Finite Group DatabasebetaFinite Group Databasebeta| order | 30690 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D15345 |
| unambiguous semidirect product | C5⋊D3069 |
| unambiguous semidirect product | C9⋊D1705 |
| unambiguous semidirect product | C11⋊D1395 |
| unambiguous semidirect product | C31⋊D495 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×C9×C11×C31 | C2 |
| Frattini | C3 | D5115 |
| socle | C3×C5×C11×C31 | D3 |
| LCS | G ▷ C5×C9×C11×C31 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 15345 |
A minimal faithful permutation representation — 2 generators on 56 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56);(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(16,25)(17,24)(18,23)(19,22)(20,21)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)Polycyclic presentation:
2,3,5,11,31,3|b^3=f^2,b^a=b^2f,c^a=c^4,d^a=d^10,e^a=e^30,f^a=f^27674 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×D15345 |
| maximal normal subgroup of | C2×D15345 |
| maximal quotient of | Dic15345, C2×D15345 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30690, 29);