Finite Group DatabasebetaFinite Group Databasebeta| order | 1764 |
| isoclinism | isoclinic class of (#64418⋊C2)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (#64418⋊C2)⋊C3 |
| unambiguous semidirect product | (#64418⋊C3)⋊C2 |
| Fitting, socle, Sylow | C72 ⋊ C2×C3×D3 |
| derived | #3862 ⋊ C22×C3 |
| Hall | ΓL(1,𝔽8)2 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #3862 | C22×C3 |
| Fitting, socle | C72 | C2×C3×D3 |
| LCS | G ▷ #3862 | |
| UCS | C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 14 |
Polycyclic presentation:
2^2,3^2,7^2|e^a=e^6,f^a=f^6,d^b=d^2,e^b=f^3,f^b=e^5,e^c=e^2,f^c=f^2,e^d=e^4,f^d=f^225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1764, 134);