Finite Group DatabasebetaFinite Group Databasebeta| order | 8820 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C49⋊C9)⋊Dic5 |
| unambiguous semidirect product | Dic245⋊C9 |
| derived, Hall | C5×C49 ⋊ C4×C9 |
| Hall | C5×C49⋊C9 ⋊ C4 |
| Hall | Dic245 ⋊ C9 |
| Hall | C49⋊C9 ⋊ Dic5 |
| Sylow | C49 ⋊ C9×Dic5 |
| Sylow | C5 ⋊ Dic49⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C5⋊(D49⋊C3) |
| commutator, hypocenter | C5×C49 | C4×C9 |
| Frattini | C2×C3×C7 | C5⋊AGL(1,𝔽7) |
| Fitting | C2×C3×C5×C49 | C2×C3 |
| socle | C2×C3×C5×C7 | AGL(1,𝔽7) |
| LCS | G ▷ C5×C49 | |
| UCS | C1 ◁ C2×C3 | |
54 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,5,7,2,3,7|a^2=e,b^3=f,d^7=g^6,c^a=c^4,d^a=d^6g,g^a=g^6,d^b=d^2g^3,g^b=g^2312 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8820, 4);