Finite Group DatabasebetaFinite Group Databasebeta| order | 490 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×C3×C4×AGL(1,𝔽7) (order 1008) · Inn(G) = D7 · Out(G) = C2×C32×C4 |
| homology | Abelianization (H1) = C2×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C5×C7×D7 |
| unambiguous semidirect product | C7⋊(C2×C5×C7) |
| Fitting, socle, Hall | C5×C72 ⋊ C2 |
| derived | C7 ⋊ C2×C5×C7 |
| Sylow | C72 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C7 | D7 |
| commutator, cosocle, hypocenter | C7 | C2×C5×C7 |
| Frattini | C1 | G |
| Fitting, socle | C5×C72 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C5×C7 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 70 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4,5)(6,7,8,9,10,11,12)(14,15)(16,17)(18,19);(13,14,16,18,19,17,15)175 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D72, C7×D5×D7, D7×D35, … |
| maximal normal subgroup of | C5×D72, C7×D5×D7, D7×D35, … |
| maximal quotient of | C5×C7×Dic7, C2×C5×C7×D7, C3×C5×C7×D7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(490, 5);