Finite Group DatabasebetaFinite Group Databasebeta| order | 31164 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C49⋊C3)⋊#6157 |
| unambiguous semidirect product | (C49⋊#6157)⋊C3 |
| Fitting, derived, Hall | C49×C53 ⋊ C3×C4 |
| Hall | C49⋊#6157 ⋊ C3 |
| Hall | C53×C49⋊C3 ⋊ C4 |
| Hall | C49⋊C3 ⋊ #6157 |
| Sylow | C53 ⋊ (C49⋊C3)⋊C4 |
| Sylow | C49 ⋊ C3×#6157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C49×C53 | C3×C4 |
| Frattini | C7 | ΓL(1,𝔽8)⋊#6157 |
| socle | C7×C53 | ΓL(1,𝔽8)⋊C4 |
| prime radical | D53×C49⋊C3 | C2 |
| cosocle | C49×D53 | C2×C3 |
| LCS | G ▷ C49×C53 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2597 |
Polycyclic presentation:
2,3,2,7,53,7|a^2=c,d^7=f^6,d^a=d^6f,e^a=e^30,f^a=f^6,d^b=d^2f^3,f^b=f^2,e^c=e^52275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31164, 14);