Finite Group DatabasebetaFinite Group Databasebeta| order | 3756 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic313⋊C3 |
| derived, Sylow | C313 ⋊ C3×C4 |
| Hall | Dic313 ⋊ C3 |
| Hall | C313⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D313⋊C3 |
| commutator, hypocenter | C313 | C3×C4 |
| Fitting, socle, cosocle | C2×C313 | C2×C3 |
| prime radical | C2×C313⋊C3 | C2 |
| LCS | G ▷ C313 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 317 |
| transitive degree | μt(G) = 1252 |
Polycyclic presentation:
2,3,313,2|a^2=d,c^a=c^312,c^b=c^214116 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic313⋊C3, C3⋊(Dic313⋊C3), C5×Dic313⋊C3, … |
| maximal normal subgroup of | C3×Dic313⋊C3, C5×Dic313⋊C3, C7×Dic313⋊C3 |
| maximal quotient of | Dic313⋊C9, C3×Dic313⋊C3, C3⋊(Dic313⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3756, 2);