Finite Group DatabasebetaFinite Group Databasebeta| order | 30690 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C9⋊(ΓL(1,𝔽32)⋊D11) |
| unambiguous semidirect product | ΓL(1,𝔽32)⋊D99 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C9×C11×C31 | C2×C5 |
| Frattini | C3 | C3⋊(ΓL(1,𝔽32)⋊D11) |
| 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)(11,12)(13,14)(15,16)(17,18)(19,20)(22,26,27,38,25,23,24,31,32,29)(28,43,44,45,30,34,35,47,49,37)(33,50,42,48,36,40,41,51,39,46);(1,9,4,7,6,5,3,2,8)(10,11,13,15,17,19,20,18,16,14,12)(21,22,25,30,32,39,49,43,27,33,42,50,44,46,34,26,24,28,36,47,41,51,40,31,35,45,48,38,37,29,23)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,e^c=e^16552 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30690, 66);