Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊((C7⋊C8)⋊C9) |
| unambiguous semidirect product | (C7⋊C9)⋊(C3⋊C8) |
| unambiguous semidirect product | (C3⋊(C7⋊C8))⋊C9 |
| derived | C3×C7 ⋊ C8×C9 |
| Hall | C3×C7⋊C9 ⋊ C8 |
| Sylow | C7 ⋊ C9×C3⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C4 | C3⋊AGL(1,𝔽7) |
| commutator, hypocenter | C3×C7 | C8×C9 |
| Fitting, cosocle | C32×C4×C7 | C2×C3 |
| socle, prime radical | C2×C32×C7 | C3×C4 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C3×C4 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 504 |
Polycyclic presentation:
2,3,2,3,2,3,7|a^2=c,b^3=d,c^2=e,f^a=f^2,g^a=g^6,g^b=g^2144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 56);