Finite Group DatabasebetaFinite Group Databasebeta| order | 560 |
| automorphism | Aut(G) = D4×AGL(1,𝔽5)×AGL(1,𝔽7) (order 6720) · Inn(G) = C2×C7⋊AGL(1,𝔽5) · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊(C7⋊AGL(1,𝔽5)) |
| Fitting | C4×C5×C7 ⋊ C4 |
| Hall | C5×C7 ⋊ C4⋊C4 |
| Sylow | C7 ⋊ C4⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4⋊Dic7 |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 140 |
Polycyclic presentation:
2^4,5,7|a^2=c,b^2=d,b^a=bd,e^a=e^3,f^a=f^6,e^c=e^462 conjugacy classes — view the full character table.
| maximal subgroup of | #11764615, #11764617, #11764622, … |
| maximal normal subgroup of | #11764615, #11764617, #11764622, … |
| maximal quotient of | #11764670, #11764671, #11764672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 105);