1 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
32 conjugacy classes — view the full character table.
| maximal subgroup of | D5×AGL(3,𝔽2), AGL(1,𝔽5)×AΓL(1,𝔽8), C2×D5×AΓL(1,𝔽8) |
| maximal normal subgroup of | AGL(1,𝔽5)×AΓL(1,𝔽8), C2×D5×AΓL(1,𝔽8) |
| maximal quotient of | C2×D5×AΓL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 933);