Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of #13094600 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C22×C32×C7 ⋊ D3 |
| derived | C3×C7×A4 ⋊ C2×C3 |
| Sylow | C7 ⋊ (C3⋊S4)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C7×A4 | C2×C3 |
| Frattini | C3 | C3×C7⋊S4 |
| Fitting | C22×C32×C7 | D3 |
| socle | C22×C3×C7 | C3×D3 |
| LCS | G ▷ C3×C7×A4 | |
| UCS | C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2,3^3,7,2^2|c^a=c^2,d^a=d^2,e^a=e^6,f^a=fg,c^b=cd^2,f^c=g,g^c=fg79 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 497);