Finite Group DatabasebetaFinite Group Databasebeta| order | 1764 |
| isoclinism | isoclinic class of #10596885⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #10596885⋊C3 |
| Fitting, socle, Sylow | C72 ⋊ C3×Dic3 |
| derived | #3862 ⋊ C3×C4 |
| Hall | ΓL(1,𝔽8)2 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | #3862 | C3×C4 |
| Fitting, socle | C72 | C3×Dic3 |
| prime radical | ΓL(1,𝔽8)⋊AGL(1,𝔽7) | C2 |
| cosocle | #64425 | C2×C3 |
| LCS | G ▷ #3862 | |
| UCS | C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 14 |
Polycyclic presentation:
2,3,2,3,7^2|a^2=c,d^a=d^2,e^a=f^3,f^a=e^2,e^b=e^2,f^b=f^2,e^c=e^6,f^c=f^6,e^d=e^4,f^d=f^222 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1764, 133);