Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5×C7 |
| unambiguous semidirect product | C3⋊((C5⋊D5)⋊#3862) |
| unambiguous semidirect product | (C3⋊(C5⋊D5))⋊#3862 |
| Fitting, socle, derived | C3×C52×C72 ⋊ C2×C3 |
| Hall | C3×C52⋊#3862 ⋊ C2 |
| Hall | C52×C72 ⋊ C3×D3 |
| Sylow | C72 ⋊ C3⋊((C5⋊D5)⋊C3) |
| Sylow | C52 ⋊ D3×#3862 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C52×C72 | C2×C3 |
| LCS | G ▷ C3×C52×C72 | |
| UCS | C1 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 315 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4)(3,13,9,7,6,10)(5,15,14,11,12,8)(17,18)(20,21,22)(23,25,24)(27,28,29)(30,32,31);(1,3,8,14,7)(2,5,6,13,11)(4,9,15,12,10)(16,17,18)(19,20,21,23,22,24,25)(26,27,29,31,28,30,32)Polycyclic presentation:
2,3^2,5^2,7^2|c^a=c^2,d^a=d^4,e^a=e^4,d^b=de^3,e^b=d^4e^3,f^b=f^2,g^b=g^4645 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 454);