Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D21⋊D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C3⋊D9)⋊Dic7 |
| derived, Hall | C3×C7×C9 ⋊ C2×C4 |
| Sylow | C3×C9 ⋊ C2×Dic7 |
| Sylow | C7 ⋊ (C3⋊D9)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D21⋊D9 |
| commutator, hypocenter | C3×C7×C9 | C2×C4 |
| Frattini | C2×C3 | (C3⋊D3)⋊D7 |
| Fitting, cosocle | C2×C3×C7×C9 | C22 |
| socle | C2×C32×C7 | C2×D3 |
| prime radical | C2×C7×C3⋊D9 | C2 |
| LCS | G ▷ C3×C7×C9 | |
| UCS | C1 ◁ C2 | |
116 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2^3,3^2,7,3|a^2=c,e^3=g,d^a=d^2,f^a=f^6,d^b=d^2,e^b=e^2g^2,g^b=g^2126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 300);