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) = C1 |
No aliases for this group.
| Hall | C5×C7 ⋊ M16 |
| Sylow | C7 ⋊ #704 |
| Sylow | C5 ⋊ #1136 |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^4,5,7|a^2=c,c^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 | #11764637, #11764638, #11764682, … |
| maximal normal subgroup of | #11764637, #11764638, #11764682, … |
| maximal quotient of | #11764678, #11764681, C22⋊(C7⋊#170), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 107);