Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#15691068 |
| 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:
(1,2,6,14,23,17,8,21,3,11,5,10,18,25,7,16,12,24,15,22,4,9,19,13,20)(26,27,30,38,45,37,28,34,39,47,50,32,31,41,49,29,33,35,42,46,44,36,40,43,48)(51,52,53,54,55,56,57,58,59,60,61);(1,3,12,23,6,16,13,8,22,24,17,25,9,11,21,4,14,10,20,19,5,15,2,7,18)(26,28,35,43,49,37,32,40,38,46,50,33,30,39,48,29,36,34,41,45,44,27,31,42,47)1639 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 11);