Finite Group DatabasebetaFinite Group Databasebeta| order | 18270 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D9135 |
| unambiguous semidirect product | C5⋊D1827 |
| unambiguous semidirect product | C7⋊D1305 |
| unambiguous semidirect product | C9⋊D1015 |
| unambiguous semidirect product | C29⋊D315 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×C7×C9×C29 | C2 |
| Frattini | C3 | D3045 |
| socle | C3×C5×C7×C29 | D3 |
| LCS | G ▷ C5×C7×C9×C29 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 9135 |
A minimal faithful permutation representation — 2 generators on 50 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50);(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(16,21)(17,20)(18,19)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)Polycyclic presentation:
2,3,5,7,29,3|b^3=f^2,b^a=b^2f,c^a=c^4,d^a=d^6,e^a=e^28,f^a=f^24569 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D29×D315, D7×D1305, D45×D203, … |
| maximal normal subgroup of | D29×D315, D7×D1305, D45×D203, … |
| maximal quotient of | Dic9135, C2×D9135, D27405 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18270, 18);