Finite Group DatabasebetaFinite Group Databasebeta| order | 1656 |
| isoclinism | isoclinic class of C3⋊(A4⋊D23) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊(A4⋊D23) |
| unambiguous semidirect product | A4⋊D69 |
| unambiguous semidirect product | C23⋊(C3⋊S4) |
| Fitting, socle | C22×C3×C23 ⋊ D3 |
| derived | C3×A4×C23 ⋊ C2 |
| Sylow | C23 ⋊ C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×A4×C23 | C2 |
| Fitting, socle | C22×C3×C23 | D3 |
| LCS | G ▷ C3×A4×C23 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 276 |
Polycyclic presentation:
2,3^2,23,2^2|b^a=b^2,c^a=c^2,d^a=d^22,e^a=ef,e^b=f,f^b=ef141 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1656, 120);