Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C7×Q8 ⋊ 3−1+2 |
| Sylow | Q8 ⋊ #4427 |
| Sylow | C7 ⋊ C9⋊SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | A4×ΓL(1,𝔽8) |
| commutator, Fitting, cosocle | C3×C7×Q8 | C32 |
| socle | C2×C3×C7 | C3×A4 |
| prime radical | C3×C7⋊SL(2,𝔽3) | C3 |
| hypocenter | C7×Q8 | 3−1+2 |
| LCS | G ▷ C3×C7×Q8 ▷ C7×Q8 | |
| UCS | C1 ◁ C2×C3 | |
54 conjugacy classes of subgroups.
| transitive degree | μt(G) = 504 |
Polycyclic presentation:
3^3,2^2,7,2|a^3=c,b^3=c,d^2=g,e^2=g,b^a=bc,f^a=f^2,d^b=de,e^b=d,e^d=eg73 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 140);