Finite Group DatabasebetaFinite Group Databasebeta| order | 30690 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C9⋊#16811784 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C9×C11×C31 | C2×C5 |
| Frattini | C3 | C3⋊#16811784 |
| socle | C3×C11×C31 | C5×D3 |
| LCS | G ▷ C9×C11×C31 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 3069 |
A minimal faithful permutation representation — 2 generators on 51 points:
(2,4)(3,7)(5,6)(8,9)(11,15,16,14,19,12,13,20,17,18)(22,26,25,34,35,23,24,28,29,44)(27,41,42,43,45,32,33,49,50,48)(30,38,39,46,47,37,31,40,36,51);(1,9,4,7,6,5,3,2,8)(10,11,14,19,13,16,20,15,18,17,12)(21,22,25,30,35,45,39,38,24,27,33,43,47,49,46,34,29,36,42,51,50,41,32,26,31,40,48,44,37,28,23)Polycyclic presentation:
2,3,5,11,31,3|b^3=f^2,b^a=b^2f,d^a=d^10,e^a=e^30,f^a=f^2,d^c=d^4,e^c=e^8336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30690, 69);