Finite Group DatabasebetaFinite Group Databasebeta| order | 744 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Q8⋊(C31⋊C3) |
| unambiguous semidirect product | C31⋊SL(2,𝔽3) |
| Fitting, derived, Hall | Q8×C31 ⋊ C3 |
| Sylow | C31 ⋊ SL(2,𝔽3) |
| Sylow | Q8 ⋊ C31⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C22⋊(C31⋊C3) |
| commutator, Fitting, cosocle, hypocenter | Q8×C31 | C3 |
| socle | C2×C31 | A4 |
| LCS | G ▷ Q8×C31 | |
| UCS | C1 ◁ C2 | |
14 conjugacy classes of subgroups.
| transitive degree | μt(G) = 248 |
Polycyclic presentation:
3,2^2,31,2|b^2=e,c^2=e,b^a=c,c^a=bce,d^a=d^25,c^b=ce57 conjugacy classes — view the full character table.
| maximal subgroup of | #13093537, D31⋊SL(2,𝔽3), C2×Q8⋊(C31⋊C3), … |
| maximal normal subgroup of | #13093537, D31⋊SL(2,𝔽3), C2×Q8⋊(C31⋊C3), … |
| maximal quotient of | C2×Q8⋊(C31⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(744, 23);