Finite Group DatabasebetaFinite Group Databasebeta| order | 1752 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Q8⋊(C73⋊C3) |
| unambiguous semidirect product | C73⋊SL(2,𝔽3) |
| Fitting, derived, Hall | Q8×C73 ⋊ C3 |
| Sylow | C73 ⋊ SL(2,𝔽3) |
| Sylow | Q8 ⋊ C73⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C22⋊(C73⋊C3) |
| commutator, Fitting, cosocle, hypocenter | Q8×C73 | C3 |
| socle | C2×C73 | A4 |
| LCS | G ▷ Q8×C73 | |
| UCS | C1 ◁ C2 | |
14 conjugacy classes of subgroups.
| transitive degree | μt(G) = 584 |
Polycyclic presentation:
3,2^2,73,2|b^2=e,c^2=e,b^a=c,c^a=bc,d^a=d^64,c^b=ce127 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1752, 26);