Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D7⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D7⋊(C22⋊C27) |
| unambiguous semidirect product | (C22⋊(C7⋊C27))⋊C2 |
| derived | C22×C7 ⋊ C2×C27 |
| Hall | C22×D7 ⋊ C27 |
| Sylow | C7 ⋊ C2×C22⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C9 | D7⋊A4 |
| commutator, hypocenter | C22×C7 | C2×C27 |
| Fitting, cosocle | C22×C7×C9 | C2×C3 |
| socle | C22×C3×C7 | C2×C9 |
| prime radical | C22×C3×D7 | C9 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C9 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 378 |
Polycyclic presentation:
2,3^3,2^2,7|b^3=c,c^3=d,g^a=g^6,e^b=ef,f^b=e,g^b=g^2108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 88);