Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D33 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | D33×C7 |
| unambiguous semidirect product | (D3×C3⋊D3)⋊(C2×C7) |
| Fitting, socle, Hall | C33×C7 ⋊ C23 |
| derived, Sylow | C33 ⋊ C23×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D33 |
| commutator, cosocle, hypocenter | C33 | C23×C7 |
| Frattini | C1 | G |
| Fitting, socle | C33×C7 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C7 | |
324 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
189 conjugacy classes — view the full character table.
| maximal subgroup of | D33×D7, D33×ΓL(1,𝔽8), D3×C7×(C3⋊D3)⋊D7 |
| maximal normal subgroup of | D33×D7, D33×ΓL(1,𝔽8) |
| maximal quotient of | D3×C7×(C3⋊D3)⋊D7 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 837);