Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #10587842⋊C3 |
| Hall | #10587842 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ D8 |
| Hall | C5×C7 ⋊ C3×D8 |
| Hall | ΓL(1,𝔽8) ⋊ #686 |
| Sylow | C7 ⋊ C3×#686 |
| Sylow | C5 ⋊ D56⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #63306⋊C3 |
| commutator, Fitting, cosocle | C4×C5×C7 | C22×C3 |
| Frattini, hypercenter | C4 | D5×AGL(1,𝔽7) |
| socle | C2×C5×C7 | C3×D4 |
| hypocenter | C5×C7 | C3×D8 |
| LCS | G ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^2,3,2^2,5,7|b^2=de,d^2=e,b^a=bd,d^a=de,g^a=g^6,f^b=f^4,g^c=g^267 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D56⋊C3, AGL(1,𝔽7)×#686, #72217938, … |
| maximal normal subgroup of | D5×D56⋊C3, AGL(1,𝔽7)×#686, #72217938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 22);