Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D7×(C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×D7×(C3⋊D3)⋊C3 |
| Fitting | C2×C7×H(3,𝔽3) ⋊ C22 |
| derived | C32×C7 ⋊ C23×C3 |
| Hall | C7×H(3,𝔽3) ⋊ C23 |
| Sylow | H(3,𝔽3) ⋊ C22×D7 |
| Sylow | C7 ⋊ C22×(C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D7×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32×C7 | C23×C3 |
| Frattini | C3 | C2×C3×D3×D7 |
| Fitting | C2×C7×H(3,𝔽3) | C22 |
| socle | C2×C3×C7 | C2×C3×D3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 244 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) = 126 |
100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 553);