Finite Group DatabasebetaFinite Group Databasebeta| order | 16758 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊#11716328 |
| Fitting, socle, derived, Hall | C72×C19 ⋊ C2×C32 |
| Hall | ΓL(1,𝔽8)×#87705 ⋊ C2 |
| Hall | C7⋊D133 ⋊ C32 |
| Sylow | C72 ⋊ C3×D19⋊C3 |
| Sylow | C19 ⋊ ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C19 | C2×C32 |
| LCS | G ▷ C72×C19 | |
| UCS | C1 | |
112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 399 |
Polycyclic presentation:
2,3^2,7^2,19|d^a=d^6,e^a=e^6,f^a=f^18,e^b=e^2,f^b=f^11,d^c=d^2131 conjugacy classes — view the full character table.
| maximal subgroup of | #45692025, AGL(1,𝔽7)×#11716328, #45692034, … |
| maximal normal subgroup of | #45692025, AGL(1,𝔽7)×#11716328, #45692034, … |
| maximal quotient of | ΓL(1,𝔽8)⋊#13097589, C2×ΓL(1,𝔽8)⋊#11716328 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16758, 186);