Finite Group DatabasebetaFinite Group Databasebeta| order | 560 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#120×AGL(1,𝔽7) (order 26880) · Inn(G) = C2×D35 · Out(G) = C2×C3×C4×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊Dic35 |
| Hall | C5×C7 ⋊ C4⋊C4 |
| Sylow | C7 ⋊ C4⋊Dic5 |
| Sylow | C5 ⋊ C4⋊Dic7 |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 560 |
Polycyclic presentation:
2^4,5,7|a^2=c,b^2=d,b^a=bd,e^a=e^4,f^a=f^6146 conjugacy classes — view the full character table.
| maximal subgroup of | #11764435, #11764437, #11764441, … |
| maximal normal subgroup of | #11764435, #11764437, #11764441, … |
| maximal quotient of | C4⋊(C7⋊#168), #11764569, #11764570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 73);