Finite Group DatabasebetaFinite Group Databasebeta| order | 8540 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊#11927413 |
| Fitting, socle, derived, Hall | C5×C7×C61 ⋊ C4 |
| Hall | C7×C61 ⋊ AGL(1,𝔽5) |
| Hall | C5×C61 ⋊ Dic7 |
| Sylow | C61 ⋊ C7⋊AGL(1,𝔽5) |
| Hall | C5×C7 ⋊ #6923 |
| Sylow | C7 ⋊ #11927413 |
| Sylow | C5 ⋊ C7⋊#6923 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C7×C61 | C4 |
| LCS | G ▷ C5×C7×C61 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,5,7,61|a^2=b,c^a=c^2,d^a=d^6,e^a=e^11,c^b=c^4,e^b=e^60542 conjugacy classes — view the full character table.
| maximal subgroup of | #38785063, #38785064, ΓL(1,𝔽8)⋊#11927413, … |
| maximal normal subgroup of | #38785063, #38785064, ΓL(1,𝔽8)⋊#11927413, … |
| maximal quotient of | C3×C7⋊#11927413, C3⋊(C7⋊#11927413) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8540, 30);