Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11×C25 · Schur multiplier (H2) = C53 |
| direct product | C5×C11×C25⋊C25 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C53×C11 | C52 |
| commutator | C5 | C52×C11×C25 |
| Frattini, cosocle | C52 | C53×C11 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C53×C11 ◁ G | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 34375 |
A minimal faithful permutation representation — 3 generators on 66 points:
(1,2,3,4,5)(56,57,58,59,60,61,62,63,64,65,66);(7,9,12,17,25)(10,14,20,13,18)(15,22,19,26,21)(23,30,29,28,27)(31,32,34,37,41,33,35,38,42,46,36,39,43,47,50,40,44,48,51,53,45,49,52,54,55);(6,7,10,15,23,8,9,13,19,27,11,12,14,21,28,16,17,18,22,29,24,25,20,26,30)7975 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 41);