Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D32×ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×D32×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C3×D3)⋊(C22×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×D32×C7)⋊C3 |
| Fitting, socle | C2×C32×C7 ⋊ C22×C3 |
| derived | C32×C7 ⋊ C23×C3 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C23 |
| Sylow | C7 ⋊ C2×C3×D32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C32×C7 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | C22×C3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
324 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 805);