Finite Group DatabasebetaFinite Group Databasebeta| order | 560 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C4×C5×D7 |
| Fitting | C2×C4×C5×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C4×C5 |
| Hall | C5×C7 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C5 | D7 |
| commutator, hypocenter | C7 | C22×C4×C5 |
| Frattini | C2 | C22×C5×D7 |
| Fitting | C2×C4×C5×C7 | C2 |
| socle | C22×C5×C7 | C22 |
| prime radical | C22×C5×D7 | C2 |
| cosocle | C2×C7 | C23×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C4×C5 | |
108 conjugacy classes of subgroups.
| transitive degree | μt(G) = 280 |
200 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D7)⋊#168, C5×(C2×D7)⋊C8, #11764747, … |
| maximal normal subgroup of | (C2×D7)⋊#168, C5×(C2×D7)⋊C8, #11764747, … |
| maximal quotient of | C4×C5×Dic14, C42×C5×D7, C5×#6435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 137);