Finite Group DatabasebetaFinite Group Databasebeta| order | 13692 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊Dic163 |
| derived, Hall | C7×C163 ⋊ C3×C4 |
| Hall | Dic1141 ⋊ C3 |
| Hall | ΓL(1,𝔽8)×C163 ⋊ C4 |
| Sylow | C163 ⋊ ΓL(1,𝔽8)⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ Dic163 |
| Sylow | C7 ⋊ C3×Dic163 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)⋊D163 |
| commutator, hypocenter | C7×C163 | C3×C4 |
| Fitting, socle, cosocle | C2×C7×C163 | C2×C3 |
| prime radical | C2×ΓL(1,𝔽8)×C163 | C2 |
| LCS | G ▷ C7×C163 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 4564 |
Polycyclic presentation:
2,3,7,163,2|a^2=e,c^a=c^6,d^a=d^162,c^b=c^2824 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13692, 22);