Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D12×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×D12) |
| unambiguous semidirect product | (D7×D12)⋊C3 |
| Fitting | C3×C4×C7 ⋊ C22×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×D4 |
| Sylow | C7 ⋊ C2×C3×D12 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C3×C7 | C23×C3 |
| Fitting | C3×C4×C7 | C22×C3 |
| hypercenter | C4 | D3×AGL(1,𝔽7) |
| hypocenter | C3×C7 | C2×C3×D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
232 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 538);