Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | AGL(1,𝔽7)×#239 |
| Fitting | C3×C7×Q8 ⋊ C22×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×#41 |
| Sylow | C7 ⋊ C2×C3×#239 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C3×C7 | C24×C3 |
| Fitting | C3×C7×Q8 | C22×C3 |
| hypercenter | Q8 | D3×AGL(1,𝔽7) |
| hypocenter | C3×C7 | C2×C3×#41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
694 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 694 conjugacy classes of subgroups — too many to draw. Only its 201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#41×AGL(1,𝔽7), AGL(1,𝔽7)×#1009, AGL(1,𝔽7)×#914, … |
| maximal normal subgroup of | D3×#41×AGL(1,𝔽7), AGL(1,𝔽7)×#1009, AGL(1,𝔽7)×#914 |
| maximal quotient of | AGL(1,𝔽7)×#3799, AGL(1,𝔽7)×#3796 |
Past the SmallGroups library coverage — no SmallGroups id.