Finite Group DatabasebetaFinite Group Databasebeta| order | 31220 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊(C223⋊AGL(1,𝔽5)) |
| unambiguous semidirect product | C223⋊(C7⋊AGL(1,𝔽5)) |
| Fitting, socle, derived, Hall | C5×C7×C223 ⋊ C4 |
| Hall | C7×C223 ⋊ AGL(1,𝔽5) |
| Hall | C5×C223 ⋊ Dic7 |
| Sylow | C223 ⋊ C7⋊AGL(1,𝔽5) |
| Hall | C5×C7 ⋊ Dic223 |
| Sylow | C7 ⋊ C223⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic1561 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C7×C223 | C4 |
| prime radical, cosocle | C7×D5×C223 | C2 |
| LCS | G ▷ C5×C7×C223 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 7805 |
Polycyclic presentation:
2^2,5,7,223|a^2=b,c^a=c^2,d^a=d^6,e^a=e^222,c^b=c^43125 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31220, 14);