Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C4×D32×C7 |
| Fitting | C32×C4×C7 ⋊ C22 |
| derived, Sylow | C32 ⋊ C22×C4×C7 |
| Hall | C32×C7 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | D32 |
| commutator, hypocenter | C32 | C22×C4×C7 |
| Frattini | C2 | C2×D32×C7 |
| Fitting | C32×C4×C7 | C22 |
| socle | C2×C32×C7 | C23 |
| prime radical | C2×D32×C7 | C2 |
| cosocle | C2×C32 | C23×C7 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4×C7 | |
232 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D32×ΓL(1,𝔽8), C4×D32×D7, C4×C7×GO+(4,𝔽2), … |
| maximal normal subgroup of | C4×D32×ΓL(1,𝔽8), C4×D32×D7, C4×C7×GO+(4,𝔽2), … |
| maximal quotient of | D32×C7×AGL(1,𝔽5), D32×AGL(1,𝔽29), C7×#63921 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 759);