Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊#20301293 |
| Fitting, socle, derived | C3×C52×C72 ⋊ C2×C3 |
| Hall | C3×C52⋊#3861 ⋊ C2 |
| Hall | C52×C72 ⋊ C3×D3 |
| Sylow | C72 ⋊ C3⋊((C5⋊D5)⋊C3) |
| Sylow | C52 ⋊ C3⋊#64423 |
| 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 | |
152 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 630 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4)(3,13,9,6,7,15)(5,10,14,11,12,8)(17,18)(20,23,24,21,22,25)(27,28,29)(30,32,31);(1,3,8,14,6)(2,5,7,13,11)(4,9,10,12,15)(16,17,18)(19,20,22,25,24,23,21)(26,27,28,30,29,31,32)Polycyclic presentation:
2,3^2,5^2,7^2|c^a=c^2,d^a=d^4,e^a=e^4,f^a=f^6,d^b=de,e^b=d^2e^3,f^b=f^2,g^b=g^2624 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 452);