Finite Group DatabasebetaFinite Group Databasebeta| order | 630 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×D7⋊C9 |
| unambiguous semidirect product | D7⋊(C5×C9) |
| unambiguous semidirect product | (C7⋊C9)⋊(C2×C5) |
| unambiguous semidirect product | (C5×D7)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C5×C9 |
| Hall | C5×C7⋊C9 ⋊ C2 |
| Hall | C5×D7 ⋊ C9 |
| Hall | C7⋊C9 ⋊ C2×C5 |
| Hall | C5×C7 ⋊ C2×C9 |
| Hall | D7 ⋊ C5×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C5×C9 |
| Frattini | C3 | C5×AGL(1,𝔽7) |
| Fitting, socle | C3×C5×C7 | C2×C3 |
| prime radical | C3×C5×D7 | C3 |
| cosocle | C3×C7 | C2×C3×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C5 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 315 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D7⋊C9, C2×C5×D7⋊C9, C5×C9×AGL(1,𝔽7), … |
| maximal normal subgroup of | D5×D7⋊C9, C2×C5×D7⋊C9, C5×C9×AGL(1,𝔽7), … |
| maximal quotient of | C5×Dic7⋊C9, C2×C5×D7⋊C9, C5×D7⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(630, 1);