Finite Group DatabasebetaFinite Group Databasebeta| order | 560 |
| automorphism | Aut(G) = C22×AGL(1,𝔽5)×AGL(1,𝔽7) (order 3360) · Inn(G) = D7×AGL(1,𝔽5) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C5×C7 ⋊ M16 |
| Sylow | C7 ⋊ #700 |
| Sylow | C5 ⋊ #1131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×AGL(1,𝔽5) |
| commutator, Fitting, socle | C2×C5×C7 | C2×C4 |
| prime radical | C2×D35 | C4 |
| cosocle | C7×Dic5 | C22 |
| hypocenter | C5×C7 | M16 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^4,5,7|a^2=d,b^2=c,c^2=d,b^a=bd,f^a=f^6,e^b=e^2,e^c=e^444 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#700, #11765317, #11765318, … |
| maximal normal subgroup of | D7×#700, #11765317, #11765318, … |
| maximal quotient of | #11764632, #11764634, Dic7⋊#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 94);