Finite Group DatabasebetaFinite Group Databasebeta| order | 16002 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| Fitting, socle, derived, Hall | C7×C127 ⋊ C2×C32 |
| Hall | ΓL(1,𝔽8)×C127⋊C3 ⋊ C2 |
| Hall | D889 ⋊ C32 |
| Sylow | C127 ⋊ C3×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C3×D127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C127 | C2×C32 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,7,127|d^a=d^6,e^a=e^126,e^b=e^107,d^c=d^2126 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×D127⋊C3, C2×ΓL(1,𝔽8)⋊(D127⋊C3) |
| maximal normal subgroup of | AGL(1,𝔽7)×D127⋊C3, C2×ΓL(1,𝔽8)⋊(D127⋊C3) |
| maximal quotient of | ΓL(1,𝔽8)⋊((C127⋊C3)⋊C4), C2×ΓL(1,𝔽8)⋊(D127⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16002, 43);