Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊#6180 |
| derived, Hall | C7×H(3,𝔽3) ⋊ C8 |
| Sylow | H(3,𝔽3) ⋊ C7⋊C8 |
| Sylow | C7 ⋊ #6180 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C4 | C3⋊D21 |
| commutator, hypocenter | C7×H(3,𝔽3) | C8 |
| Fitting, cosocle | C4×C7×H(3,𝔽3) | C2 |
| socle | C2×C3×C7 | C3⋊Dic3 |
| prime radical | C2×C7×H(3,𝔽3) | C4 |
| LCS | G ▷ C7×H(3,𝔽3) | |
| UCS | C1 ◁ C3×C4 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 504 |
Polycyclic presentation:
2^3,3^2,7,3|a^2=b,b^2=c,d^a=d^2,e^a=e^2,f^a=f^6,e^d=eg172 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 78);