Finite Group DatabasebetaFinite Group Databasebeta| order | 31164 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #64418⋊D53 |
| Fitting, socle, Hall | C72×C53 ⋊ C2×D3 |
| derived, Hall | C53×#3862 ⋊ C22 |
| Hall | #3862 ⋊ C2×D53 |
| Sylow | C53 ⋊ #64418⋊C2 |
| Sylow | C72 ⋊ D3×D53 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C53×#3862 | C22 |
| Fitting, socle | C72×C53 | C2×D3 |
| LCS | G ▷ C53×#3862 | |
| UCS | C1 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 742 |
Polycyclic presentation:
2^2,3,53,7^2|c^a=c^2,d^a=d^52,e^a=e^4f,f^a=e^6f^3,c^b=c^2,e^b=e^3f^6,f^b=ef^4,e^c=e^2f^6,f^c=f^4539 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31164, 85);