Finite Group DatabasebetaFinite Group Databasebeta| order | 31164 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #23780749⋊C3 |
| Fitting, socle, derived, Hall | C72×C53 ⋊ C3×C4 |
| Hall | #23780749 ⋊ C3 |
| Hall | C53×#3861 ⋊ C4 |
| Hall | #3861 ⋊ #6157 |
| Sylow | C53 ⋊ #5997⋊C3 |
| Sylow | C72 ⋊ C3×#6157 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C72×C53 | C3×C4 |
| prime radical | C53⋊#64424 | C2 |
| cosocle | C7⋊D371 | C2×C3 |
| LCS | G ▷ C72×C53 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 742 |
Polycyclic presentation:
2,3,2,7^2,53|a^2=c,d^a=d^3e^5,e^a=d^5e^4,f^a=f^30,d^b=d^2,e^b=e^2,d^c=d^6,e^c=e^6,f^c=f^52263 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31164, 82);