Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11 · Schur multiplier (H2) = C5 |
| direct product | C11×#15691064 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C11 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C52×C11 |
| prime radical | C53×C11 | C52 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C11 ◁ C53×C11 ◁ G | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 6875 |
A minimal faithful permutation representation — 2 generators on 61 points:
(1,2,6,14,11,5,7,17,20,24,13,18,15,21,23,12,19,25,9,10,4,8,16,22,3)(26,27,30,37,45,29,28,33,38,46,36,35,41,43,49,44,42,31,39,47,50,32,34,40,48)(51,52,53,54,55,56,57,58,59,60,61);(1,3,9,6,8,4,10,21,16,19,12,23,20,25,18,13,24,14,15,7,5,11,22,17,2)(26,28,34,37,45,29,35,30,38,46,36,42,33,43,49,44,32,41,39,47,50,27,31,40,48)1639 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 7);