Finite Group DatabasebetaFinite Group Databasebeta| order | 1656 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C4⋊(C3⋊D69) |
| Fitting | C32×C4×C23 ⋊ C2 |
| Hall | C32×C23 ⋊ D4 |
| Sylow | C23 ⋊ C4⋊(C3⋊D3) |
| Sylow | C32 ⋊ D92 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊D69 |
| commutator, socle, cosocle | C2×C32×C23 | C22 |
| Fitting | C32×C4×C23 | C2 |
| hypercenter | C4 | C3⋊D69 |
| hypocenter | C32×C23 | D4 |
| LCS | G ▷ C2×C32×C23 ▷ C32×C23 | |
| UCS | C1 ◁ C2 ◁ C4 | |
96 conjugacy classes of subgroups.
| transitive degree | μt(G) = 828 |
Polycyclic presentation:
2^3,3^2,23|b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=f^22417 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1656, 102);