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 | #3861⋊(C19⋊C3) ⋊ C2 |
| Hall | C7×D7×C19 ⋊ 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 | |
100 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 798 |
Polycyclic presentation:
2,3^2,7^2,19|d^a=d^6,e^b=e^4,f^b=f^11,d^c=d^2,e^c=e^2137 conjugacy classes — view the full character table.
| maximal subgroup of | #45692021, #45692039, (#64423⋊C2)⋊(C19⋊C3), … |
| maximal normal subgroup of | #45692021, #45692039, (#64423⋊C2)⋊(C19⋊C3), … |
| maximal quotient of | #45691769, C2×#30660685 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16758, 212);