Finite Group DatabasebetaFinite Group Databasebeta| order | 16758 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, socle, derived, Hall | C72×C19 ⋊ C2×C32 |
| Hall | #22552727 ⋊ C2 |
| Hall | C19×C7⋊D7 ⋊ C32 |
| Hall | C7⋊D7 ⋊ C3×C19⋊C3 |
| Sylow | C72 ⋊ C2×C3×C19⋊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 24 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 24 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,d^b=d^2,f^b=f^7,d^c=d^4,e^c=e^2110 conjugacy classes — view the full character table.
| maximal subgroup of | #45691996, #45691998, #45692004, … |
| maximal normal subgroup of | #45691996, #45691998, #45692004, … |
| maximal quotient of | #45691750, C2×#30660666 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16758, 193);