Finite Group DatabasebetaFinite Group Databasebeta| order | 1656 |
| isoclinism | isoclinic class of D32⋊D23 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D32⋊D23 |
| Fitting, socle, Hall | C32×C23 ⋊ D4 |
| Sylow | C23 ⋊ GO+(4,𝔽2) |
| Sylow | C32 ⋊ C22⋊D23 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C3⋊D3×C23 | C22 |
| Fitting, socle, hypocenter | C32×C23 | D4 |
| LCS | G ▷ C3⋊D3×C23 ▷ C32×C23 | |
| UCS | C1 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 138 |
Polycyclic presentation:
2^3,3^2,23|b^a=bc,d^a=e,e^a=d,f^a=f^22,d^b=d^2,d^c=d^2,e^c=e^2108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1656, 112);