Finite Group DatabasebetaFinite Group Databasebeta| order | 31164 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C7⋊#6157) |
| Fitting, socle, derived, Hall | C72×C53 ⋊ C3×C4 |
| Hall | C7⋊(C7⋊#6157) ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8)×C53 ⋊ C4 |
| Hall | C7×ΓL(1,𝔽8) ⋊ #6157 |
| Sylow | C53 ⋊ ΓL(1,𝔽8)⋊Dic7 |
| Sylow | C72 ⋊ C3×#6157 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C72×C53 | C3×C4 |
| prime radical | C7×ΓL(1,𝔽8)×D53 | C2 |
| cosocle | C72×D53 | C2×C3 |
| LCS | G ▷ C72×C53 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1113 |
Polycyclic presentation:
2,3,2,7^2,53|a^2=c,d^a=d^6,e^a=e^6,f^a=f^30,d^b=d^2,f^c=f^52499 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31164, 90);