Finite Group DatabasebetaFinite Group Databasebeta| order | 30450 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C29⋊#11760386 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C7×C29 | C2×C3 |
| LCS | G ▷ C52×C7×C29 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 3045 |
A minimal faithful permutation representation — 2 generators on 51 points:
(2,5,6,3,4,7)(8,9,11)(10,20,16,13,14,22)(12,17,21,18,19,15)(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);(1,2,4,7,6,5,3)(8,10,15,21,13)(9,12,14,20,18)(11,16,17,19,22)(23,24,26,28,30,32,34,36,38,40,42,44,46,48,50,51,49,47,45,43,41,39,37,35,33,31,29,27,25)Polycyclic presentation:
2,3,5^2,7,29|c^a=c^4,d^a=d^4,e^a=e^6,f^a=f^28,c^b=cd,d^b=c^2d^3,e^b=e^2889 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30450, 65);