Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C7⋊#6258 |
| Fitting | C22×C32×C7 ⋊ D3 |
| derived | C7×C22⋊H(3,𝔽3) ⋊ C2 |
| Sylow | C7 ⋊ #6258 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C3⋊(C7⋊S4) |
| commutator, cosocle, hypocenter | C7×C22⋊H(3,𝔽3) | C2 |
| Fitting | C22×C32×C7 | D3 |
| socle | C22×C3×C7 | C3⋊D3 |
| LCS | G ▷ C7×C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
114 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2,3^2,7,3,2^2|b^a=b^2,c^a=c^2,d^a=d^6,f^a=fg,c^b=ce,f^b=g,g^b=fg79 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 503);