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) = C22 |
| unambiguous semidirect product | C22⋊Dic35 |
| Hall | C5×C7 ⋊ C22⋊C4 |
| Sylow | C7 ⋊ C22⋊Dic5 |
| Sylow | C5 ⋊ C22⋊Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D35 |
| commutator | C2×C5×C7 | C2×C4 |
| Fitting, prime radical | C23×C5×C7 | C2 |
| socle, cosocle | C22×C5×C7 | C22 |
| hypercenter | C23 | D35 |
| hypocenter | C5×C7 | C22⋊C4 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^4,5,7|a^2=c,b^a=bd,e^a=e^4,f^a=f^6146 conjugacy classes — view the full character table.
| maximal subgroup of | #11764465, #11764466, #11764559, … |
| maximal normal subgroup of | #11764465, #11764466, #11764559, … |
| maximal quotient of | C22⋊(C7⋊#168), #11764584, #11764585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 79);