Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of #66665 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×D7⋊SL(2,𝔽3) |
| unambiguous semidirect product | D7⋊(C5×SL(2,𝔽3)) |
| unambiguous semidirect product | (C5×D7)⋊SL(2,𝔽3) |
| Fitting | C5×C7×Q8 ⋊ C2×C3 |
| derived | C7×Q8 ⋊ C2×C3×C5 |
| Hall | C5×Q8×D7 ⋊ C3 |
| Hall | Q8×D7 ⋊ C3×C5 |
| Hall | C5×C7 ⋊ C2×SL(2,𝔽3) |
| Sylow | C7 ⋊ C2×C5×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C2×C3×C5 |
| Frattini | C2 | C5×D7⋊A4 |
| Fitting | C5×C7×Q8 | C2×C3 |
| socle | C2×C5×C7 | C2×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2×C5 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4,5)(7,8,10)(11,13,12)(15,19,18,16,17,20);(6,10,9,12)(7,13,11,8)(14,15,18,19,17,20,16)95 conjugacy classes — view the full character table.
| maximal subgroup of | C5×SL(2,𝔽3)×AGL(1,𝔽7), D5×D7⋊SL(2,𝔽3) |
| maximal normal subgroup of | C5×SL(2,𝔽3)×AGL(1,𝔽7), D5×D7⋊SL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 609);