Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of (C3⋊D3)⋊(C22⋊ΓL(1,𝔽8)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | (C3⋊D3)⋊(C22⋊ΓL(1,𝔽8)) |
| Fitting, derived | C22×C32×C7 ⋊ C2×C3 |
| Sylow | C7 ⋊ (C3⋊D3)⋊A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C32×C7 | C2×C3 |
| Frattini | C3 | D3×C22⋊ΓL(1,𝔽8) |
| socle | C22×C3×C7 | C3×D3 |
| LCS | G ▷ C22×C32×C7 | |
| UCS | C1 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2,3,2^2,3^2,7|e^a=e^2,f^a=f^2,c^b=cd,d^b=c,e^b=ef,g^b=g^464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 543);